Finite Group Databasebeta

← #27174449

Subgroups — #27174449

Order 12928.

1056 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108074C211yesmaximal, characteristic, complemented
2C2×#15692225C211yesmaximal, characteristic, complemented
3C2×#15692223C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D101C211yesmaximal, characteristic, complemented
5#20265090C211yesmaximal, characteristic, complemented
6#20265089C211yesmaximal, characteristic, complemented
7#20265086C211yesmaximal, characteristic, complemented
8#20265085C211yesmaximal, characteristic, complemented
9#20265056C211yesmaximal, characteristic, complemented
10#20265055C211yesmaximal, characteristic, complemented
11#20265055C211yesmaximal, characteristic, complemented
12#20265053C211yesmaximal, characteristic, complemented
13#20265053C211yesmaximal, characteristic, complemented
14#20265039C211yesmaximal, characteristic, complemented
15#386×C101C211yesmaximal, characteristic, complemented
16C24×D101C2211yescharacteristic
17C2×#13108074—24no—
18C2×#13108074—24no—
19C2×#13108074—24no—
20C22×Dic202C2211yescharacteristic, complemented
21C22×D404C2211yescharacteristic
22C22×D404C2211yescharacteristic
23C22×C22⋊D101C2211yescharacteristic
24C22×C22⋊D101C2211yescharacteristic
25C22×C22⋊D101C2211yescharacteristic
26C22×C22⋊D101C2211yescharacteristic, complemented
27C22×C4×D101C2211yescharacteristic
28C22×C4×D101C2211yescharacteristic
29C22×C4×D101C2211yescharacteristic, complemented
30#15692225—24no—
31#15692223—24no—
32C2×C22⋊Dic101C2211yescharacteristic
33C2×C22⋊Dic101C2211yescharacteristic
34C2×C22⋊Dic101C2211yescharacteristic, complemented
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C2×#13108051C2211yescharacteristic
39C2×#13108051C2211yescharacteristic
40C2×#13108051C2211yescharacteristic
41C2×#13108051C2211yescharacteristic, complemented
42C2×#13108051C2211yescharacteristic, complemented
43C22⋊C4×D101—24no—
44#15692162C2211yescharacteristic, complemented
45#15692162C2211yescharacteristic, complemented
46#15692162C2211yescharacteristic, complemented
47#15692162C2211yescharacteristic, complemented
48#15692162C2211yescharacteristic, complemented
49#15692162C2211yescharacteristic, complemented
50C22×D4×C101C2211yescharacteristic
51C23×C4×C101C2211yescharacteristic, complemented
52C2×C22⋊C4×C101C2211yescharacteristic
53C2×C22⋊C4×C101C2211yescharacteristic
54C2×C22⋊C4×C101C2211yescharacteristic, complemented
55#95×C101C2211yescharacteristic, complemented
56#95×C101C2211yescharacteristic, complemented
57C23×D101—22no—
58C23×D101—22no—
59C23×D101—22no—
60C23×D101—22no—
61C23×D101—22no—
62C23×D101—41no—
63C23×D101—41no—
64C23×D101C2311yescharacteristic
65C23×D101C2311yescharacteristic
66C23×D101C2311yescharacteristic
67#13108074—232no—
68C2×Dic202—22no—
69C2×Dic202—22no—
70C2×Dic202—22no—
71C2×D404—22no—
72C2×D404—22no—
73C2×D404—22no—
74C2×D404—22no—
75C2×D404—41no—
76C2×D404D414yescomplemented
77C2×C22⋊D101—22no—
78C2×C22⋊D101—22no—
79C2×C22⋊D101—22no—
80C2×C22⋊D101—22no—
81C2×C22⋊D101—22no—
82C2×C22⋊D101—22no—
83C2×C22⋊D101—22no—
84C2×C22⋊D101—22no—
85C2×C22⋊D101—22no—
86C2×C22⋊D101—41no—
87C2×C22⋊D101—41no—
88C2×C22⋊D101D414yescomplemented
89C22×Dic101C2311yescharacteristic
90C22×Dic101C2311yescharacteristic
91C22×Dic101C2311yescharacteristic
92C22×Dic101C2311yescharacteristic
93C22×Dic101C2311yescharacteristic
94C2×C4×D101—22no—
95C2×C4×D101—22no—
96C2×C4×D101—22no—
97C2×C4×D101—22no—
98C2×C4×D101—22no—
99C2×C4×D101—22no—
100C2×C4×D101—41no—
101C2×C4×D101—41no—
102C2×C4×D101D414yescomplemented
103C22⋊Dic101—22no—
104C22⋊Dic101—22no—
105C22⋊Dic101—41no—
106#13108051—22no—
107#13108051—22no—
108#13108051—22no—
109#13108051—22no—
110#13108051—22no—
111#13108051—22no—
112#13108051—41no—
113#13108051—41no—
114C24×C101C2311yescharacteristic
115C24×C101C2311yescharacteristic
116C2×D4×C101—22no—
117C2×D4×C101—22no—
118C2×D4×C101—41no—
119C22×C4×C101—22no—
120C22×C4×C101—22no—
121C22×C4×C101—22no—
122C22×C4×C101C2311yescharacteristic
123C22×C4×C101C2311yescharacteristic
124C22×C4×C101C2311yescharacteristic
125C22×C4×C101C2311yescharacteristic
126C22×C4×C101C2311yescharacteristic
127C22⋊C4×C101—22no—
128C22⋊C4×C101—22no—
129C22⋊C4×C101—41no—
130C23×C101—21no—
131C23×C101—21no—
132C23×C101—21no—
133C23×C101—21no—
134C23×C101—21no—
135C23×C101—21no—
136C23×C101—21no—
137C23×C101—21no—
138C23×C101—21no—
139C23×C101—21no—
140C23×C101—21no—
141C23×C101—21no—
142C23×C101C2×D412yes—
143C23×C101#4112yescomplemented
144C23×C101C2411yescharacteristic
145C22×D101—21no—
146C22×D101—21no—
147C22×D101—21no—
148C22×D101—21no—
149C22×D101—28no—
150C22×D101—21no—
151C22×D101—21no—
152C22×D101—21no—
153C22×D101—21no—
154C22×D101—21no—
155C22×D101—21no—
156C22×D101—21no—
157C22×D101—21no—
158C22×D101—21no—
159C22×D101—21no—
160C22×D101—21no—
161C22×D101—21no—
162C22×D101—44no—
163C22×D101—44no—
164C22×D101—44no—
165C22×D101—44no—
166C22×D101—44no—
167C22×D101—44no—
168C22×D101C2×D412yes—
169C22×D101C2×D412yes—
170C22×D101C2×D412yes—
171C22×D101C2×D412yes—
172C22×D101#4112yescomplemented
173D4×C101—44no—
174C2×C4×C101—28no—
175C2×C4×C101—21no—
176C2×C4×C101—21no—
177C2×C4×C101—21no—
178C2×C4×C101—21no—
179C2×C4×C101—21no—
180C2×C4×C101—21no—
181C2×C4×C101—21no—
182C2×C4×C101—21no—
183C2×C4×C101—21no—
184C2×C4×C101—21no—
185C2×C4×C101—21no—
186C2×C4×C101—21no—
187C2×C4×C101—21no—
188C2×C4×C101C2×D412yes—
189C2×C4×C101C2×D412yes—
190C22⋊D101—28no—
191C22⋊D101—28no—
192C22⋊D101—44no—
193C22⋊D101—44no—
194C2×Dic101—21no—
195C2×Dic101—21no—
196C2×Dic101—21no—
197C2×Dic101—21no—
198C2×Dic101—21no—
199C2×Dic101—21no—
200C2×Dic101—21no—
201C2×Dic101—21no—
202C2×Dic101—21no—
203C2×Dic101—21no—
204C2×Dic101—21no—
205C2×Dic101—21no—
206C2×Dic101—21no—
207C2×Dic101C2×D412yes—
208C2×Dic101C2×D412yes—
209D404—28no—
210D404—44no—
211C4×D101—28no—
212C4×D101—28no—
213C4×D101—44no—
214Dic202—28no—
215C22×C101—22no—
216C22×C101—22no—
217C22×C101—22no—
218C22×C101—22no—
219C22×C101—22no—
220C22×C101—22no—
221C22×C101—41no—
222C22×C101—41no—
223C22×C101—41no—
224C22×C101—41no—
225C22×C101—41no—
226C22×C101—41no—
227C22×C101—41no—
228C22×C101—41no—
229C22×C101C22×D411yescharacteristic
230C22×C101C22×D411yescharacteristic
231C22×C101C22×D411yescharacteristic
232C22×C101C2×#4111yescharacteristic
233C22×C101C2×#4111yescharacteristic
234C22×C101H(5,𝔽2)11yescharacteristic
235C22×C101H(5,𝔽2)11yescharacteristic
236C2×D101—22no—
237C2×D101—22no—
238C2×D101—22no—
239C2×D101—22no—
240C2×D101—22no—
241C2×D101—22no—
242C2×D101—22no—
243C2×D101—22no—
244C2×D101—22no—
245C2×D101—22no—
246C2×D101—22no—
247C2×D101—22no—
248C2×D101—416no—
249C2×D101—41no—
250C2×D101—41no—
251C2×D101—41no—
252C2×D101—41no—
253C2×D101—41no—
254C2×D101—41no—
255C2×D101—41no—
256C2×D101—41no—
257C2×D101#12014yes—
258C4×C101—22no—
259C4×C101—22no—
260C4×C101—41no—
261C4×C101—41no—
262C4×C101—41no—
263Dic101—22no—
264Dic101—22no—
265Dic101—41no—
266Dic101—41no—
267Dic101—41no—
268C2×C101—42no—
269C2×C101—42no—
270C2×C101C2×#12011yescharacteristic
271C2×C101#52511yescharacteristic
272C2×C101#53511yescharacteristic
273C2×C101#54611yescharacteristic
274C2×C101#54611yescharacteristic
275C2×C101#54611yescharacteristic
276C2×C101#54611yescharacteristic
277D101—24no—
278D101—42no—
279D101—42no—
280#2404—1011nomaximal, Sylow
281C101#240411yescharacteristic, complemented, Sylow
282C22×#41—1011no—
283C2×#120—1011no—
284C2×#120—1011no—
285C22×C22⋊C4—1011no—
286#393—1011no—
287#393—1011no—
288#392—1011no—
289#392—1011no—
290#386—1011no—
291#386—1011no—
292#386—1011no—
293#386—1011no—
294#386—1011no—
295#386D10111yescharacteristic, complemented
296#381—1011no—
297C25—1011no—
298C2×#41—2024no—
299C2×#41—2024no—
300C2×#41—2024no—
301C22×Q8—1011no—
302C22×D4—1011no—
303C22×D4—1011no—
304C22×D4—1011no—
305C22×D4—1011no—
306C22×D4—1011no—
307C22×D4—1011no—
308C22×D4C2×D10111yescharacteristic
309C23×C4—1011no—
310C23×C4—1011no—
311C23×C4—1011no—
312C23×C4C2×D10111yescharacteristic, complemented
313#120—2024no—
314#120—2024no—
315C2×C22⋊C4—1011no—
316C2×C22⋊C4—1011no—
317C2×C22⋊C4—1011no—
318C2×C22⋊C4—1011no—
319C2×C22⋊C4—1011no—
320C2×C22⋊C4—1011no—
321C2×C22⋊C4—1011no—
322C2×C22⋊C4—1011no—
323C2×C22⋊C4—1011no—
324C2×C22⋊C4—1011no—
325C2×C22⋊C4—1011no—
326C2×C22⋊C4—2024no—
327C2×C22⋊C4C2×D10111yescharacteristic
328C2×C22⋊C4C2×D10111yescharacteristic
329C2×C22⋊C4C2×D10111yescharacteristic, complemented
330#95—1011no—
331#95—1011no—
332#95—1011no—
333#95—1011no—
334#95—1011no—
335#95—1011no—
336#95C2×D10111yescharacteristic, complemented
337#95C2×D10111yescharacteristic, complemented
338C24—1011no—
339C24—1011no—
340C24—1011no—
341C24—4041no—
342C24—4041no—
343C24—2022no—
344C24—2022no—
345C24—2022no—
346C24—2022no—
347C24—2022no—
348C24C22×D10111yescharacteristic
349C24C22×D10111yescharacteristic
350#41—20232no—
351C2×Q8—2022no—
352C2×Q8—2022no—
353C2×Q8—2022no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—1014no—
358C2×D4—1014no—
359C2×D4—4041no—
360C2×D4—4041no—
361C2×D4—4041no—
362C2×D4—2022no—
363C2×D4—2022no—
364C2×D4—2022no—
365C2×D4—2022no—
366C2×D4—2022no—
367C2×D4—2022no—
368C2×D4—2022no—
369C2×D4—2022no—
370C2×D4—2022no—
371C2×D4—2022no—
372C2×D4—2022no—
373C2×D4—2022no—
374C2×D4—2022no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—1011no—
379C22×C4—1011no—
380C22×C4—1014no—
381C22×C4—1011no—
382C22×C4—1011no—
383C22×C4—1011no—
384C22×C4—4041no—
385C22×C4—4041no—
386C22×C4—2022no—
387C22×C4—2022no—
388C22×C4—2022no—
389C22×C4—2022no—
390C22×C4—2022no—
391C22×C4—2022no—
392C22×C4C22×D10111yescharacteristic
393C22×C4C22×D10111yescharacteristic
394C22×C4C22×D10111yescharacteristic
395C22×C4C22×D10111yescharacteristic
396C22×C4C22×D10111yescharacteristic
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—4041no—
401C22⋊C4—4041no—
402C22⋊C4—4041no—
403C22⋊C4—2022no—
404C22⋊C4—2022no—
405C22⋊C4—2022no—
406C22⋊C4—2022no—
407C22⋊C4—2022no—
408C22⋊C4—2022no—
409C22⋊C4—2022no—
410C22⋊C4—2022no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—1012no—
424C23—1012no—
425C23—1012no—
426C23—1012no—
427C23—1012no—
428C23—4044no—
429C23—4044no—
430C23—4044no—
431C23—4044no—
432C23—4044no—
433C23—4044no—
434C23—2028no—
435C23—2021no—
436C23—2021no—
437C23—2021no—
438C23—2021no—
439C23—2021no—
440C23—2021no—
441C23—2021no—
442C23—2021no—
443C23—2021no—
444C23—2021no—
445C23—2021no—
446C23—2021no—
447C23—2021no—
448C23—2021no—
449C23—2021no—
450C23—2021no—
451C23D4×D10112yes—
452C23#1310807412yescomplemented
453C23C23×D10111yescharacteristic
454Q8—2028no—
455D4—44no—
456D4—4044no—
457D4—4044no—
458D4—4044no—
459D4—2028no—
460D4—2028no—
461D4—2028no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—1012no—
477C2×C4—1012no—
478C2×C4—4044no—
479C2×C4—2028no—
480C2×C4—2028no—
481C2×C4—2021no—
482C2×C4—2021no—
483C2×C4—2021no—
484C2×C4—2021no—
485C2×C4—2021no—
486C2×C4—2021no—
487C2×C4—2021no—
488C2×C4—2021no—
489C2×C4—2021no—
490C2×C4—2021no—
491C2×C4—2021no—
492C2×C4—2021no—
493C2×C4—2021no—
494C2×C4D4×D10112yes—
495C2×C4D4×D10112yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—1014no—
511C22—40416no—
512C22—4041no—
513C22—4041no—
514C22—4041no—
515C22—4041no—
516C22—4041no—
517C22—4041no—
518C22—4041no—
519C22—4041no—
520C22—2022no—
521C22—2022no—
522C22—2022no—
523C22—2022no—
524C22—2022no—
525C22—2022no—
526C22—2022no—
527C22—2022no—
528C22—2022no—
529C22—2022no—
530C22—2022no—
531C22—2022no—
532C22C2×D4×D10111yescharacteristic
533C22C2×D4×D10111yescharacteristic
534C22C2×D4×D10111yescharacteristic
535C22#41×D10111yescharacteristic
536C22C2×#1310807411yescharacteristic
537C22#1569230011yescharacteristic
538C22#1569230111yescharacteristic
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—4041no—
545C4—4041no—
546C4—4041no—
547C4—2022no—
548C4—2022no—
549C2—42no—
550C2—42no—
551C2—4042no—
552C2—4042no—
553C2—2024no—
554C2#120×D10111yescharacteristic
555C2#2026568611yescharacteristic
556C2#2026574311yescharacteristic
557C2#2026581711yescharacteristic
558C2#2026581911yescharacteristic
559C2#2026582311yescharacteristic
560C2#2026582411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#386×C101C2
hypercenter#386D101
hypocenterC101#2404
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#13108074C2semidirect
2. C2×#15692225C2semidirect
3. C2×#15692223C2semidirect
4. C2×C22⋊C4×D101C2semidirect
5. #20265090C2semidirect
6. #20265089C2semidirect
7. #20265086C2semidirect
8. #20265085C2semidirect
9. #20265056C2semidirect
10. #20265055C2semidirect
11. #20265055C2semidirect
12. #20265053C2semidirect
13. #20265053C2semidirect
14. #20265039C2semidirect
15. #386×C101C2semidirect
20. C22×Dic202C22semidirect
26. C22×C22⋊D101C22semidirect
29. C22×C4×D101C22semidirect
34. C2×C22⋊Dic101C22semidirect
41. C2×#13108051C22semidirect
42. C2×#13108051C22semidirect
44. #15692162C22semidirect
45. #15692162C22semidirect
46. #15692162C22semidirect
47. #15692162C22semidirect
48. #15692162C22semidirect
49. #15692162C22semidirect
51. C23×C4×C101C22semidirect
54. C2×C22⋊C4×C101C22semidirect
55. #95×C101C22semidirect
56. #95×C101C22semidirect
76. C2×D404D4semidirect
88. C2×C22⋊D101D4semidirect
102. C2×C4×D101D4semidirect
143. C23×C101#41semidirect
172. C22×D101#41semidirect
281. C101#2404semidirect
295. #386D101semidirect
312. C23×C4C2×D101semidirect
329. C2×C22⋊C4C2×D101semidirect
336. #95C2×D101semidirect
337. #95C2×D101semidirect
452. C23#13108074semidirect

Maximal subgroups

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

subgroupindex
C22×#131080742
C2×#156922252
C2×#156922232
C2×C22⋊C4×D1012
#202650902
#202650892
#202650862
#202650852
#202650562
#202650552
#202650532
#202650392
#386×C1012
#2404101