Finite Group Databasebeta

← #30544254

Subgroups — #30544254

Order 15616.

2658 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840577C211yesmaximal, characteristic
22−1+4×Dic61C211yesmaximal, characteristic, complemented
3#21459030C211yesmaximal, characteristic, complemented
4#21454690C213yesmaximal, complemented
5#21454683C213yesmaximal, complemented
6#21454665C213yesmaximal, complemented
7#21454656C213yesmaximal, complemented
8#21454643C213yesmaximal, complemented
9#21454634C213yesmaximal, complemented
10#21454587C213yesmaximal, complemented
11C2×#16839854C213yesmaximal, complemented
12#21454417C211yesmaximal, characteristic, complemented
13#21454353C211yesmaximal, characteristic, complemented
14#131×Dic61C211yesmaximal, characteristic, complemented
15C61×#2898C211yesmaximal, characteristic, complemented
16C2×#14519824C2211yescharacteristic
17#16840577C4116yescomplemented
18C2×#14519808C2213yes—
19C2×#14519807C2213yes—
20C2×#14519797C2213yes—
21C22×#11758202C2213yes—
22C2×#14519784C2211yescharacteristic
23#16840245—23no—
24#16840245C2213yes—
25#16840245C2211yescharacteristic
26#16840243C2213yes—
27#16840241—23no—
28#16840240—21no—
29#16840240C2213yes—
30#16840239C2213yes—
31#16840238—21no—
32#41×Dic61—23no—
33#41×Dic61C2213yes—
34#41×Dic61C2211yescharacteristic
35#16840230C2213yes—
36C4×#11758224—23no—
37C2×Q8×Dic61—21no—
38C2×Q8×Dic61C2213yes—
39C2×#14519742C2213yes—
40C4×Q8×D61—21no—
41#16839866C2216yes—
42#16839866C2216yes—
43#16839865C2216yes—
44#16839859C2216yes—
45#16839854C2216yes—
46#16839854C2216yes—
47#16839854C2213yes—
48#16839854C2216yes—
49#16839854C2213yes—
50#16839853C2213yes—
51#16839853C2213yes—
52#16839847C2213yes—
53#16839847C2213yes—
54#16839842C2213yes—
55#16839842C2211yescharacteristic
56#16839805C2213yes—
57#16839802C2213yes—
58#16839798C2213yes—
59C2×#14519710C2213yes—
60#16839772C2213yes—
61#16839764C2213yes—
62#16839759C2213yes—
63C2×#14519702C2213yes—
64#16839720C2213yes—
65#16839717C2213yes—
66M16×Dic61C2213yes—
67C2×C8×Dic61C2213yes—
68C2×2−1+4×C61C2211yescharacteristic
69C2×#131×C61C2211yescharacteristic
70C61×#520C2211yescharacteristic
71C61×#419C2213yes—
72C61×#417C2213yes—
73C61×#416C2213yes—
74C2×#103×C61C2213yes—
75#14519824—24no—
76#14519824C2×C418yescomplemented
77C2×#11758222C2313yes—
78C2×#11758224—23no—
79C22×Dic122C2313yes—
80C2×Q8×D61—21no—
81#14519808C2×C4124yescomplemented
82#14519807C2×C4124yescomplemented
83#14519797C2×C4124yescomplemented
84C2×#11758202C2×C4124yescomplemented
85C2×#11758202C2313yes—
86C2×#11758202C2316yes—
87C2×#11758200C2313yes—
88C2×#11758201C2313yes—
89C2×#11758199C2311yescharacteristic
90#14519784C2×C418yescomplemented
91C2×#11758194C2313yes—
92C22×#92418C2313yes—
93Q8×Dic61—21no—
94Q8×Dic61—23no—
95Q8×Dic61—23no—
96Q8×Dic61—23no—
97Q8×Dic61—23no—
98Q8×Dic61D411yescharacteristic
99Q8×Dic61D411yescharacteristic
100Q8×Dic61C2313yes—
101Q8×Dic61C2316yes—
102Q8×Dic61C2313yes—
103C4×Dic122—23no—
104C4×Dic122—23no—
105#14519742—23no—
106#14519742—23no—
107#14519742C2313yes—
108#14519742C2316yes—
109#14519742C2313yes—
110D4×Dic61—23no—
111D4×Dic61—23no—
112D4×Dic61—23no—
113D4×Dic61—26no—
114D4×Dic61D413yes—
115D4×Dic61D413yes—
116D4×Dic61C2311yescharacteristic
117D4×Dic61C2313yes—
118#14519739C2311yescharacteristic
119#14519739C2313yes—
120C4×C22⋊D61—26no—
121#14519737—23no—
122#14519737—23no—
123#14519736—26no—
124#14519736—23no—
125#14519736—23no—
126#14519736C2313yes—
127#14519736C2313yes—
128#14519735—23no—
129#14519734—23no—
130#14519733—26no—
131C2×C4⋊Dic61—23no—
132C2×C4⋊Dic61—23no—
133C2×C4⋊Dic61C2313yes—
134C2×#11758196—23no—
135C4⋊C4×D61—23no—
136C2×C4×Dic61—23no—
137C2×C4×Dic61—23no—
138C2×C4×Dic61—23no—
139C2×C4×Dic61C2313yes—
140C42×D61—23no—
141#14519710C2313yes—
142#14519710C2316yes—
143#14519710C2311yescharacteristic
144#14519708C2313yes—
145#14519708C2313yes—
146#14519702C2313yes—
147#14519702C2316yes—
148#14519702C2313yes—
149#14519699C2311yescharacteristic
150#14519699C2313yes—
151#14519687C2316yes—
152C8×Dic61C2311yescharacteristic
153C8×Dic61C2313yes—
154C8×Dic61C2313yes—
155C8×Dic61C2313yes—
1562−1+4×C61—24no—
1572−1+4×C61C2×C418yescomplemented
158C2×#41×C61—23no—
159C2×#41×C61C2313yes—
160C2×#41×C61C2311yescharacteristic
161C22×Q8×C61—21no—
162C22×Q8×C61C2313yes—
163#131×C61—24no—
164C2×M16×C61C2313yes—
165C22×C8×C61C2313yes—
166C4×Q8×C61—21no—
167C4×D4×C61—23no—
168#117×C61C2313yes—
169C2×C4⋊C4×C61C2313yes—
170#103×C61C2313yes—
171#103×C61C2316yes—
172#103×C61C2313yes—
173#102×C61C2311yescharacteristic
174#102×C61C2313yes—
175#41×C61—224no—
176#41×C61—26no—
177#41×C61C22×C4112yes—
178#41×C61C22×C414yes—
179#41×C61#4114yes—
180C2×Q8×C61—28no—
181C2×Q8×C61—23no—
182C2×Q8×C61—26no—
183C2×Q8×C61C22×C4112yes—
184C2×Q8×C61C2×D411yescharacteristic
185C2×Q8×C61C2413yes—
186C2×D4×C61—23no—
187C2×D4×C61C2×D413yes—
188C2×D4×C61C2411yescharacteristic
189C22×C4×C61—23no—
190C22×C4×C61—23no—
191C22×C4×C61—23no—
192C22×C4×C61C2413yes—
193C2×C22⋊D61—23no—
194C22×Dic61—23no—
195C22×Dic61—23no—
196C22×Dic61—21no—
197C22×Dic61—23no—
198C22×Dic61C2×D413yes—
199C22×Dic61C2×D413yes—
200Q8×D61—28no—
201#11758224—224no—
202#11758222—26no—
203#11758222C22×C4112yes—
204C2×D244C2411yescharacteristic
205C2×C4×D61—23no—
206C2×C4×D61—21no—
207C2×C4×D61—23no—
208C2×Dic122—23no—
209C2×Dic122—26no—
210C2×Dic122C22×C4112yes—
211C2×Dic122C2413yes—
212M16×C61—26no—
213C2×C8×C61—26no—
214C2×C8×C61C2411yescharacteristic
215C2×C8×C61C2413yes—
216C4⋊C4×C61—23no—
217C4⋊C4×C61C2413yes—
218C4⋊C4×C61C2411yescharacteristic
219C22⋊C4×C61—23no—
220C42×C61—23no—
221C22⋊Dic61—23no—
222C22⋊Dic61—21no—
223C22⋊Dic61—23no—
224C22⋊Dic61C2×D416yes—
225#11758202C22×C4124yes—
226#11758202C22×C4112yes—
227#11758201C22×C4112yes—
228#11758200C22×C4112yes—
229#11758199C22×C414yes—
230#11758198—21no—
231#11758198—23no—
232C4⋊Dic61—23no—
233C4⋊Dic61—26no—
234C4⋊Dic61—23no—
235C4⋊Dic61—23no—
236C4⋊Dic61C2×D413yes—
237C4⋊Dic61C2×D413yes—
238C4⋊Dic61C2411yescharacteristic
239C4⋊Dic61C2413yes—
240#11758196—26no—
241#11758196—23no—
242#11758196—23no—
243C4×Dic61—23no—
244C4×Dic61—23no—
245C4×Dic61—23no—
246C4×Dic61—23no—
247C4×Dic61—26no—
248C4×Dic61—23no—
249C4×Dic61—26no—
250C4×Dic61C2×D413yes—
251C4×Dic61C2×D413yes—
252C4×Dic61C2411yescharacteristic
253C4×Dic61C2413yes—
254#11758194C22×C4112yes—
255C2×#92418C22×C4112yes—
256C2×#92418C2411yescharacteristic
257C2×#92418C2413yes—
258C23×C61—21no—
259C23×C61C22×D413yes—
260C22×D61—21no—
261Q8×C61—212no—
262Q8×C61—23no—
263Q8×C61C4×D414yes—
264Q8×C61C23×C416yes—
265D4×C61—212no—
266D4×C61—21no—
267D4×C61C4×D4112yes—
268D4×C61C23×C412yes—
269C2×C4×C61—212no—
270C2×C4×C61—23no—
271C2×C4×C61—212no—
272C2×C4×C61—21no—
273C2×C4×C61—23no—
274C2×C4×C61—83no—
275C2×C4×C61C23×C416yes—
276C2×C4×C61C22×D413yes—
277C2×C4×C61C2×#4116yes—
278C2×C4×C61C2511yescharacteristic
279C22⋊D61—212no—
280C2×Dic61—212no—
281C2×Dic61—23no—
282C2×Dic61—26no—
283C2×Dic61—21no—
284C2×Dic61—23no—
285C2×Dic61—43no—
286C2×Dic61—42no—
287C2×Dic61—83no—
288C2×Dic61—83no—
289C2×Dic61C22×D413yes—
290C2×Dic61C22×D413yes—
291C2×Dic61C22×D411yescharacteristic
292C2×Dic61C22×D411yescharacteristic
293D244—21no—
294D244C23×C412yes—
295C4×D61—212no—
296C4×D61—42no—
297C4×D61—83no—
298Dic122—212no—
299Dic122—23no—
300Dic122C23×C416yes—
301C8×C61—21no—
302C8×C61—23no—
303#92418C23×C412yes—
304#92418C23×C416yes—
305C4×#578—611nomaximal, Sylow
306C22×C61—23no—
307C22×C61—23no—
308C22×C61—22no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61C2×C4×D416yes—
312C22×C61C23×D411yescharacteristic
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—26no—
317C4×C61—43no—
318C4×C61—41no—
319C4×C61C2×C4×D416yes—
320C4×C61C24×C411yescharacteristic
321C4×C61C22×#4111yescharacteristic
322Dic61—23no—
323Dic61—21no—
324Dic61—26no—
325Dic61—23no—
326Dic61—43no—
327Dic61—41no—
328Dic61#57812yescomplemented
329C2×#578—611no—
330C4×2−1+4—611no—
331C4×2−1+4—611no—
332#2951—613no—
333#2950—613no—
334#2947—613no—
335#2947—613no—
336C4×#137—613no—
337C4×#137—613no—
338C4×#135—613no—
339C2×C4×Q16—613no—
340#2913—611no—
341#2898—611no—
342#2898D6111yescharacteristic, complemented
343C4×#131—611no—
344C2×C61—26no—
345C2×C61—42no—
346C2×C61C22×C4×D411yescharacteristic
347C2×C61C2×#57811yescharacteristic
348C2×C61#358411yescharacteristic
349D61—42no—
350C2×2−1+4—611no—
351C2×2−1+4C2×D6111yescharacteristic
352#578—6116no—
353C2×#137—613no—
354C2×#137—613no—
355C2×#135—613no—
356C22×Q16—613no—
357C2×#131—611no—
358C2×#131C2×D6111yescharacteristic
359#520—611no—
360#520—613no—
361#520—613no—
362#520—1223no—
363#520—1223no—
364#520C2×D6111yescharacteristic
365#519—613no—
366#519—613no—
367#519—1221no—
368#519—1221no—
369C4×#41—611no—
370C4×#41—613no—
371C4×#41—613no—
372C4×#41—1223no—
373C4×#41—1223no—
374C2×C4×Q8—613no—
375C2×C4×Q8—613no—
376C2×C4×Q8—1221no—
377C2×C4×Q8—1221no—
378#441—616no—
379#441—616no—
380#440—616no—
381#440—616no—
382C4×Q16—613no—
383C4×Q16—616no—
384C4×Q16—616no—
385C4×Q16—616no—
386C4×Q16—613no—
387C4×SD16—613no—
388C4×SD16—613no—
389C4×SD16—613no—
390C4×SD16—613no—
391C4×D8—611no—
392C4×D8—613no—
393#428—613no—
394#428—613no—
395#427—613no—
396C2×#107—613no—
397#419—613no—
398#419C2×D6113yes—
399#417—613no—
400#417C2×D6113yes—
401#416—613no—
402#416C2×D6113yes—
403C2×#103—613no—
404C2×#103C2×D6113yes—
405#405—613no—
406#405—613no—
407C4×M16—613no—
408C2×C4×C8—613no—
409C61C4×#57811yescharacteristic, complemented, Sylow
4102−1+4—24no—
4112−1+4—618no—
4122−1+4—1224no—
4132−1+4C4×D6118yescomplemented
414C2×#41—23no—
415C2×#41—613no—
416C2×#41—1223no—
417C2×#41C22×D6113yes—
418C2×#41C22×D6111yescharacteristic
419C22×Q8—21no—
420C22×Q8—613no—
421C22×Q8—1221no—
422C22×Q8C22×D6113yes—
423#137—6124no—
424#137—6124no—
425#135—6124no—
426C2×Q16—6124no—
427C2×Q16—613no—
428C2×Q16—616no—
429C2×SD16—613no—
430C2×SD16—613no—
431C2×D8—611no—
432#131—24no—
433#131—618no—
434C2×M16—613no—
435C2×M16C22×D6113yes—
436C22×C8—613no—
437C22×C8C22×D6113yes—
438C4×Q8—21no—
439C4×Q8—611no—
440C4×Q8—611no—
441C4×Q8—613no—
442C4×Q8—616no—
443C4×Q8—613no—
444C4×Q8—613no—
445C4×Q8—616no—
446C4×Q8—613no—
447C4×Q8—1223no—
448C4×Q8—1223no—
449C4×Q8—1223no—
450C4×Q8—1223no—
451C4×Q8—1223no—
452C4×Q8—1223no—
453C4×Q8—1223no—
454C4×Q8—1223no—
455C4×Q8—1221no—
456C4×D4—23no—
457C4×D4—613no—
458C4×D4—613no—
459C4×D4—611no—
460C4×D4—613no—
461C4×D4—611no—
462C4×D4—613no—
463C4×D4—1223no—
464C4×D4—1223no—
465C4×D4—1226no—
466C4×D4—1226no—
467C4×D4—1223no—
468C4×D4—1223no—
469C4×D4—1223no—
470#117—613no—
471#117—613no—
472#117—1223no—
473#117—1223no—
474#117—1226no—
475#117—1226no—
476#117—1223no—
477#117—1223no—
478#117C22×D6113yes—
479C2×C4⋊C4—613no—
480C2×C4⋊C4—1223no—
481C2×C4⋊C4—1223no—
482C2×C4⋊C4—1223no—
483C2×C4⋊C4—1223no—
484C2×C4⋊C4C22×D6113yes—
485C2×C42—613no—
486C2×C42—1223no—
487C2×C42—1223no—
488C2×C42—1223no—
489C2×C42—1223no—
490#107—613no—
491#107—616no—
492#107—611no—
493#106—613no—
494#106—613no—
495#103—613no—
496#103—616no—
497#103—613no—
498#103C22×D6113yes—
499#103C22×D6116yes—
500#103C22×D6113yes—
501#102—611no—
502#102—613no—
503#102C22×D6111yescharacteristic
504#102C22×D6113yes—
505#97—616no—
506C4×C8—611no—
507C4×C8—613no—
508C4×C8—613no—
509C4×C8—613no—
510#41—224no—
511#41—26no—
512#41—6112no—
513#41—12224no—
514#41—1226no—
515#41C2×C4×D61112yes—
516#41C2×C4×D6114yes—
517#41#1175822414yes—
518C2×Q8—28no—
519C2×Q8—23no—
520C2×Q8—26no—
521C2×Q8—6112no—
522C2×Q8—613no—
523C2×Q8—1228no—
524C2×Q8—1223no—
525C2×Q8—1226no—
526C2×Q8C2×C4×D61112yes—
527C2×Q8D4×D6111yescharacteristic
528C2×Q8C23×D6113yes—
529C2×D4—23no—
530C2×D4—611no—
531C2×D4—1223no—
532C2×D4D4×D6113yes—
533C2×D4C23×D6111yescharacteristic
534C22×C4—23no—
535C22×C4—23no—
536C22×C4—23no—
537C22×C4—613no—
538C22×C4—613no—
539C22×C4—1223no—
540C22×C4—1221no—
541C22×C4—1223no—
542C22×C4—1221no—
543C22×C4—1223no—
544C22×C4—1223no—
545C22×C4—1223no—
546C22×C4C23×D6113yes—
547Q16—6112no—
548Q16—6124no—
549SD16—6112no—
550SD16—6112no—
551D8—614no—
552M16—26no—
553M16—6112no—
554C2×C8—26no—
555C2×C8—6112no—
556C2×C8—611no—
557C2×C8—613no—
558C2×C8C23×D6111yescharacteristic
559C2×C8C23×D6113yes—
560C4⋊C4—23no—
561C4⋊C4—613no—
562C4⋊C4—613no—
563C4⋊C4—611no—
564C4⋊C4—613no—
565C4⋊C4—1223no—
566C4⋊C4—1226no—
567C4⋊C4—1223no—
568C4⋊C4—1226no—
569C4⋊C4—1223no—
570C4⋊C4—1223no—
571C4⋊C4—1223no—
572C4⋊C4C23×D6111yescharacteristic
573C4⋊C4C23×D6113yes—
574C22⋊C4—23no—
575C22⋊C4—616no—
576C22⋊C4—1221no—
577C22⋊C4—1223no—
578C22⋊C4—1221no—
579C22⋊C4—1223no—
580C22⋊C4—1223no—
581C42—23no—
582C42—613no—
583C42—613no—
584C42—611no—
585C42—613no—
586C42—1223no—
587C42—1223no—
588C42—1226no—
589C42—1223no—
590C42—1223no—
591C42—1226no—
592C42—1223no—
593C23—21no—
594C23—1221no—
595C23C2×D4×D6113yes—
596Q8—212no—
597Q8—23no—
598Q8—616no—
599Q8—12212no—
600Q8—1223no—
601Q8#1451973714yes—
602Q8C22×C4×D6116yes—
603D4—212no—
604D4—21no—
605D4—612no—
606D4—12212no—
607D4—1221no—
608D4#14519737112yes—
609D4C22×C4×D6112yes—
610C2×C4—212no—
611C2×C4—23no—
612C2×C4—212no—
613C2×C4—21no—
614C2×C4—23no—
615C2×C4—83no—
616C2×C4—611no—
617C2×C4—611no—
618C2×C4—613no—
619C2×C4—613no—
620C2×C4—1226no—
621C2×C4—12212no—
622C2×C4—1223no—
623C2×C4—12212no—
624C2×C4—1221no—
625C2×C4—1223no—
626C2×C4—4883no—
627C2×C4—4883no—
628C2×C4—4883no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2443no—
632C2×C4C22×C4×D6116yes—
633C2×C4C2×D4×D6113yes—
634C2×C4C2×#1175822416yes—
635C2×C4C24×D6111yescharacteristic
636C8—21no—
637C8—23no—
638C8—612no—
639C8—616no—
640C22—23no—
641C22—23no—
642C22—22no—
643C22—41no—
644C22—41no—
645C22—1222no—
646C22—2441no—
647C22—2441no—
648C22C2×#1451973716yes—
649C22C22×D4×D6111yescharacteristic
650C4—26no—
651C4—41no—
652C4—43no—
653C4—612no—
654C4—1226no—
655C4—1223no—
656C4—1223no—
657C4—1221no—
658C4—2441no—
659C4—2443no—
660C4C2×#1451973716yes—
661C4C23×C4×D6111yescharacteristic
662C4C22×#1175822411yescharacteristic
663C2—26no—
664C2—42no—
665C2—2442no—
666C2C22×#1451973711yescharacteristic
667C2#578×D6111yescharacteristic
668C2#2146059111yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutatorC4×C61C24×C4
FrattiniC2×C4C24×D61
FittingC61×#2898C2
socleC22×C61C23×D4
prime radicalC2×#16840577C2
cosocleC2×C4×C61C25
hypercenter#2898D61
hypocenterC61C4×#578
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2898

Complements

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

NH ≅ G/Ntype
2. 2−1+4×Dic61C2semidirect
3. #21459030C2semidirect
4. #21454690C2semidirect
5. #21454683C2semidirect
6. #21454665C2semidirect
7. #21454656C2semidirect
8. #21454643C2semidirect
9. #21454634C2semidirect
10. #21454587C2semidirect
11. C2×#16839854C2semidirect
12. #21454417C2semidirect
13. #21454353C2semidirect
14. #131×Dic61C2semidirect
15. C61×#2898C2semidirect
17. #16840577C4semidirect
76. #14519824C2×C4semidirect
81. #14519808C2×C4semidirect
82. #14519807C2×C4semidirect
83. #14519797C2×C4semidirect
84. C2×#11758202C2×C4semidirect
90. #14519784C2×C4semidirect
157. 2−1+4×C61C2×C4semidirect
328. Dic61#578semidirect
342. #2898D61semidirect
409. C61C4×#578semidirect
413. 2−1+4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168405772
2−1+4×Dic612
#214590302
#214546902
#214546832
#214546652
#214546562
#214546432
#214546342
#214545872
C2×#168398542
#214544172
#214543532
#131×Dic612
C61×#28982
C4×#57861