Finite Group Databasebeta

← #27144714

Subgroups — #27144714

Order 12416.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690780C211yesmaximal, characteristic
2C2×C4×D388C211yesmaximal, characteristic, complemented
3C2×#15690772C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
6#20260635C211yesmaximal, characteristic, complemented
7#20260628C211yesmaximal, characteristic, complemented
8#20260613C211yesmaximal, characteristic, complemented
9#20260613C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11#20260609C211yesmaximal, characteristic, complemented
12#20260606C211yesmaximal, characteristic, complemented
13C4×#13095488C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×C22⋊D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C22×C4×D97C2211yescharacteristic
24C22×C4×D97C2211yescharacteristic, complemented
25#15690780C418yescomplemented
26C4×D388—24no—
27#15690772—24no—
28C2×C4⋊Dic97C2211yescharacteristic, complemented
29C2×#13095490C2211yescharacteristic, complemented
30C2×C22⋊Dic97C2211yescharacteristic
31C2×C22⋊Dic97C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic, complemented
39C2×#13095488C2211yescharacteristic, complemented
40C2×#13095488C2211yescharacteristic, complemented
41C22⋊C4×D97—24no—
42C22⋊C4×D97—24no—
43C2×C4×Dic97C2211yescharacteristic, complemented
44C2×C4×Dic97C2211yescharacteristic, complemented
45#15690717C2211yescharacteristic, complemented
46#15690717C2211yescharacteristic, complemented
47#15690717C2211yescharacteristic, complemented
48#15690717C2211yescharacteristic, complemented
49C2×C4⋊C4×C97C2211yescharacteristic, complemented
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97C2211yescharacteristic
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C42×C97C2211yescharacteristic, complemented
54#95×C97C2211yescharacteristic, complemented
55#95×C97C2211yescharacteristic, complemented
56C23×D97—22no—
57C23×D97—22no—
58C23×D97—22no—
59C23×D97—22no—
60C23×D97—41no—
61C23×D97—41no—
62C23×D97C2×C414yescomplemented
63C23×D97C2311yescharacteristic
64C23×D97C2311yescharacteristic
65C23×D97C2311yescharacteristic
66C2×D388—22no—
67C2×D388—22no—
68C2×D388—22no—
69C2×D388—22no—
70C2×D388C2×C414yescomplemented
71C2×D388C2×C414yescomplemented
72C2×C22⋊D97—41no—
73C2×C22⋊D97—41no—
74C2×C22⋊D97C2×C414yescomplemented
75C22×Dic97C2311yescharacteristic
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2311yescharacteristic
79C22×Dic97C2311yescharacteristic
80C2×C4×D97—22no—
81C2×C4×D97—22no—
82C2×C4×D97—22no—
83C2×C4×D97—22no—
84C2×C4×D97—22no—
85C2×C4×D97—22no—
86C2×C4×D97—22no—
87C2×C4×D97—22no—
88C2×C4×D97—22no—
89C2×C4×D97—22no—
90C2×C4×D97—41no—
91C2×C4×D97—41no—
92C2×C4×D97—41no—
93C2×C4×D97—41no—
94C2×C4×D97—41no—
95C4⋊Dic97—22no—
96#13095490—41no—
97C22⋊Dic97—22no—
98C22⋊Dic97—22no—
99#13095488—22no—
100#13095488—22no—
101#13095488—22no—
102#13095488—22no—
103#13095488—41no—
104#13095488C2×C414yescomplemented
105#13095488C2×C414yescomplemented
106#13095488D414yescomplemented
107#13095488D414yescomplemented
108C4×Dic97—22no—
109C4×Dic97—41no—
110C24×C97C2311yescharacteristic
111C22×C4×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113C22×C4×C97C2311yescharacteristic
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C4⋊C4×C97—22no—
118C22⋊C4×C97—22no—
119C22⋊C4×C97—22no—
120C22⋊C4×C97C2×C414yescomplemented
121C42×C97—22no—
122C23×C97—21no—
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C23×C97—21no—
128C23×C97C22×C412yes—
129C23×C97C2411yescharacteristic
130C22×D97—21no—
131C22×D97—21no—
132C22×D97—21no—
133C22×D97—21no—
134C22×D97—21no—
135C22×D97—21no—
136C22×D97—21no—
137C22×D97—21no—
138C22×D97—21no—
139C22×D97—21no—
140C22×D97—21no—
141C22×D97—21no—
142C22×D97—21no—
143C22×D97—21no—
144C22×D97—28no—
145C22×D97—28no—
146C22×D97—44no—
147C22×D97—44no—
148C22×D97—44no—
149C22×D97—44no—
150C22×D97—44no—
151C22×D97C22×C412yes—
152C22×D97C22×C412yes—
153C22×D97C22×C412yes—
154C22×D97C2×D412yes—
155C22×D97C2×D412yes—
156C22×D97#4112yes—
157C22×D97#4112yes—
158C2×C4×C97—21no—
159C2×C4×C97—21no—
160C2×C4×C97—21no—
161C2×C4×C97—21no—
162C2×C4×C97—21no—
163C2×C4×C97—21no—
164C2×C4×C97—21no—
165C2×C4×C97—21no—
166C2×C4×C97—21no—
167C2×C4×C97—21no—
168C2×C4×C97—21no—
169C2×C4×C97—21no—
170C2×C4×C97—21no—
171C2×C4×C97—21no—
172C2×C4×C97C22×C412yes—
173C2×C4×C97C22×C412yes—
174C2×C4×C97C2×D412yes—
175C2×C4×C97C2×D412yes—
176C22⋊D97—44no—
177C2×Dic97—21no—
178C2×Dic97—21no—
179C2×Dic97—21no—
180C2×Dic97—21no—
181C2×Dic97—21no—
182C2×Dic97—21no—
183C2×Dic97—21no—
184C2×Dic97—21no—
185C2×Dic97—21no—
186C2×Dic97—21no—
187C2×Dic97—21no—
188C2×Dic97—21no—
189C2×Dic97C22×C412yes—
190C2×Dic97C2×D412yes—
191C2×Dic97C2×D412yes—
192D388—28no—
193D388—28no—
194C4×D97—44no—
195C4×D97—44no—
196C4×D97—44no—
197C4×D97—44no—
198C4×D97—44no—
199C22×C97—22no—
200C22×C97—22no—
201C22×C97—22no—
202C22×C97—41no—
203C22×C97—41no—
204C22×C97—41no—
205C22×C97—41no—
206C22×C97C23×C411yescharacteristic
207C22×C97C22×D411yescharacteristic
208C22×C97C22×D411yescharacteristic
209C22×C97C2×#4111yescharacteristic
210C22×C97C2×#4111yescharacteristic
211C22×C97H(5,𝔽2)11yescharacteristic
212C22×C97H(5,𝔽2)11yescharacteristic
213C2×D97—22no—
214C2×D97—22no—
215C2×D97—22no—
216C2×D97—22no—
217C2×D97—22no—
218C2×D97—22no—
219C2×D97—22no—
220C2×D97—22no—
221C2×D97—22no—
222C2×D97—22no—
223C2×D97—22no—
224C2×D97—22no—
225C2×D97—22no—
226C2×D97—22no—
227C2×D97—22no—
228C2×D97—416no—
229C2×D97—41no—
230C2×D97—41no—
231C2×D97—41no—
232C2×D97—41no—
233C2×D97C4×D414yescomplemented
234C2×D97C4×D414yescomplemented
235C4×C97—22no—
236C4×C97—22no—
237C4×C97—22no—
238C4×C97—22no—
239C4×C97—41no—
240C4×C97—41no—
241Dic97—22no—
242Dic97—22no—
243Dic97—22no—
244Dic97—41no—
245Dic97—41no—
246C2×C97—42no—
247C2×C97C2×C4×D411yescharacteristic
248C2×C97C2×C4×D411yescharacteristic
249C2×C97#51811yescharacteristic
250C2×C97D4211yescharacteristic
251C2×C97#54611yescharacteristic
252C2×C97#54611yescharacteristic
253C2×C97#55111yescharacteristic
254D97—24no—
255D97—24no—
256D97—42no—
257#2358—971nomaximal, Sylow
258C97#235811yescharacteristic, complemented, Sylow
259C2×#120—971no—
260C2×C4×D4—971no—
261C2×C4×D4—971no—
262C22×C22⋊C4—971no—
263C22×C22⋊C4—971no—
264#390—971no—
265#388—971no—
266#388D9711yescharacteristic, complemented
267#386—971no—
268#386—971no—
269#386—971no—
270#385—971no—
271#385—971no—
272C4×C22⋊C4—971no—
273C4×C22⋊C4—971no—
274C25—971no—
275C22×D4—971no—
276C22×D4—971no—
277C22×D4—971no—
278C23×C4—971no—
279C23×C4—971no—
280C23×C4—971no—
281C23×C4—971no—
282C23×C4—971no—
283#120—978no—
284C4×D4—1944no—
285C4×D4—1944no—
286C2×C4⋊C4—971no—
287C2×C4⋊C4—971no—
288C2×C4⋊C4C2×D9711yescharacteristic, complemented
289C2×C22⋊C4—971no—
290C2×C22⋊C4—971no—
291C2×C22⋊C4—971no—
292C2×C22⋊C4—971no—
293C2×C22⋊C4—971no—
294C2×C22⋊C4—971no—
295C2×C22⋊C4—971no—
296C2×C22⋊C4—971no—
297C2×C22⋊C4—971no—
298C2×C22⋊C4—971no—
299C2×C22⋊C4—971no—
300C2×C22⋊C4—1944no—
301C2×C22⋊C4—1944no—
302C2×C22⋊C4C2×D9711yescharacteristic
303C2×C22⋊C4C2×D9711yescharacteristic
304C2×C22⋊C4C2×D9711yescharacteristic
305C2×C42—971no—
306C2×C42—971no—
307C2×C42C2×D9711yescharacteristic, complemented
308#95—971no—
309#95—971no—
310#95—971no—
311#95—971no—
312#95C2×D9711yescharacteristic, complemented
313#95C2×D9711yescharacteristic, complemented
314C24—971no—
315C24—971no—
316C24—974no—
317C24—971no—
318C24—3881no—
319C24—3881no—
320C24—1942no—
321C24—1942no—
322C24—1942no—
323C24—1942no—
324C24C22×D9711yescharacteristic
325C2×D4—974no—
326C2×D4—974no—
327C2×D4—974no—
328C2×D4—3881no—
329C2×D4—3881no—
330C2×D4—1942no—
331C2×D4—1942no—
332C2×D4—1942no—
333C2×D4—1942no—
334C22×C4—971no—
335C22×C4—971no—
336C22×C4—971no—
337C22×C4—971no—
338C22×C4—971no—
339C22×C4—3881no—
340C22×C4—3881no—
341C22×C4—3881no—
342C22×C4—3881no—
343C22×C4—3881no—
344C22×C4—1942no—
345C22×C4—1942no—
346C22×C4—1942no—
347C22×C4—1942no—
348C22×C4—1942no—
349C22×C4—1942no—
350C22×C4—1942no—
351C22×C4—1942no—
352C22×C4—1942no—
353C22×C4—1942no—
354C22×C4C22×D9711yescharacteristic
355C22×C4C22×D9711yescharacteristic
356C22×C4C22×D9711yescharacteristic
357C22×C4C22×D9711yescharacteristic
358C22×C4C22×D9711yescharacteristic
359C22×C4C22×D9711yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—3881no—
362C4⋊C4—1942no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—974no—
366C22⋊C4—974no—
367C22⋊C4—974no—
368C22⋊C4—974no—
369C22⋊C4—3881no—
370C22⋊C4—1942no—
371C22⋊C4—1942no—
372C22⋊C4—1942no—
373C22⋊C4—1942no—
374C22⋊C4—1942no—
375C22⋊C4—1942no—
376C22⋊C4C4×D9714yescomplemented
377C42—22no—
378C42—3881no—
379C42—1942no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—972no—
387C23—972no—
388C23—972no—
389C23—972no—
390C23—972no—
391C23—972no—
392C23—972no—
393C23—3884no—
394C23—3884no—
395C23—3884no—
396C23—3884no—
397C23—3884no—
398C23—1948no—
399C23—1948no—
400C23—1941no—
401C23—1941no—
402C23—1941no—
403C23—1941no—
404C23—1941no—
405C23—1941no—
406C23—1941no—
407C23—1941no—
408C23—1941no—
409C23—1941no—
410C23—1941no—
411C23—1941no—
412C23—1941no—
413C23—1941no—
414C23C2×C4×D9712yes—
415C23C23×D9711yescharacteristic
416D4—3884no—
417D4—1948no—
418D4—1948no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—972no—
434C2×C4—972no—
435C2×C4—972no—
436C2×C4—3884no—
437C2×C4—3884no—
438C2×C4—3884no—
439C2×C4—3884no—
440C2×C4—3884no—
441C2×C4—1941no—
442C2×C4—1941no—
443C2×C4—1941no—
444C2×C4—1941no—
445C2×C4—1941no—
446C2×C4—1941no—
447C2×C4—1941no—
448C2×C4—1941no—
449C2×C4—1941no—
450C2×C4—1941no—
451C2×C4—1941no—
452C2×C4—1941no—
453C2×C4C2×C4×D9712yes—
454C2×C4C2×C4×D9712yes—
455C2×C4D4×D9712yes—
456C2×C4D4×D9712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—974no—
465C22—974no—
466C22—38816no—
467C22—3881no—
468C22—3881no—
469C22—3881no—
470C22—3881no—
471C22—1942no—
472C22—1942no—
473C22—1942no—
474C22—1942no—
475C22—1942no—
476C22—1942no—
477C22—1942no—
478C22—1942no—
479C22—1942no—
480C22—1942no—
481C22—1942no—
482C22—1942no—
483C22—1942no—
484C22—1942no—
485C22—1942no—
486C22C22×C4×D9711yescharacteristic
487C22C2×D4×D9711yescharacteristic
488C22C2×D4×D9711yescharacteristic
489C22#41×D9711yescharacteristic
490C22#41×D9711yescharacteristic
491C22#1569085611yescharacteristic
492C22#1569085611yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—41no—
499C4—3881no—
500C4—3881no—
501C4—1942no—
502C4—1942no—
503C4—1942no—
504C2—42no—
505C2—3882no—
506C2—1944no—
507C2—1944no—
508C2C4×D4×D9711yescharacteristic
509C2C4×D4×D9711yescharacteristic
510C2#2026120911yescharacteristic
511C2#2026136911yescharacteristic
512C2#2026137711yescharacteristic
513C2#2026137811yescharacteristic
514C2#2026141011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690780C2
hypercenter#388D97
hypocenterC97#2358
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D388C2semidirect
3. C2×#15690772C2semidirect
4. C2×C22⋊C4×D97C2semidirect
5. C2×C22⋊C4×D97C2semidirect
6. #20260635C2semidirect
7. #20260628C2semidirect
8. #20260613C2semidirect
9. #20260613C2semidirect
10. #20260611C2semidirect
11. #20260609C2semidirect
12. #20260606C2semidirect
13. C4×#13095488C2semidirect
14. #20260582C2semidirect
15. #388×C97C2semidirect
24. C22×C4×D97C22semidirect
25. #15690780C4semidirect
28. C2×C4⋊Dic97C22semidirect
29. C2×#13095490C22semidirect
38. C2×#13095488C22semidirect
39. C2×#13095488C22semidirect
40. C2×#13095488C22semidirect
43. C2×C4×Dic97C22semidirect
44. C2×C4×Dic97C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
47. #15690717C22semidirect
48. #15690717C22semidirect
49. C2×C4⋊C4×C97C22semidirect
53. C2×C42×C97C22semidirect
54. #95×C97C22semidirect
55. #95×C97C22semidirect
62. C23×D97C2×C4semidirect
70. C2×D388C2×C4semidirect
71. C2×D388C2×C4semidirect
74. C2×C22⋊D97C2×C4semidirect
104. #13095488C2×C4semidirect
105. #13095488C2×C4semidirect
106. #13095488D4semidirect
107. #13095488D4semidirect
120. C22⋊C4×C97C2×C4semidirect
233. C2×D97C4×D4semidirect
234. C2×D97C4×D4semidirect
258. C97#2358semidirect
266. #388D97semidirect
288. C2×C4⋊C4C2×D97semidirect
307. C2×C42C2×D97semidirect
312. #95C2×D97semidirect
313. #95C2×D97semidirect
376. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907802
C2×C4×D3882
C2×#156907722
C2×C22⋊C4×D972
#202606352
#202606282
#202606132
#202606112
#202606092
#202606062
C4×#130954882
#202605822
#388×C972
#235897