Finite Group Databasebeta

← #27144716

Subgroups — #27144716

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×#15690778C211yesmaximal, characteristic
2C2×D4×Dic97C211yesmaximal, characteristic, complemented
3C2×#15690770C211yesmaximal, characteristic, complemented
4C22×#13095488C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
6#20260635C211yesmaximal, characteristic, complemented
7#20260627C211yesmaximal, characteristic, complemented
8#20260624C211yesmaximal, characteristic, complemented
9#20260612C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11#20260609C211yesmaximal, characteristic, complemented
12#20260607C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#386×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C23×Dic97C2211yescharacteristic
20C23×Dic97C2211yescharacteristic, complemented
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23#15690778C418yescomplemented
24D4×Dic97—24no—
25#15690770—24no—
26C2×C4⋊Dic97C2211yescharacteristic, complemented
27C2×#13095490C2211yescharacteristic, complemented
28C2×#13095490C2211yescharacteristic, complemented
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×C22⋊Dic97C2211yescharacteristic, complemented
32C2×#13095488—24no—
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic
39C2×#13095488C2211yescharacteristic
40C2×#13095488C2211yescharacteristic
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
49C22×D4×C97C2211yescharacteristic
50C23×C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97C2211yescharacteristic
52C2×C22⋊C4×C97C2211yescharacteristic, complemented
53C2×C22⋊C4×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
65C2×C22⋊D97—22no—
66C2×C22⋊D97—22no—
67C2×C22⋊D97—41no—
68C2×C22⋊D97—41no—
69C2×C22⋊D97C2×C414yescomplemented
70C2×C22⋊D97C2×C414yescomplemented
71C22×Dic97—22no—
72C22×Dic97—22no—
73C22×Dic97—22no—
74C22×Dic97—22no—
75C22×Dic97—41no—
76C22×Dic97—41no—
77C22×Dic97C2311yescharacteristic
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—41no—
88C2×C4×D97—41no—
89C4⋊Dic97—22no—
90#13095490—22no—
91#13095490—41no—
92C22⋊Dic97—22no—
93C22⋊Dic97C2×C414yescomplemented
94C22⋊Dic97D414yescomplemented
95#13095488—22no—
96#13095488—22no—
97#13095488—22no—
98#13095488—22no—
99#13095488—22no—
100#13095488C2×C414yescomplemented
101#13095488C2×C414yescomplemented
102#13095488D414yescomplemented
103C4×Dic97—22no—
104C4×Dic97—22no—
105C4×Dic97—41no—
106C24×C97C2311yescharacteristic
107C24×C97C2311yescharacteristic
108C2×D4×C97—22no—
109C2×D4×C97—22no—
110C2×D4×C97C2×C414yescomplemented
111C22×C4×C97—22no—
112C22×C4×C97—22no—
113C22×C4×C97—41no—
114C22×C4×C97C2311yescharacteristic
115C22×C4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118C22×C4×C97C2311yescharacteristic
119C22⋊C4×C97—22no—
120C22⋊C4×C97—22no—
121C22⋊C4×C97—41no—
122C23×C97—21no—
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C23×C97—21no—
128C23×C97—21no—
129C23×C97—21no—
130C23×C97—21no—
131C23×C97—21no—
132C23×C97C22×C412yes—
133C23×C97C22×C412yes—
134C23×C97C2×D412yes—
135C23×C97#4112yes—
136C23×C97C2411yescharacteristic
137C22×D97—21no—
138C22×D97—21no—
139C22×D97—21no—
140C22×D97—21no—
141C22×D97—28no—
142C22×D97—21no—
143C22×D97—21no—
144C22×D97—21no—
145C22×D97—21no—
146C22×D97—21no—
147C22×D97—21no—
148C22×D97—28no—
149C22×D97—44no—
150C22×D97—44no—
151C22×D97—44no—
152C22×D97—44no—
153C22×D97—44no—
154C22×D97C22×C412yes—
155C22×D97C22×C412yes—
156C22×D97C2×D412yes—
157C22×D97#4112yes—
158D4×C97—28no—
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—44no—
173C2×C4×C97C22×C412yes—
174C2×C4×C97C2×D412yes—
175C22⋊D97—28no—
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×Dic97—21no—
190C2×Dic97—44no—
191C2×Dic97—44no—
192C2×Dic97C22×C412yes—
193C2×Dic97C22×C412yes—
194C2×Dic97C2×D412yes—
195C2×Dic97C2×D412yes—
196C2×Dic97C2×D412yes—
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×C97—22no—
206C22×C97—22no—
207C22×C97—22no—
208C22×C97—41no—
209C22×C97—41no—
210C22×C97—41no—
211C22×C97—41no—
212C22×C97C4×D414yescomplemented
213C22×C97C23×C411yescharacteristic
214C22×C97C22×D411yescharacteristic
215C22×C97C22×D411yescharacteristic
216C22×C97C2×#4111yescharacteristic
217C22×C97C2×#4111yescharacteristic
218C22×C97H(5,𝔽2)11yescharacteristic
219C22×C97H(5,𝔽2)11yescharacteristic
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—22no—
229C2×D97—416no—
230C2×D97—41no—
231C2×D97—41no—
232C2×D97—41no—
233C2×D97—41no—
234C2×D97C4×D414yescomplemented
235C4×C97—22no—
236C4×C97—22no—
237C4×C97—41no—
238C4×C97—41no—
239C4×C97—41no—
240Dic97—22no—
241Dic97—22no—
242Dic97—22no—
243Dic97—22no—
244Dic97—22no—
245Dic97—41no—
246C2×C97—24no—
247C2×C97—42no—
248C2×C97C2×C4×D411yescharacteristic
249C2×C97C2×C4×D411yescharacteristic
250C2×C97#51811yescharacteristic
251C2×C97D4211yescharacteristic
252C2×C97#54611yescharacteristic
253C2×C97#54611yescharacteristic
254C2×C97#55111yescharacteristic
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#388—971no—
267#386—971no—
268#386—971no—
269#386D9711yescharacteristic, complemented
270#385—971no—
271#385—971no—
272C4×C22⋊C4—971no—
273C4×C22⋊C4—971no—
274C25—971no—
275C22×D4—971no—
276C22×D4—971no—
277C22×D4C2×D9711yescharacteristic
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⋊C4—971no—
288C2×C4⋊C4—971no—
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, complemented
304C2×C22⋊C4C2×D9711yescharacteristic, complemented
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—3881no—
318C24—3881no—
319C24—1942no—
320C24—1942no—
321C24—1942no—
322C24—1942no—
323C24C22×D9711yescharacteristic
324C24C22×D9711yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—974no—
328C2×D4—974no—
329C2×D4—3881no—
330C2×D4—3881no—
331C2×D4—1942no—
332C2×D4—1942no—
333C2×D4C4×D9714yescomplemented
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—971no—
343C22×C4—3881no—
344C22×C4—3881no—
345C22×C4—3881no—
346C22×C4—3881no—
347C22×C4—1942no—
348C22×C4—1942no—
349C22×C4—1942no—
350C22×C4—1942no—
351C22×C4—1942no—
352C22×C4—1942no—
353C22×C4—1942no—
354C22×C4—1942no—
355C22×C4C22×D9711yescharacteristic
356C22×C4C22×D9711yescharacteristic
357C22×C4C22×D9711yescharacteristic
358C22×C4C22×D9711yescharacteristic
359C22×C4C22×D9711yescharacteristic
360C4⋊C4—3881no—
361C4⋊C4—1942no—
362C4⋊C4—1942no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—41no—
366C22⋊C4—974no—
367C22⋊C4—974no—
368C22⋊C4—974no—
369C22⋊C4—974no—
370C22⋊C4—974no—
371C22⋊C4—1942no—
372C22⋊C4—1942no—
373C22⋊C4—1942no—
374C22⋊C4—1942no—
375C22⋊C4—1942no—
376C22⋊C4—1942no—
377C42—3881no—
378C42—1942no—
379C42—1942no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—972no—
391C23—972no—
392C23—972no—
393C23—972no—
394C23—3884no—
395C23—3884no—
396C23—3884no—
397C23—3884no—
398C23—3884no—
399C23—1948no—
400C23—1948no—
401C23—1941no—
402C23—1941no—
403C23—1941no—
404C23—1941no—
405C23—1941no—
406C23—1941no—
407C23—1941no—
408C23—1941no—
409C23—1941no—
410C23—1941no—
411C23C2×C4×D9712yes—
412C23C2×C4×D9712yes—
413C23D4×D9712yes—
414C23#1309551012yes—
415C23C23×D9711yescharacteristic
416D4—28no—
417D4—3884no—
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—44no—
433C2×C4—972no—
434C2×C4—972no—
435C2×C4—972no—
436C2×C4—972no—
437C2×C4—972no—
438C2×C4—3884no—
439C2×C4—3884no—
440C2×C4—3884no—
441C2×C4—3884no—
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×C4—1941no—
454C2×C4—1941no—
455C2×C4C2×C4×D9712yes—
456C2×C4D4×D9712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—974no—
471C22—38816no—
472C22—3881no—
473C22—3881no—
474C22—3881no—
475C22—3881no—
476C22—1942no—
477C22—1942no—
478C22—1942no—
479C22—1942no—
480C22—1942no—
481C22—1942no—
482C22—1942no—
483C22—1942no—
484C22—1942no—
485C22#1569077014yescomplemented
486C22C22×C4×D9711yescharacteristic
487C22C2×D4×D9711yescharacteristic
488C22C2×D4×D9711yescharacteristic
489C22#41×D9711yescharacteristic
490C22C2×#1309551011yescharacteristic
491C22#1569085511yescharacteristic
492C22#1569085611yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—3881no—
499C4—1942no—
500C4—1942no—
501C4—1942no—
502C4—1942no—
503C4—1942no—
504C2—24no—
505C2—42no—
506C2—3882no—
507C2—1944no—
508C2C4×D4×D9711yescharacteristic
509C2C2×#1569077011yescharacteristic
510C2#2026120711yescharacteristic
511C2#2026136911yescharacteristic
512C2#2026137211yescharacteristic
513C2#2026137411yescharacteristic
514C2#2026140811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#386×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690778C2
hypercenter#386D97
hypocenterC97#2358
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic97C2semidirect
3. C2×#15690770C2semidirect
4. C22×#13095488C2semidirect
5. C2×C22⋊C4×D97C2semidirect
6. #20260635C2semidirect
7. #20260627C2semidirect
8. #20260624C2semidirect
9. #20260612C2semidirect
10. #20260611C2semidirect
11. #20260609C2semidirect
12. #20260607C2semidirect
13. C4×C22⋊Dic97C2semidirect
14. #20260582C2semidirect
15. #386×C97C2semidirect
20. C23×Dic97C22semidirect
23. #15690778C4semidirect
26. C2×C4⋊Dic97C22semidirect
27. C2×#13095490C22semidirect
28. C2×#13095490C22semidirect
31. C2×C22⋊Dic97C22semidirect
42. C2×C4×Dic97C22semidirect
43. C2×C4×Dic97C22semidirect
44. C2×C4×Dic97C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
47. #15690717C22semidirect
48. #15690717C22semidirect
52. C2×C22⋊C4×C97C22semidirect
53. C2×C22⋊C4×C97C22semidirect
54. #95×C97C22semidirect
55. #95×C97C22semidirect
62. C23×D97C2×C4semidirect
69. C2×C22⋊D97C2×C4semidirect
70. C2×C22⋊D97C2×C4semidirect
93. C22⋊Dic97C2×C4semidirect
94. C22⋊Dic97D4semidirect
100. #13095488C2×C4semidirect
101. #13095488C2×C4semidirect
102. #13095488D4semidirect
110. C2×D4×C97C2×C4semidirect
212. C22×C97C4×D4semidirect
234. C2×D97C4×D4semidirect
258. C97#2358semidirect
269. #386D97semidirect
303. C2×C22⋊C4C2×D97semidirect
304. C2×C22⋊C4C2×D97semidirect
312. #95C2×D97semidirect
313. #95C2×D97semidirect
333. C2×D4C4×D97semidirect
485. C22#15690770semidirect

Maximal subgroups

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

subgroupindex
C2×#156907782
C2×D4×Dic972
C2×#156907702
C22×#130954882
C2×C22⋊C4×D972
#202606352
#202606272
#202606242
#202606122
#202606112
#202606092
#202606072
C4×C22⋊Dic972
#202605822
#386×C972
#235897