Finite Group Databasebeta

← #29739937

Subgroups — #29739937

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840102C211yesmaximal, characteristic, complemented
2C2×#16839803C211yesmaximal, characteristic, complemented
3#21449251C211yesmaximal, characteristic
4#21448751C211yesmaximal, characteristic
5#21447367C211yesmaximal, characteristic, complemented
6#21447302C211yesmaximal, characteristic, complemented
7#21447251C211yesmaximal, characteristic, complemented
8#21447014C211yesmaximal, characteristic, complemented
9#21446404C211yesmaximal, characteristic, complemented
10#21446249C211yesmaximal, characteristic, complemented
11#21446123C211yesmaximal, characteristic, complemented
12#21446115C211yesmaximal, characteristic
13#21446021C211yesmaximal, characteristic, complemented
14#21446015C211yesmaximal, characteristic, complemented
15C61×#1928C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519753C2211yescharacteristic
18C2×#14519753C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519735C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16840102—24no—
24#16839803—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519709C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×#14519689C2211yescharacteristic
35#16839691—21no—
36#16839680—21no—
37#16839660C2211yescharacteristic
38#16839659—21no—
39#16839653—21no—
40#16839651—21no—
41#16839647—21no—
42#16839640—21no—
43#16839628C2211yescharacteristic
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47C4×#11758198—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×D244D411yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758189—22no—
66C2×#11758189—22no—
67C2×#11758189—22no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519753—22no—
71#14519753—22no—
72#14519753—22no—
73#14519753—41no—
74C4×D244—22no—
75C4×D244—22no—
76#14519735—22no—
77#14519735—22no—
78#14519735—41no—
79#14519735—41no—
80C2×C4⋊Dic61D411yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—41no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198D411yescharacteristic
99C2×#11758198D411yescharacteristic
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102#14519711—22no—
103#14519710—22no—
104#14519709—22no—
105#14519708—41no—
106#14519708D414yescomplemented
107#14519701—22no—
108#14519701—41no—
109#14519689—22no—
110#14519689—22no—
111#14519689—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118C22×Q8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#107×C61—22no—
127#103×C61—22no—
128#103×C61—22no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C23×D61C2×D411yescharacteristic
133C23×D61C2×D411yescharacteristic
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×D244—22no—
154C2×D244—41no—
155C2×D244—41no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—81no—
169C2×C4×D61#4112yes—
170C2×C4×D61#4112yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61#4112yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61C2×D412yes—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—81no—
186#11758198—22no—
187#11758198—22no—
188#11758198—42no—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194#11758198—81no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C2×D412yes—
204#11758196—42no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208C4×Dic61—42no—
209C4×Dic61—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216#11758189—28no—
217C23×C61C22×D411yescharacteristic
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61#12312yes—
232Q8×C61—41no—
233Q8×C61—41no—
234Q8×C61—41no—
235Q8×C61—41no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61#12312yes—
278D244—44no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—84no—
288C8×C61—22no—
289C8×C61—41no—
290#92418—22no—
291#92418—41no—
292#26154—611nomaximal, Sylow
293C22×C61C2×#12311yescharacteristic
294C22×C61D4211yescharacteristic
295C22×C61#54711yescharacteristic
296C22×C61#57511yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57811yescharacteristic
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—42no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61#52511yescharacteristic
321C4×C61#53911yescharacteristic
322C4×C61#54811yescharacteristic
323C4×C61#55311yescharacteristic
324Dic61—22no—
325Dic61—41no—
326Dic61—42no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330C2×#485—611no—
331C2×#428—611no—
332#2415—611no—
333#2352—611no—
334#2097—611no—
335#2085—611no—
336#2076—611no—
337#2035—611no—
338#1928D6111yescharacteristic, complemented
339#1920—611no—
340#1894—611no—
341#1872—611no—
342#1870—611no—
343#1854—611no—
344#1854—611no—
345C2×C61#244311yescharacteristic
346C2×C61#309311yescharacteristic
347C2×C61#309711yescharacteristic
348C2×C61#326911yescharacteristic
349C2×C61#327511yescharacteristic
350C2×C61#335011yescharacteristic
351C2×C61#335211yescharacteristic
352D61—42no—
353D61—81no—
354C22×M16—611no—
355C2×C22⋊Q8—611no—
356C2×C22⋊Q8—611no—
357C2×C4×D4—611no—
358C2×#117—611no—
359C2×#117—611no—
360C22×C4⋊C4—611no—
361#485—1224no—
362#428—1224no—
363C2×#107—611no—
364C2×#107—611no—
365C2×#107C2×D6111yescharacteristic
366C2×#106—611no—
367C2×#106—611no—
368C2×#106—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103C2×D6111yescharacteristic
372C2×#103C2×D6111yescharacteristic
373C2×C22⋊C8—611no—
374C2×C22⋊C8—611no—
375C2×C22⋊C8—611no—
376#397—1221no—
377#394—1221no—
378#391C2×D6111yescharacteristic
379#389—611no—
380#389—1221no—
381#388—1221no—
382#388—1221no—
383#387—1221no—
384#386—1221no—
385#382—611no—
386#380—611no—
387#380—611no—
388#380—611no—
389C4×C4⋊C4C2×D6111yescharacteristic
390C4×C22⋊C4—1221no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340C2×D6111yescharacteristic
397#336—611no—
398#336C2×D6111yescharacteristic
399C61#2615411yescharacteristic, complemented, Sylow
400C22×Q8C22×D6111yescharacteristic
401C22×D4—611no—
402C23×C4—611no—
403C23×C4—611no—
404C23×C4—1221no—
405C2×M16—1222no—
406C2×M16—1222no—
407C2×M16—1222no—
408C22×C8—611no—
409C22×C8—611no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414C22⋊Q8—1222no—
415C22⋊Q8—2441no—
416C4×D4—1222no—
417C4×D4—1222no—
418#117—1222no—
419#117—1222no—
420#117—2441no—
421#117—2441no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—2441no—
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—611no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C42—21no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#107—22no—
450#107—1222no—
451#107—1222no—
452#106—614no—
453#106—1222no—
454#106—2441no—
455#103—22no—
456#103—22no—
457#103—1222no—
458#103—2441no—
459C22⋊C8—1222no—
460C22⋊C8—1222no—
461C22⋊C8—1222no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24—611no—
472C24—611no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×D4—1222no—
480C2×D4—2441no—
481C2×D4—2441no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—611no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—611no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—4881no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510M16—1228no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—612no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8#1175822212yes—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—612no—
529C4⋊C4—1222no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C4⋊C4D4×D6112yes—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—4881no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C42—21no—
552C42—21no—
553C42—81no—
554C42—2442no—
555C42—2442no—
556C23—612no—
557C23—1221no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23C2×D4×D6111yescharacteristic
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575D4—2444no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—612no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1222no—
604C2×C4—1221no—
605C2×C4—4884no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4#41×D6111yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C2×C4#1451982311yescharacteristic
626C2×C4#1451982411yescharacteristic
627C8—22no—
628C8—41no—
629C8—1222no—
630C8—2441no—
631C22—1222no—
632C22—1222no—
633C22—1222no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22—2441no—
642C22—2441no—
643C22—2441no—
644C22—2441no—
645C22#123×D6111yescharacteristic
646C22#1684039611yescharacteristic
647C22#1684041711yescharacteristic
648C22#1684055911yescharacteristic
649C22#1684056111yescharacteristic
650C22#1684057211yescharacteristic
651C22#1684057511yescharacteristic
652C4—42no—
653C4—41no—
654C4—41no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—1222no—
659C4—4881no—
660C4—4881no—
661C4—4881no—
662C4—2441no—
663C4—2442no—
664C4#1684027111yescharacteristic
665C4#1684035311yescharacteristic
666C4#1684042011yescharacteristic
667C4#1684045611yescharacteristic
668C2—4881no—
669C2—2442no—
670C2#2144956111yescharacteristic
671C2#2145575111yescharacteristic
672C2#2145578711yescharacteristic
673C2#2145736911yescharacteristic
674C2#2145745611yescharacteristic
675C2#2145836911yescharacteristic
676C2#2145839511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1928C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenter#1928D61
hypocenterC61#26154
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
1. C2×#16840102C2semidirect
2. C2×#16839803C2semidirect
5. #21447367C2semidirect
6. #21447302C2semidirect
7. #21447251C2semidirect
8. #21447014C2semidirect
9. #21446404C2semidirect
10. #21446249C2semidirect
11. #21446123C2semidirect
13. #21446021C2semidirect
14. #21446015C2semidirect
15. C61×#1928C2semidirect
106. #14519708D4semidirect
338. #1928D61semidirect
399. C61#26154semidirect

Maximal subgroups

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

subgroupindex
C2×#168401022
C2×#168398032
#214492512
#214487512
#214473672
#214473022
#214472512
#214470142
#214464042
#214462492
#214461232
#214461152
#214460212
#214460152
C61×#19282
#2615461