Finite Group Databasebeta

← #27144712

Subgroups — #27144712

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×#15690772C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
4C22×#13095488C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
6#20260636C211yesmaximal, characteristic, complemented
7#20260625C211yesmaximal, characteristic, complemented
8#20260624C211yesmaximal, characteristic, complemented
9#20260613C211yesmaximal, characteristic, complemented
10#20260613C211yesmaximal, characteristic, complemented
11#20260612C211yesmaximal, characteristic, complemented
12#20260606C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#385×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×D388C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×C22⋊D97C2211yescharacteristic
20C23×Dic97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C22×C4×D97C2211yescharacteristic, complemented
24#15690780C418yescomplemented
25#15690772—24no—
26C4×C22⋊D97—24no—
27C2×#13095490C2211yescharacteristic, complemented
28C2×C22⋊Dic97C2211yescharacteristic
29C2×#13095488—24no—
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
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—
42C2×C4×Dic97C2211yescharacteristic, complemented
43C2×C4×Dic97C2211yescharacteristic, complemented
44C2×C4×Dic97C2211yescharacteristic, complemented
45#15690717C2211yescharacteristic, complemented
46#15690717C2211yescharacteristic, complemented
47#15690717C2211yescharacteristic, complemented
48#15690717C2211yescharacteristic, complemented
49C23×C4×C97C2211yescharacteristic
50C2×C4⋊C4×C97C2211yescharacteristic, complemented
51C2×C4⋊C4×C97C2211yescharacteristic, complemented
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C22⋊C4×C97C2211yescharacteristic
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—41no—
69C2×D388—41no—
70C2×D388C2×C414yescomplemented
71C2×D388C2×C414yescomplemented
72C2×C22⋊D97—22no—
73C2×C22⋊D97—22no—
74C2×C22⋊D97C2×C414yescomplemented
75C22×Dic97—22no—
76C22×Dic97—22no—
77C22×Dic97—41no—
78C22×Dic97C2311yescharacteristic
79C22×Dic97C2311yescharacteristic
80C22×Dic97C2311yescharacteristic
81C22×Dic97C2311yescharacteristic
82C22×Dic97C2311yescharacteristic
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—41no—
90C2×C4×D97—41no—
91C2×C4×D97—41no—
92#13095490—22no—
93C22⋊Dic97D414yescomplemented
94#13095488—22no—
95#13095488—22no—
96#13095488—22no—
97#13095488—22no—
98#13095488—22no—
99#13095488—22no—
100#13095488—22no—
101#13095488—41no—
102#13095488C2×C414yescomplemented
103#13095488C2×C414yescomplemented
104#13095488D414yescomplemented
105C4×Dic97—22no—
106C4×Dic97—22no—
107C4×Dic97—41no—
108C24×C97C2311yescharacteristic
109C22×C4×C97—22no—
110C22×C4×C97—22no—
111C22×C4×C97—41no—
112C22×C4×C97C2311yescharacteristic
113C22×C4×C97C2311yescharacteristic
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118C4⋊C4×C97—22no—
119C4⋊C4×C97—41no—
120C22⋊C4×C97—22no—
121C22⋊C4×C97C2×C414yescomplemented
122C23×C97—21no—
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97C22×C412yes—
127C23×C97C2×D412yes—
128C23×C97#4112yes—
129C23×C97C2411yescharacteristic
130C22×D97—21no—
131C22×D97—21no—
132C22×D97—21no—
133C22×D97—21no—
134C22×D97—28no—
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—21no—
145C22×D97—21no—
146C22×D97—21no—
147C22×D97—28no—
148C22×D97—44no—
149C22×D97—44no—
150C22×D97—44no—
151C22×D97—44no—
152C22×D97—44no—
153C22×D97C22×C412yes—
154C22×D97C22×C412yes—
155C22×D97C22×C412yes—
156C22×D97C2×D412yes—
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×C97—21no—
173C2×C4×C97—44no—
174C2×C4×C97C22×C412yes—
175C2×C4×C97C22×C412yes—
176C2×C4×C97C2×D412yes—
177C22⋊D97—28no—
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×Dic97—44no—
190C2×Dic97C22×C412yes—
191C2×Dic97C2×D412yes—
192C2×Dic97C2×D412yes—
193C2×Dic97C2×D412yes—
194D388—28no—
195D388—44no—
196C4×D97—44no—
197C4×D97—44no—
198C4×D97—44no—
199C22×C97—22no—
200C22×C97—22no—
201C22×C97—22no—
202C22×C97—22no—
203C22×C97—22no—
204C22×C97—22no—
205C22×C97C4×D414yescomplemented
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—416no—
226C2×D97—41no—
227C2×D97—41no—
228C2×D97—41no—
229C2×D97—41no—
230C2×D97—41no—
231C2×D97—41no—
232C2×D97—41no—
233C2×D97—41no—
234C2×D97C4×D414yescomplemented
235C4×C97—22no—
236C4×C97—22no—
237C4×C97—22no—
238C4×C97—41no—
239C4×C97—41no—
240C4×C97—41no—
241Dic97—22no—
242Dic97—22no—
243Dic97—22no—
244Dic97—22no—
245Dic97—41no—
246C2×C97—24no—
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—42no—
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#388—971no—
267#386—971no—
268#386—971no—
269#386—971no—
270#385—971no—
271#385D9711yescharacteristic, complemented
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×C4C2×D9711yescharacteristic
283#120—978no—
284C4×D4—1944no—
285C4×D4—1944no—
286C2×C4⋊C4—971no—
287C2×C4⋊C4C2×D9711yescharacteristic, complemented
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—971no—
301C2×C22⋊C4—1944no—
302C2×C22⋊C4—1944no—
303C2×C22⋊C4C2×D9711yescharacteristic
304C2×C22⋊C4C2×D9711yescharacteristic
305C2×C42—971no—
306C2×C42—971no—
307C2×C42—971no—
308#95—971no—
309#95—971no—
310#95—971no—
311#95—971no—
312#95C2×D9711yescharacteristic, complemented
313#95C2×D9711yescharacteristic, complemented
314C24—971no—
315C24—974no—
316C24—971no—
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—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—971no—
338C22×C4—971no—
339C22×C4—971no—
340C22×C4—971no—
341C22×C4—971no—
342C22×C4—3881no—
343C22×C4—3881no—
344C22×C4—3881no—
345C22×C4—3881no—
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—41no—
362C4⋊C4—1942no—
363C22⋊C4—22no—
364C22⋊C4—974no—
365C22⋊C4—974no—
366C22⋊C4—974no—
367C22⋊C4—974no—
368C22⋊C4—3881no—
369C22⋊C4—1942no—
370C22⋊C4—1942no—
371C22⋊C4—1942no—
372C22⋊C4—1942no—
373C22⋊C4—1942no—
374C22⋊C4—1942no—
375C22⋊C4—1942no—
376C22⋊C4C4×D9714yescomplemented
377C42—3881no—
378C42—1942no—
379C42—1942no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—972no—
385C23—972no—
386C23—972no—
387C23—972no—
388C23—972no—
389C23—3884no—
390C23—3884no—
391C23—3884no—
392C23—3884no—
393C23—3884no—
394C23—1948no—
395C23—1948no—
396C23—1941no—
397C23—1941no—
398C23—1941no—
399C23—1941no—
400C23—1941no—
401C23—1941no—
402C23—1941no—
403C23—1941no—
404C23—1941no—
405C23—1941no—
406C23—1941no—
407C23—1941no—
408C23—1941no—
409C23—1941no—
410C23—1941no—
411C23—1941no—
412C23C2×C4×D9712yes—
413C23D4×D9712yes—
414C23#1309551212yes—
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—21no—
434C2×C4—44no—
435C2×C4—972no—
436C2×C4—972no—
437C2×C4—972no—
438C2×C4—972no—
439C2×C4—3884no—
440C2×C4—3884no—
441C2×C4—3884no—
442C2×C4—3884no—
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×C4—1941no—
454C2×C4C2×C4×D9712yes—
455C2×C4C2×C4×D9712yes—
456C2×C4D4×D9712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—974no—
464C22—38816no—
465C22—3881no—
466C22—3881no—
467C22—3881no—
468C22—3881no—
469C22—3881no—
470C22—3881no—
471C22—3881no—
472C22—3881no—
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#1569077214yescomplemented
486C22C22×C4×D9711yescharacteristic
487C22C2×D4×D9711yescharacteristic
488C22C2×D4×D9711yescharacteristic
489C22#41×D9711yescharacteristic
490C22C2×#1309551211yescharacteristic
491C22#1569085511yescharacteristic
492C22#1569085611yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—3881no—
500C4—1942no—
501C4—1942no—
502C4—1942no—
503C4—1942no—
504C2—24no—
505C2—3882no—
506C2—3882no—
507C2—1944no—
508C2C4×D4×D9711yescharacteristic
509C2C2×#1569077211yescharacteristic
510C2#2026120711yescharacteristic
511C2#2026136711yescharacteristic
512C2#2026137611yescharacteristic
513C2#2026137711yescharacteristic
514C2#2026141211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#385×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690780C2
hypercenter#385D97
hypocenterC97#2358
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#15690772C2semidirect
3. C2×C4×C22⋊D97C2semidirect
4. C22×#13095488C2semidirect
5. C2×C22⋊C4×D97C2semidirect
6. #20260636C2semidirect
7. #20260625C2semidirect
8. #20260624C2semidirect
9. #20260613C2semidirect
10. #20260613C2semidirect
11. #20260612C2semidirect
12. #20260606C2semidirect
13. C22⋊C4×Dic97C2semidirect
14. #20260582C2semidirect
15. #385×C97C2semidirect
23. C22×C4×D97C22semidirect
24. #15690780C4semidirect
27. C2×#13095490C22semidirect
38. C2×#13095488C22semidirect
39. C2×#13095488C22semidirect
40. C2×#13095488C22semidirect
42. C2×C4×Dic97C22semidirect
43. C2×C4×Dic97C22semidirect
44. C2×C4×Dic97C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
47. #15690717C22semidirect
48. #15690717C22semidirect
50. C2×C4⋊C4×C97C22semidirect
51. C2×C4⋊C4×C97C22semidirect
54. #95×C97C22semidirect
55. #95×C97C22semidirect
62. C23×D97C2×C4semidirect
70. C2×D388C2×C4semidirect
71. C2×D388C2×C4semidirect
74. C2×C22⋊D97C2×C4semidirect
93. C22⋊Dic97D4semidirect
102. #13095488C2×C4semidirect
103. #13095488C2×C4semidirect
104. #13095488D4semidirect
121. C22⋊C4×C97C2×C4semidirect
205. C22×C97C4×D4semidirect
234. C2×D97C4×D4semidirect
258. C97#2358semidirect
271. #385D97semidirect
287. C2×C4⋊C4C2×D97semidirect
288. C2×C4⋊C4C2×D97semidirect
312. #95C2×D97semidirect
313. #95C2×D97semidirect
376. C22⋊C4C4×D97semidirect
485. C22#15690772semidirect

Maximal subgroups

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

subgroupindex
C2×#156907802
C2×#156907722
C2×C4×C22⋊D972
C22×#130954882
C2×C22⋊C4×D972
#202606362
#202606252
#202606242
#202606132
#202606122
#202606062
C22⋊C4×Dic972
#202605822
#385×C972
#235897