Finite Group Databasebeta

← #27144392

Subgroups — #27144392

Order 12416.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690778C211yesmaximal, characteristic
2C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
4#20260638C211yesmaximal, characteristic, complemented
5#20260635C211yesmaximal, characteristic, complemented
6#20260627C211yesmaximal, characteristic, complemented
7#20260624C211yesmaximal, characteristic, complemented
8#20260623C211yesmaximal, characteristic, complemented
9#20260623C211yesmaximal, characteristic, complemented
10#20260614C211yesmaximal, characteristic, complemented
11#20260611C211yesmaximal, characteristic, complemented
12#20260609C211yesmaximal, characteristic, complemented
13#20260590C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#386×C97C211yesmaximal, characteristic, complemented
16C24×D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21#15690778C418yescomplemented
22C4×C22⋊D97—24no—
23C2×C4⋊Dic97C2211yescharacteristic, complemented
24C2×#13095490C2211yescharacteristic, complemented
25C2×#13095490C2211yescharacteristic, complemented
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic, complemented
30C2×C22⋊Dic97C2211yescharacteristic, complemented
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic
39C22⋊C4×D97—24no—
40C2×C4×Dic97C2211yescharacteristic, complemented
41C2×C4×Dic97C2211yescharacteristic, complemented
42C2×C4×Dic97C2211yescharacteristic, complemented
43#15690717C2211yescharacteristic, complemented
44#15690717C2211yescharacteristic, complemented
45#15690717C2211yescharacteristic, complemented
46#15690717C2211yescharacteristic, complemented
47C22×D4×C97C2211yescharacteristic
48C23×C4×C97C2211yescharacteristic, complemented
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97C2211yescharacteristic, complemented
52#95×C97C2211yescharacteristic, complemented
53#95×C97C2211yescharacteristic, complemented
54C23×D97—22no—
55C23×D97—22no—
56C23×D97—41no—
57C23×D97—41no—
58C23×D97—41no—
59C23×D97—41no—
60C23×D97C2×C414yescomplemented
61C23×D97C2311yescharacteristic
62C23×D97C2311yescharacteristic
63C2×C22⋊D97—22no—
64C2×C22⋊D97—22no—
65C2×C22⋊D97—41no—
66C2×C22⋊D97—41no—
67C2×C22⋊D97C2×C414yes—
68C2×C22⋊D97C2×C414yescomplemented
69C22×Dic97C2311yescharacteristic
70C22×Dic97C2311yescharacteristic
71C22×Dic97C2311yescharacteristic
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2311yescharacteristic
74C22×Dic97C2311yescharacteristic
75C2×C4×D97—22no—
76C2×C4×D97—41no—
77C2×C4×D97—41no—
78C2×C4×D97—41no—
79C2×C4×D97—41no—
80C2×C4×D97D414yescomplemented
81C4⋊Dic97—41no—
82#13095490—22no—
83#13095490—41no—
84C22⋊Dic97—22no—
85C22⋊Dic97—22no—
86C22⋊Dic97—41no—
87C22⋊Dic97—41no—
88C22⋊Dic97C2×C414yescomplemented
89#13095488—22no—
90#13095488—22no—
91#13095488—22no—
92#13095488—22no—
93#13095488—41no—
94#13095488—41no—
95#13095488C2×C414yescomplemented
96#13095488C2×C414yescomplemented
97C4×Dic97—22no—
98C4×Dic97—41no—
99C4×Dic97—41no—
100C24×C97C2311yescharacteristic
101C24×C97C2311yescharacteristic
102C2×D4×C97—41no—
103C2×D4×C97—41no—
104C2×D4×C97C2×C414yescomplemented
105C22×C4×C97—22no—
106C22×C4×C97—41no—
107C22×C4×C97—41no—
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111C22×C4×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113C22⋊C4×C97—22no—
114C22⋊C4×C97—41no—
115C22⋊C4×C97—41no—
116C23×C97—21no—
117C23×C97—21no—
118C23×C97—21no—
119C23×C97—21no—
120C23×C97—21no—
121C23×C97—21no—
122C23×C97—21no—
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—21no—
126C23×C97—21no—
127C23×C97—21no—
128C23×C97C22×C412yes—
129C23×C97C22×C412yes—
130C23×C97C2411yescharacteristic
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—28no—
142C22×D97—44no—
143C22×D97—44no—
144C22×D97—44no—
145C22×D97—44no—
146C22×D97—44no—
147C22×D97—44no—
148C22×D97C22×C412yes—
149C22×D97C22×C412yes—
150C22×D97C2×D412yes—
151C22×D97#4112yes—
152D4×C97—44no—
153C2×C4×C97—21no—
154C2×C4×C97—21no—
155C2×C4×C97—21no—
156C2×C4×C97—21no—
157C2×C4×C97—21no—
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—44no—
167C2×C4×C97C22×C412yes—
168C2×C4×C97C2×D412yes—
169C22⋊D97—28no—
170C22⋊D97—44no—
171C2×Dic97—21no—
172C2×Dic97—21no—
173C2×Dic97—21no—
174C2×Dic97—21no—
175C2×Dic97—21no—
176C2×Dic97—21no—
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×Dic97C22×C412yes—
187C2×Dic97C22×C412yes—
188C2×Dic97C2×D412yes—
189C4×D97—44no—
190C4×D97—44no—
191C22×C97—22no—
192C22×C97—22no—
193C22×C97—22no—
194C22×C97—22no—
195C22×C97—22no—
196C22×C97—22no—
197C22×C97—41no—
198C22×C97—41no—
199C22×C97—41no—
200C22×C97—41no—
201C22×C97—41no—
202C22×C97—41no—
203C22×C97—41no—
204C22×C97—41no—
205C22×C97C23×C411yescharacteristic
206C22×C97C22×D411yescharacteristic
207C22×C97C2×#4111yescharacteristic
208C22×C97H(5,𝔽2)11yescharacteristic
209C22×C97H(5,𝔽2)11yescharacteristic
210C22×C97H(5,𝔽2)11yescharacteristic
211C22×C97H(5,𝔽2)11yescharacteristic
212C2×D97—22no—
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—416no—
222C2×D97—41no—
223C2×D97—41no—
224C2×D97—41no—
225C2×D97—41no—
226C2×D97C4×D414yes—
227C4×C97—22no—
228C4×C97—22no—
229C4×C97—41no—
230C4×C97—41no—
231C4×C97—41no—
232Dic97—22no—
233Dic97—22no—
234Dic97—22no—
235Dic97—41no—
236Dic97—41no—
237Dic97—41no—
238C2×C97—42no—
239C2×C97—42no—
240C2×C97C2×C4×D411yescharacteristic
241C2×C97#51811yescharacteristic
242C2×C97#51811yescharacteristic
243C2×C97#53411yescharacteristic
244C2×C97#53511yescharacteristic
245C2×C97#53811yescharacteristic
246C2×C97#53811yescharacteristic
247D97—24no—
248D97—42no—
249#2321—971nomaximal, Sylow
250C97#232111yescharacteristic, complemented, Sylow
251C2×#120—971no—
252C2×C4×D4—971no—
253C22×C22⋊C4—971no—
254#390—971no—
255#390—971no—
256#388—971no—
257#388—971no—
258#388—971no—
259#388—971no—
260#386—971no—
261#386—971no—
262#386D9711yescharacteristic, complemented
263#385—971no—
264#379—971no—
265C4×C22⋊C4—971no—
266C25—971no—
267C22×D4—971no—
268C22×D4—971no—
269C22×D4C2×D9711yescharacteristic
270C23×C4—971no—
271C23×C4—971no—
272C23×C4C2×D9711yescharacteristic, complemented
273#120—978no—
274C4×D4—1944no—
275C2×C4⋊C4—971no—
276C2×C4⋊C4—971no—
277C2×C4⋊C4—971no—
278C2×C22⋊C4—971no—
279C2×C22⋊C4—971no—
280C2×C22⋊C4—971no—
281C2×C22⋊C4—971no—
282C2×C22⋊C4—971no—
283C2×C22⋊C4—971no—
284C2×C22⋊C4—971no—
285C2×C22⋊C4—971no—
286C2×C22⋊C4—971no—
287C2×C22⋊C4—971no—
288C2×C22⋊C4—971no—
289C2×C22⋊C4—971no—
290C2×C22⋊C4—971no—
291C2×C22⋊C4—1944no—
292C2×C22⋊C4C2×D9711yescharacteristic
293C2×C22⋊C4C2×D9711yescharacteristic
294C2×C22⋊C4C2×D9711yescharacteristic, complemented
295C2×C42—971no—
296C2×C42—971no—
297C2×C42—971no—
298#95—971no—
299#95—971no—
300#95—971no—
301#95—971no—
302#95C2×D9711yescharacteristic, complemented
303#95C2×D9711yescharacteristic, complemented
304C24—971no—
305C24—974no—
306C24—971no—
307C24—3881no—
308C24—3881no—
309C24—3881no—
310C24—3881no—
311C24—1942no—
312C24—1942no—
313C24C22×D9711yescharacteristic
314C24C22×D9711yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—974no—
318C2×D4—974no—
319C2×D4—3881no—
320C2×D4—3881no—
321C2×D4—1942no—
322C2×D4—1942no—
323C2×D4C4×D9714yescomplemented
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—971no—
328C22×C4—971no—
329C22×C4—971no—
330C22×C4—974no—
331C22×C4—971no—
332C22×C4—971no—
333C22×C4—971no—
334C22×C4—3881no—
335C22×C4—3881no—
336C22×C4—3881no—
337C22×C4—3881no—
338C22×C4—1942no—
339C22×C4C22×D9711yescharacteristic
340C22×C4C22×D9711yescharacteristic
341C22×C4C22×D9711yescharacteristic
342C22×C4C22×D9711yescharacteristic
343C22×C4C22×D9711yescharacteristic
344C4⋊C4—3881no—
345C4⋊C4—3881no—
346C4⋊C4—1942no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—974no—
351C22⋊C4—974no—
352C22⋊C4—974no—
353C22⋊C4—3881no—
354C22⋊C4—3881no—
355C22⋊C4—3881no—
356C22⋊C4—3881no—
357C22⋊C4—1942no—
358C22⋊C4—1942no—
359C22⋊C4—1942no—
360C22⋊C4—1942no—
361C22⋊C4—1942no—
362C22⋊C4—1942no—
363C42—3881no—
364C42—3881no—
365C42—1942no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—972no—
379C23—972no—
380C23—972no—
381C23—972no—
382C23—3884no—
383C23—3884no—
384C23—3884no—
385C23—3884no—
386C23—3884no—
387C23—3884no—
388C23—1948no—
389C23—1941no—
390C23—1941no—
391C23—1941no—
392C23—1941no—
393C23—1941no—
394C23—1941no—
395C23—1941no—
396C23—1941no—
397C23—1941no—
398C23—1941no—
399C23C2×C4×D9712yes—
400C23C2×C4×D9712yes—
401C23C23×D9711yescharacteristic
402D4—44no—
403D4—3884no—
404D4—1948no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—44no—
419C2×C4—972no—
420C2×C4—972no—
421C2×C4—972no—
422C2×C4—3884no—
423C2×C4—3884no—
424C2×C4—1941no—
425C2×C4—1941no—
426C2×C4—1941no—
427C2×C4—1941no—
428C2×C4—1941no—
429C2×C4—1941no—
430C2×C4—1941no—
431C2×C4—1941no—
432C2×C4—1941no—
433C2×C4—1941no—
434C2×C4—1941no—
435C2×C4—1941no—
436C2×C4—1941no—
437C2×C4—1941no—
438C2×C4—1941no—
439C2×C4C2×C4×D9712yes—
440C2×C4D4×D9712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—974no—
456C22—38816no—
457C22—3881no—
458C22—3881no—
459C22—3881no—
460C22—3881no—
461C22—1942no—
462C22—1942no—
463C22—1942no—
464C22—1942no—
465C22—1942no—
466C22—1942no—
467C22—1942no—
468C22—1942no—
469C22—1942no—
470C22C22×C4×D9711yescharacteristic
471C22C2×D4×D9711yescharacteristic
472C22#41×D9711yescharacteristic
473C22#1569085511yescharacteristic
474C22#1569085511yescharacteristic
475C22#1569085511yescharacteristic
476C22#1569085611yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—3881no—
483C4—3881no—
484C4—3881no—
485C4—1942no—
486C4—1942no—
487C4—1942no—
488C2—42no—
489C2—42no—
490C2—3882no—
491C2—1944no—
492C2C4×D4×D9711yescharacteristic
493C2#2026120711yescharacteristic
494C2#2026120811yescharacteristic
495C2#2026129511yescharacteristic
496C2#2026130111yescharacteristic
497C2#2026131711yescharacteristic
498C2#2026132111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#386×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690778C2
hypercenter#386D97
hypocenterC97#2321
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D97C2semidirect
3. C2×C22⋊C4×D97C2semidirect
4. #20260638C2semidirect
5. #20260635C2semidirect
6. #20260627C2semidirect
7. #20260624C2semidirect
8. #20260623C2semidirect
9. #20260623C2semidirect
10. #20260614C2semidirect
11. #20260611C2semidirect
12. #20260609C2semidirect
13. #20260590C2semidirect
14. #20260582C2semidirect
15. #386×C97C2semidirect
21. #15690778C4semidirect
23. C2×C4⋊Dic97C22semidirect
24. C2×#13095490C22semidirect
25. C2×#13095490C22semidirect
29. C2×C22⋊Dic97C22semidirect
30. C2×C22⋊Dic97C22semidirect
40. C2×C4×Dic97C22semidirect
41. C2×C4×Dic97C22semidirect
42. C2×C4×Dic97C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
48. C23×C4×C97C22semidirect
51. C2×C22⋊C4×C97C22semidirect
52. #95×C97C22semidirect
53. #95×C97C22semidirect
60. C23×D97C2×C4semidirect
68. C2×C22⋊D97C2×C4semidirect
80. C2×C4×D97D4semidirect
88. C22⋊Dic97C2×C4semidirect
95. #13095488C2×C4semidirect
96. #13095488C2×C4semidirect
104. C2×D4×C97C2×C4semidirect
250. C97#2321semidirect
262. #386D97semidirect
272. C23×C4C2×D97semidirect
294. C2×C22⋊C4C2×D97semidirect
302. #95C2×D97semidirect
303. #95C2×D97semidirect
323. C2×D4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907782
C2×C4×C22⋊D972
C2×C22⋊C4×D972
#202606382
#202606352
#202606272
#202606242
#202606232
#202606142
#202606112
#202606092
#202605902
#202605822
#386×C972
#232197