Finite Group Databasebeta

← #29697696

Subgroups — #29697696

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839797C211yesmaximal, characteristic
2#21448708C211yesmaximal, characteristic
3#21448700C211yesmaximal, characteristic
4#21446562C211yesmaximal, characteristic, complemented
5#21446560C211yesmaximal, characteristic, complemented
6#21446411C211yesmaximal, characteristic, complemented
7#21446409C211yesmaximal, characteristic, complemented
8#21446390C211yesmaximal, characteristic, complemented
9#21446388C211yesmaximal, characteristic, complemented
10#21445913C211yesmaximal, characteristic
11#21445912C211yesmaximal, characteristic
12#21445845C211yesmaximal, characteristic
13#21445719C211yesmaximal, characteristic, complemented
14#21445596C211yesmaximal, characteristic
15C61×#1776C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C2×#14519735C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19#16839797C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519689C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26#16839660C2211yescharacteristic
27#16839660C2211yescharacteristic
28#16839658C2211yescharacteristic
29#16839658C2211yescharacteristic
30#16839653—21no—
31#16839653—21no—
32#16839651—21no—
33#16839651—21no—
34#16839629—21no—
35#16839626—21no—
36#16839619C2211yescharacteristic
37#16839619C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×#11758196—21no—
41C4×#11758198C2211yescharacteristic
42#16839608—21no—
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C2×C8×D61—22no—
64C2×C8×D61—22no—
65C2×C8×D61C2×C414yes—
66#14519735—22no—
67#14519735—22no—
68#14519735C2×C414yes—
69#14519735C2×C414yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198D411yescharacteristic
83C2×#11758198D411yescharacteristic
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89#14519711—22no—
90#14519710C2×C414yes—
91#14519710C2×C414yes—
92#14519709C2×C414yes—
93#14519689—22no—
94#14519687—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#107×C61—22no—
115#106×C61C2×C414yes—
116#97×C61—22no—
117#95×C61—21no—
118#95×C61—21no—
119C23×D61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61C22×C412yes—
139C2×C4×D61#4112yes—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61C22×C412yes—
144C2×C8×C61#4112yes—
145C2×C8×C61#4112yes—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C22×C412yes—
153C4⋊C4×C61C22×C412yes—
154C42×C61—21no—
155C42×C61—21no—
156C42×C61—42no—
157C42×C61—81no—
158#11758198—24no—
159#11758198—24no—
160#11758198—41no—
161#11758198—41no—
162#11758198—81no—
163#11758198—81no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61C22×C412yes—
171C4⋊Dic61C22×C412yes—
172#11758196—42no—
173#11758196—42no—
174#11758196—81no—
175#11758196—81no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—24no—
179C4×Dic61—24no—
180C4×Dic61—42no—
181C4×Dic61—81no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418C22×C412yes—
188C8×D61—28no—
189C23×C61C22×D411yescharacteristic
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61C4×D412yes—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61C23×C411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C612−1+411yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61C4×D412yes—
242C4×D61—22no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C8×C61—41no—
247C8×C61#11714yes—
248#92418—22no—
249#92418—41no—
250#22616—611nomaximal, Sylow
251C22×C61C2×C4×D411yescharacteristic
252C22×C61#54711yescharacteristic
253C22×C61#54711yescharacteristic
254C22×C61#57511yescharacteristic
255C22×C61#57511yescharacteristic
256C22×C61#57611yescharacteristic
257C22×C61#57811yescharacteristic
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—41no—
264C2×D61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61C2×#11711yescharacteristic
272C4×C61#51911yescharacteristic
273C4×C61#55211yescharacteristic
274C4×C61#55211yescharacteristic
275Dic61—22no—
276Dic61—41no—
277Dic61—41no—
278Dic61—42no—
279Dic61—42no—
280Dic61—81no—
281Dic61—81no—
282C2×#427—611no—
283#2347—611no—
284#2346—611no—
285#1944—611no—
286#1944—611no—
287#1921—611no—
288#1921—611no—
289#1918—611no—
290#1918—611no—
291#1825—611no—
292#1825—611no—
293#1808—611no—
294#1776—611no—
295#1776D6111yescharacteristic, complemented
296#1752—611no—
297C2×C61#234411yescharacteristic
298C2×C61#294311yescharacteristic
299C2×C61#295111yescharacteristic
300C2×C61#334711yescharacteristic
301C2×C61#334711yescharacteristic
302C2×C61#334911yescharacteristic
303C2×C61#334911yescharacteristic
304D61—42no—
305C23×C8—611no—
306C2×#117—611no—
307C2×#117—611no—
308#427—618no—
309C2×#107—611no—
310C2×#107—611no—
311C2×#107—611no—
312C2×#107C2×D6111yescharacteristic
313C2×#106—611no—
314C2×#106C2×D6111yescharacteristic
315C2×C22⋊C8—611no—
316C2×#97—611no—
317C2×#97C2×D6111yescharacteristic
318#389—611no—
319#389—611no—
320#389—611no—
321#389—611no—
322#388—1221no—
323#388—1221no—
324#388—1221no—
325#388—1221no—
326#383—1221no—
327#382—1221no—
328#380—611no—
329#380—611no—
330C4×C4⋊C4—611no—
331C4×C4⋊C4—611no—
332C4×C4⋊C4—1221no—
333C4×C4⋊C4C2×D6111yescharacteristic
334C4×C4⋊C4C2×D6111yescharacteristic
335C4×C22⋊C4—611no—
336#376—1221no—
337#340—611no—
338#340—611no—
339#340—611no—
340#340—611no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340—611no—
345#340C2×D6111yescharacteristic
346#340C2×D6111yescharacteristic
347#336—611no—
348#336—611no—
349C61#2261611yescharacteristic, complemented, Sylow
350C23×C4—611no—
351C22×C8—611no—
352C22×C8—614no—
353C22×C8—611no—
354C22×C8—1222no—
355C22×C8—1222no—
356C22×C8C22×D6111yescharacteristic
357C22×C8C22×D6111yescharacteristic
358#117—614no—
359#117—614no—
360#117—1222no—
361#117—1222no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—611no—
375C2×C22⋊C4—611no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—611no—
383C2×C42—1221no—
384C2×C42—1221no—
385C2×C42—1221no—
386C2×C42—1221no—
387C2×C42C22×D6111yescharacteristic
388#107—22no—
389#107—614no—
390#107—614no—
391#107—1222no—
392#106—614no—
393#106C4×D6114yes—
394C22⋊C8—1222no—
395#97—22no—
396#97—1222no—
397#95—21no—
398#95—21no—
399#95—611no—
400#95—611no—
401#95—1221no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409C24—611no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—611no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—612no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1228no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8C2×C4×D6112yes—
441C2×C8#1175822212yes—
442C2×C8#1175822212yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—612no—
450C4⋊C4—612no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—4881no—
458C4⋊C4—4881no—
459C4⋊C4—2442no—
460C4⋊C4—2442no—
461C4⋊C4C2×C4×D6112yes—
462C4⋊C4C2×C4×D6112yes—
463C22⋊C4—1224no—
464C22⋊C4—1224no—
465C22⋊C4—4881no—
466C22⋊C4—4881no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—81no—
473C42—1224no—
474C42—1224no—
475C42—1221no—
476C42—1221no—
477C42—4881no—
478C42—2442no—
479C23—612no—
480C23—1221no—
481C23—1221no—
482C23—1221no—
483C23—1221no—
484C23—1221no—
485C23—1221no—
486C23C2×D4×D6111yescharacteristic
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—612no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C22×C4×D6111yescharacteristic
531C2×C4#41×D6111yescharacteristic
532C2×C4C2×#1175822211yescharacteristic
533C2×C4C2×#1175822211yescharacteristic
534C2×C4#1451982411yescharacteristic
535C2×C4#1451982511yescharacteristic
536C8—41no—
537C8—1222no—
538C8—2441no—
539C8#1451973414yes—
540C22—1222no—
541C22—1222no—
542C22—1222no—
543C22—2441no—
544C22—2441no—
545C22—2441no—
546C22—2441no—
547C22C4×D4×D6111yescharacteristic
548C22#1684041311yescharacteristic
549C22#1684041711yescharacteristic
550C22#1684055811yescharacteristic
551C22#1684056111yescharacteristic
552C22#1684056611yescharacteristic
553C22#1684057811yescharacteristic
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—1222no—
561C4—4881no—
562C4—4881no—
563C4—2442no—
564C4—2442no—
565C4—2441no—
566C4—2441no—
567C4C2×#1451973411yescharacteristic
568C4#1684023811yescharacteristic
569C4#1684045111yescharacteristic
570C4#1684045311yescharacteristic
571C2—2442no—
572C2#2144868411yescharacteristic
573C2#2145461411yescharacteristic
574C2#2145469111yescharacteristic
575C2#2145832011yescharacteristic
576C2#2145832411yescharacteristic
577C2#2145835011yescharacteristic
578C2#2145835411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1776C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1776D61
hypocenterC61#22616
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #21446562C2semidirect
5. #21446560C2semidirect
6. #21446411C2semidirect
7. #21446409C2semidirect
8. #21446390C2semidirect
9. #21446388C2semidirect
13. #21445719C2semidirect
15. C61×#1776C2semidirect
19. #16839797C4semidirect
295. #1776D61semidirect
349. C61#22616semidirect

Maximal subgroups

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

subgroupindex
C2×#168397972
#214487082
#214487002
#214465622
#214465602
#214464112
#214464092
#214463902
#214463882
#214459132
#214459122
#214458452
#214457192
#214455962
C61×#17762
#2261661