Finite Group Databasebeta

← #30543693

Subgroups — #30543693

Order 15616.

2694 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840527C211yesmaximal, characteristic
2#21459019C211yesmaximal, characteristic
3#520×D61C211yesmaximal, characteristic
4#21454419C211yesmaximal, characteristic, complemented
5#21454418C213yesmaximal, complemented
6#21454417C211yesmaximal, characteristic, complemented
7#21454416C213yesmaximal, complemented
8#21454413C213yesmaximal, complemented
9#21454412C211yesmaximal, characteristic, complemented
10#21454410C213yesmaximal, complemented
11#21454407C213yesmaximal, complemented
12#21454406C213yesmaximal, complemented
13C2×#16839801C213yesmaximal, complemented
14#428×D61C213yesmaximal, complemented
15C61×#2912C211yesmaximal, characteristic, complemented
16C2×#41×D61C2211yescharacteristic
17#16840527C4116yescomplemented
18C2×#14519785C2213yes—
19C2×#14519784C2211yescharacteristic
20C2×#14519783C2213yes—
21C22×#11758189C2213yes—
22C2×M16×D61C2213yes—
23#16840245C2211yescharacteristic
24#16840245C2211yescharacteristic
25#16840244—21no—
26#16840243C2213yes—
27#16840243C2213yes—
28#16840242C2213yes—
29#16840242C2213yes—
30#16840241—23no—
31#16840238—21no—
32#16840234—23no—
33C4×#11758226—21no—
34C4×#11758224—23no—
35C4×Q8×D61—21no—
36C4×D4×D61—23no—
37C2×#14519735C2213yes—
38#117×D61C2213yes—
39#117×D61C2213yes—
40C2×C4⋊C4×D61C2213yes—
41#16839805C2213yescomplemented
42#16839805C2213yescomplemented
43#16839804C2213yescomplemented
44#16839804C2213yescomplemented
45#16839803C2213yes—
46#16839803C2213yes—
47#16839803C2213yescomplemented
48#16839803C2213yescomplemented
49#16839802C2213yescomplemented
50#16839802C2213yescomplemented
51#16839801C2216yes—
52#16839801C2213yes—
53#16839801C2211yescharacteristic
54#16839801C2216yescomplemented
55#16839801C2216yescomplemented
56#16839801C2213yescomplemented
57#16839801C2213yescomplemented
58#16839800C2213yescomplemented
59#16839799C2216yescomplemented
60#16839798C2213yescomplemented
61#16839798C2213yescomplemented
62#16839797C2216yescomplemented
63C2×#14519711C2213yescomplemented
64C2×#14519710C2213yescomplemented
65#107×D61C2216yescomplemented
66C2×#14519708C2213yescomplemented
67#106×D61C2216yescomplemented
68C2×#131×C61C2211yescharacteristic
69C61×#520C2211yescharacteristic
70C61×#520C2211yescharacteristic
71C61×#428C2213yescomplemented
72C61×#428C2213yescomplemented
73C61×#427C2213yescomplemented
74C2×#106×C61C2213yescomplemented
75C2×#11758226D411yescharacteristic
76C2×#11758222C2313yes—
77C2×#11758224D413yes—
78#41×D61C2×C418yes—
79#41×D61Q818yes—
80C2×Q8×D61D411yescharacteristic
81C2×D4×D61D413yes—
82C22×C4×D61—23no—
83C22×C4×D61C2313yes—
84#14519785C2×C4124yescomplemented
85#14519784C2×C418yescomplemented
86#14519783C2×C4124yescomplemented
87C2×#11758194C2313yes—
88C2×#11758189C2×C4124yescomplemented
89C2×#11758189C2313yes—
90C2×#11758189C2311yescharacteristic
91C2×#11758189C2313yes—
92C2×#11758189C2313yes—
93M16×D61C2×C4124yescomplemented
94C22×#92418C2313yes—
95C2×C8×D61C2316yes—
96Q8×Dic61—21no—
97Q8×Dic61—21no—
98C4×Dic122—23no—
99C4×Dic122—23no—
100#14519742—23no—
101#14519742—23no—
102C4×D244—23no—
103C4×D244—23no—
104D4×Dic61—23no—
105D4×Dic61—23no—
106#14519739—23no—
107#14519739—23no—
108C4×C22⋊D61—26no—
109C4×C22⋊D61—26no—
110#14519737—26no—
111#14519737—26no—
112#14519736C2313yes—
113#14519736C2313yes—
114#14519735—23no—
115#14519735—23no—
116#14519735C2311yescharacteristic
117#14519735C2313yes—
118#14519735C2313yes—
119#14519735C2313yes—
120#14519735C2316yes—
121#14519734—23no—
122#14519734—23no—
123#14519733—23no—
124#14519733—23no—
125C2×C4⋊Dic61C2313yes—
126C2×C4⋊Dic61C2313yes—
127C2×#11758196—23no—
128C4⋊C4×D61—23no—
129C4⋊C4×D61—23no—
130C4⋊C4×D61C2313yes—
131C4⋊C4×D61C2316yes—
132C4⋊C4×D61C2311yescharacteristic
133C4⋊C4×D61C2313yes—
134C4⋊C4×D61C2313yes—
135C2×#11758198—23no—
136C22⋊C4×D61—23no—
137C22⋊C4×D61—23no—
138C2×C4×Dic61—23no—
139C42×D61—23no—
140C42×D61—23no—
141#14519711C2316yes—
142#14519711C2313yes—
143#14519711C2311yescharacteristic
144#14519710C2313yes—
145#14519710C2316yes—
146#14519710C2313yes—
147#14519710C2313yes—
148#14519710C2311yescharacteristic
149#14519709C2313yes—
150#14519709C2313yes—
151#14519708C2313yes—
152#14519708C2316yes—
153#14519708C2313yes—
154#14519708C2311yescharacteristic
155#14519708C2313yes—
156C2×#41×C61C2311yescharacteristic
157#131×C61C2×C418yescomplemented
158C2×M16×C61C2313yes—
159C22×C8×C61C2313yes—
160C4×Q8×C61—21no—
161C4×Q8×C61—21no—
162C4×D4×C61—23no—
163C4×D4×C61—23no—
164#117×C61C2313yes—
165#117×C61C2313yes—
166C2×C4⋊C4×C61C2313yes—
167C2×C4⋊C4×C61C2313yes—
168#107×C61C2313yes—
169#107×C61C2313yes—
170#106×C61C2316yes—
171#106×C61C2313yes—
172#106×C61C2311yescharacteristic
173C23×D61C2×D413yes—
174#41×C61C22×C414yes—
175#41×C61C2×Q814yes—
176C2×Q8×C61C2×D411yescharacteristic
177C2×D4×C61C2×D413yes—
178C22×C4×C61—23no—
179C22×C4×C61—23no—
180C22×C4×C61C2413yes—
181C2×C22⋊D61C2×D416yes—
182C22×Dic61—23no—
183C22×Dic61—23no—
184C22×Dic61C2×D413yes—
185#11758226C4⋊C418yes—
186Q8×D61C4⋊C418yes—
187#11758224C4⋊C4124yes—
188D4×D61C4⋊C4124yes—
189#11758222C22×C4112yes—
190#11758222C2×Q8112yes—
191C2×D244C2×D413yes—
192C2×C4×D61—23no—
193C2×C4×D61—21no—
194C2×C4×D61—26no—
195C2×C4×D61—23no—
196C2×C4×D61—23no—
197C2×C4×D61—86no—
198C2×C4×D61C22×C4112yes—
199C2×C4×D61C2×D413yes—
200C2×C4×D61C2×D413yes—
201C2×C4×D61C2×Q8112yes—
202C2×C4×D61C2411yescharacteristic
203C2×C4×D61C2413yes—
204C2×Dic122C2×D413yes—
205M16×C61C22×C4112yescomplemented
206C2×C8×C61C22×C4112yescomplemented
207C2×C8×C61C2411yescharacteristic
208C2×C8×C61C2413yes—
209C4⋊C4×C61—23no—
210C4⋊C4×C61—23no—
211C4⋊C4×C61C2413yes—
212C4⋊C4×C61C2413yes—
213C4⋊C4×C61C2411yescharacteristic
214C4⋊C4×C61C2411yescharacteristic
215C22⋊C4×C61—23no—
216C22⋊C4×C61—23no—
217C42×C61—23no—
218C42×C61—23no—
219C22⋊Dic61—23no—
220C22⋊Dic61—23no—
221#11758198—21no—
222#11758198—23no—
223#11758198—23no—
224#11758198—23no—
225#11758198—26no—
226C4⋊Dic61—23no—
227C4⋊Dic61—23no—
228C4⋊Dic61C2411yescharacteristic
229C4⋊Dic61C2413yes—
230C4⋊Dic61C2411yescharacteristic
231C4⋊Dic61C2413yes—
232#11758196—23no—
233#11758196—26no—
234#11758196—21no—
235#11758196—23no—
236#11758196—23no—
237C4×Dic61—23no—
238C4×Dic61—23no—
239C4×Dic61—21no—
240C4×Dic61—23no—
241C4×Dic61—23no—
242C4×Dic61—23no—
243C4×Dic61—26no—
244#11758194C22×C4112yescomplemented
245C2×#92418C22×C4112yescomplemented
246C2×#92418C2413yes—
247C2×#92418C2411yescharacteristic
248#11758189C22×C4112yes—
249#11758189C22×C414yes—
250#11758189C22×C4112yescomplemented
251#11758189C22×C4112yescomplemented
252C8×D61C22×C4124yescomplemented
253C23×C61C22×D413yes—
254C22×D61—43no—
255C22×D61—43no—
256C22×D61C2×C4⋊C4112yes—
257C22×D61C22×D411yescharacteristic
258C22×D61C22×D413yes—
259Q8×C61C2×C4⋊C414yes—
260D4×C61C2×C4⋊C4112yes—
261C2×C4×C61—21no—
262C2×C4×C61—23no—
263C2×C4×C61—21no—
264C2×C4×C61—23no—
265C2×C4×C61—83no—
266C2×C4×C61—83no—
267C2×C4×C61C23×C416yes—
268C2×C4×C61C22×D413yes—
269C2×C4×C61C22×Q816yes—
270C2×C4×C61C2511yescharacteristic
271C22⋊D61C2×C4⋊C4124yes—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—23no—
275C2×Dic61—23no—
276C2×Dic61—83no—
277C2×Dic61—83no—
278C2×Dic61C2×C4⋊C4112yes—
279C2×Dic61C22×D413yes—
280C2×Dic61C22×D411yescharacteristic
281D244C2×C4⋊C4112yes—
282C4×D61—42no—
283C4×D61—46no—
284C4×D61—83no—
285C4×D61—83no—
286C4×D61—86no—
287C4×D61C2×C4⋊C4112yes—
288C4×D61C2×C4⋊C4112yes—
289C4×D61C23×C416yes—
290C4×D61C23×C412yes—
291C4×D61C22×Q812yes—
292C4×D61C22×Q816yes—
293Dic122C2×C4⋊C4112yes—
294C8×C61C23×C416yes—
295C8×C61C23×C412yes—
296#92418C23×C416yes—
297#92418C23×C412yes—
298#60109—611nomaximal, Sylow
299C22×C61—23no—
300C22×C61—23no—
301C22×C61C22×C4⋊C416yes—
302C22×C61C23×D411yescharacteristic
303C2×D61—21no—
304C2×D61—23no—
305C2×D61—21no—
306C2×D61—23no—
307C2×D61—412no—
308C2×D61C22×C4⋊C412yes—
309C2×D61C22×C4⋊C416yes—
310C4×C61—43no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—43no—
314C4×C61C22×C4⋊C416yes—
315C4×C61C24×C411yescharacteristic
316C4×C61C23×Q811yescharacteristic
317Dic61—43no—
318Dic61—43no—
319Dic61—41no—
320Dic61—41no—
321Dic61C22×C4⋊C416yes—
322Dic61C22×C4⋊C412yes—
323C2×#568—611no—
324#3428—611no—
325C2×#520—611no—
326#2913—613no—
327#2913—613no—
328#2913—611no—
329#2913—611no—
330#2912—613no—
331#2912—613no—
332#2912—611no—
333#2912D6111yescharacteristic, complemented
334#2911—613no—
335#2911—613no—
336C2×#428—613no—
337C2×#428—613no—
338C2×C61—26no—
339C2×C61C23×C4⋊C411yescharacteristic
340C2×C61#358511yescharacteristic
341C2×C61#358611yescharacteristic
342D61—22no—
343D61—26no—
344C22×#41—611no—
345#568—6116no—
346C2×#131—611no—
347C2×#131—613no—
348C2×#131—613no—
349C2×#131C2×D6111yescharacteristic
350C22×M16—613no—
351C22×M16—613no—
352#520—611no—
353#520—611no—
354#520—613no—
355#520—613no—
356#520—613no—
357#520—613no—
358#520—1223no—
359#520—1221no—
360#520C2×D6111yescharacteristic
361#520C2×D6111yescharacteristic
362#519—1221no—
363#518—1223no—
364C4×#41—1223no—
365C4×#41—1221no—
366C2×C4×Q8—1221no—
367C2×C4×D4—1223no—
368C2×#117—613no—
369C2×#117—613no—
370C2×#117—613no—
371C22×C4⋊C4—613no—
372#428—613no—
373#428—613no—
374#428—613no—
375#428—613no—
376#428—613no—
377#428—613no—
378#428—613no—
379#428—613no—
380#428—613no—
381#428—613no—
382#428—616no—
383#428—613no—
384#428—616no—
385#428—616no—
386#428—611no—
387#428—613no—
388#428—613no—
389#428C2×D6113yescomplemented
390#428C2×D6113yescomplemented
391#427—613no—
392#427—613no—
393#427—613no—
394#427—616no—
395#427—616no—
396#427C2×D6113yescomplemented
397C2×#107—613no—
398C2×#107—613no—
399C2×#107—616no—
400C2×#106—613no—
401C2×#106—616no—
402C2×#106C2×D6113yescomplemented
403C61#6010911yescharacteristic, complemented, Sylow
404C2×#41—618no—
405C2×#41—618no—
406C2×#41—611no—
407C2×#41—613no—
408C2×#41—613no—
409C2×#41C22×D6111yescharacteristic
410C22×Q8—611no—
411C22×D4—613no—
412C23×C4—613no—
413C23×C4—1223no—
414#131—618no—
415#131—6124no—
416#131—6124no—
417#131C4×D6118yescomplemented
418C2×M16—613no—
419C2×M16—6124no—
420C2×M16—6124no—
421C2×M16—613no—
422C2×M16—611no—
423C2×M16—613no—
424C2×M16—613no—
425C2×M16C22×D6113yes—
426C22×C8—613no—
427C22×C8—616no—
428C22×C8C22×D6113yes—
429C4×Q8—21no—
430C4×Q8—21no—
431C4×Q8—1223no—
432C4×Q8—1223no—
433C4×Q8—1223no—
434C4×Q8—1223no—
435C4×Q8—1221no—
436C4×Q8—1221no—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—1223no—
440C4×D4—1223no—
441C4×D4—1226no—
442C4×D4—1226no—
443C4×D4—1223no—
444C4×D4—1223no—
445C4×D4—1226no—
446C4×D4—1226no—
447C4×D4—1223no—
448C4×D4—1223no—
449#117—613no—
450#117—613no—
451#117—611no—
452#117—613no—
453#117—613no—
454#117—613no—
455#117—616no—
456#117—1223no—
457#117—1223no—
458#117—1223no—
459#117—1223no—
460#117—1223no—
461#117—1223no—
462#117C22×D6113yes—
463#117C22×D6113yes—
464C2×C4⋊C4—613no—
465C2×C4⋊C4—613no—
466C2×C4⋊C4—613no—
467C2×C4⋊C4—616no—
468C2×C4⋊C4—611no—
469C2×C4⋊C4—613no—
470C2×C4⋊C4—613no—
471C2×C4⋊C4—1223no—
472C2×C4⋊C4—1223no—
473C2×C4⋊C4—1223no—
474C2×C4⋊C4C22×D6113yes—
475C2×C4⋊C4C22×D6113yes—
476C2×C22⋊C4—1223no—
477C2×C22⋊C4—1223no—
478C2×C22⋊C4—1223no—
479C2×C42—1223no—
480C2×C42—1223no—
481C2×C42—1223no—
482#107—616no—
483#107—613no—
484#107—611no—
485#107—613no—
486#107—616no—
487#107—613no—
488#107—611no—
489#107—613no—
490#107C22×D6113yes—
491#107C22×D6113yes—
492#106—613no—
493#106—613no—
494#106—613no—
495#106—616no—
496#106—613no—
497#106—611no—
498#106—613no—
499#106C22×D6116yes—
500#106C22×D6113yes—
501#106C22×D6111yescharacteristic
502C24—613no—
503#41—618no—
504#41—6124no—
505#41—6112no—
506#41—6112no—
507#41C2×C4×D6114yes—
508#41Q8×D6114yes—
509C2×Q8—618no—
510C2×Q8—613no—
511C2×Q8D4×D6111yescharacteristic
512C2×D4—616no—
513C2×D4—6124no—
514C2×D4—613no—
515C2×D4D4×D6113yes—
516C22×C4—23no—
517C22×C4—23no—
518C22×C4—613no—
519C22×C4—6112no—
520C22×C4—613no—
521C22×C4—613no—
522C22×C4—6112no—
523C22×C4—613no—
524C22×C4—611no—
525C22×C4—1223no—
526C22×C4—1223no—
527C22×C4—1226no—
528C22×C4—1221no—
529C22×C4—1223no—
530C22×C4—1223no—
531C22×C4—1223no—
532C22×C4—4886no—
533C22×C4C23×D6113yes—
534M16—6112no—
535M16—6112no—
536M16—614no—
537M16—6112no—
538M16—6112no—
539M16C2×C4×D61112yescomplemented
540C2×C8—6112no—
541C2×C8—613no—
542C2×C8—611no—
543C2×C8—6124no—
544C2×C8C2×C4×D61112yescomplemented
545C2×C8C23×D6113yes—
546C2×C8C23×D6111yescharacteristic
547C4⋊C4—23no—
548C4⋊C4—23no—
549C4⋊C4—611no—
550C4⋊C4—613no—
551C4⋊C4—611no—
552C4⋊C4—613no—
553C4⋊C4—1223no—
554C4⋊C4—1226no—
555C4⋊C4—1221no—
556C4⋊C4—1223no—
557C4⋊C4—1223no—
558C4⋊C4—1223no—
559C4⋊C4—1223no—
560C4⋊C4C23×D6113yes—
561C4⋊C4C23×D6111yescharacteristic
562C4⋊C4C23×D6113yes—
563C4⋊C4C23×D6111yescharacteristic
564C22⋊C4—23no—
565C22⋊C4—23no—
566C22⋊C4—1221no—
567C22⋊C4—1223no—
568C22⋊C4—1223no—
569C22⋊C4—1226no—
570C22⋊C4—1223no—
571C22⋊C4—1223no—
572C22⋊C4—1223no—
573C42—23no—
574C42—23no—
575C42—1221no—
576C42—1223no—
577C42—1223no—
578C42—1226no—
579C42—1223no—
580C42—1223no—
581C42—1223no—
582C23—611no—
583C23—613no—
584C23—6112no—
585C23—2443no—
586C23—2443no—
587C23C2×D4×D6113yes—
588Q8—6112no—
589Q8C4⋊C4×D6114yes—
590D4—6124no—
591D4—6112no—
592D4C4⋊C4×D61112yes—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—23no—
596C2×C4—23no—
597C2×C4—83no—
598C2×C4—83no—
599C2×C4—6112no—
600C2×C4—611no—
601C2×C4—613no—
602C2×C4—6112no—
603C2×C4—6112no—
604C2×C4—616no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—616no—
608C2×C4—1221no—
609C2×C4—1223no—
610C2×C4—1221no—
611C2×C4—1223no—
612C2×C4—4883no—
613C2×C4—4883no—
614C2×C4—4886no—
615C2×C4—4883no—
616C2×C4—4883no—
617C2×C4—2442no—
618C2×C4—2446no—
619C2×C4C22×C4×D6116yes—
620C2×C4C2×D4×D6113yes—
621C2×C4C2×Q8×D6116yes—
622C2×C4C24×D6111yescharacteristic
623C8—616no—
624C8—612no—
625C8C22×C4×D6116yes—
626C8C22×C4×D6112yes—
627C22—23no—
628C22—23no—
629C22—612no—
630C22—616no—
631C22—1223no—
632C22—1221no—
633C22—1223no—
634C22—1221no—
635C22—24412no—
636C22C2×C4⋊C4×D6116yes—
637C22C22×D4×D6111yescharacteristic
638C4—43no—
639C4—41no—
640C4—43no—
641C4—41no—
642C4—612no—
643C4—616no—
644C4—2443no—
645C4—2443no—
646C4—2441no—
647C4—2441no—
648C4C2×C4⋊C4×D6116yes—
649C4C23×C4×D6111yescharacteristic
650C4C22×Q8×D6111yescharacteristic
651C2—26no—
652C2—1222no—
653C2—1226no—
654C2C22×C4⋊C4×D6111yescharacteristic
655C2#2146059711yescharacteristic
656C2#2146060111yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutatorC4×C61C24×C4
FrattiniC2×C4C24×D61
FittingC61×#2912C2
socleC22×C61C23×D4
prime radicalC2×#41×D61C22
cosocleC2×C4×C61C25
hypercenter#2912D61
hypocenterC61#60109
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2912

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #21454419C2semidirect
5. #21454418C2semidirect
6. #21454417C2semidirect
7. #21454416C2semidirect
8. #21454413C2semidirect
9. #21454412C2semidirect
10. #21454410C2semidirect
11. #21454407C2semidirect
12. #21454406C2semidirect
13. C2×#16839801C2semidirect
14. #428×D61C2semidirect
15. C61×#2912C2semidirect
17. #16840527C4semidirect
41. #16839805C22semidirect
42. #16839805C22semidirect
43. #16839804C22semidirect
44. #16839804C22semidirect
47. #16839803C22semidirect
48. #16839803C22semidirect
49. #16839802C22semidirect
50. #16839802C22semidirect
54. #16839801C22semidirect
55. #16839801C22semidirect
56. #16839801C22semidirect
57. #16839801C22semidirect
58. #16839800C22semidirect
59. #16839799C22semidirect
60. #16839798C22semidirect
61. #16839798C22semidirect
62. #16839797C22semidirect
63. C2×#14519711C22semidirect
64. C2×#14519710C22semidirect
65. #107×D61C22semidirect
66. C2×#14519708C22semidirect
67. #106×D61C22semidirect
71. C61×#428C22semidirect
72. C61×#428C22semidirect
73. C61×#427C22semidirect
74. C2×#106×C61C22semidirect
84. #14519785C2×C4semidirect
85. #14519784C2×C4semidirect
86. #14519783C2×C4semidirect
88. C2×#11758189C2×C4semidirect
93. M16×D61C2×C4semidirect
157. #131×C61C2×C4semidirect
205. M16×C61C22×C4semidirect
206. C2×C8×C61C22×C4semidirect
244. #11758194C22×C4semidirect
245. C2×#92418C22×C4semidirect
250. #11758189C22×C4semidirect
251. #11758189C22×C4semidirect
252. C8×D61C22×C4semidirect
333. #2912D61semidirect
389. #428C2×D61semidirect
390. #428C2×D61semidirect
396. #427C2×D61semidirect
402. C2×#106C2×D61semidirect
403. C61#60109semidirect
417. #131C4×D61semidirect
539. M16C2×C4×D61semidirect
544. C2×C8C2×C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168405272
#214590192
#520×D612
#214544192
#214544182
#214544172
#214544162
#214544132
#214544122
#214544102
#214544072
#214544062
C2×#168398012
#428×D612
C61×#29122
#6010961