Finite Group Databasebeta

← #27144398

Subgroups — #27144398

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×#15690780C211yesmaximal, characteristic
2C2×C4×D388C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D97C211yesmaximal, characteristic, complemented
4#20260636C211yesmaximal, characteristic, complemented
5#20260635C211yesmaximal, characteristic, complemented
6#20260628C211yesmaximal, characteristic, complemented
7#20260627C211yesmaximal, characteristic, complemented
8#20260626C211yesmaximal, characteristic, complemented
9#20260613C211yesmaximal, characteristic, complemented
10#20260613C211yesmaximal, characteristic, complemented
11#20260611C211yesmaximal, characteristic, complemented
12#20260606C211yesmaximal, characteristic, complemented
13#20260590C211yesmaximal, 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, complemented
23#15690780C418yescomplemented
24C4×D388—24no—
25C2×C4⋊Dic97C2211yescharacteristic, complemented
26C2×#13095490C2211yescharacteristic, complemented
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic, complemented
38C2×#13095488C2211yescharacteristic, complemented
39C2×#13095488C2211yescharacteristic, complemented
40C22⋊C4×D97—24no—
41C2×C4×Dic97C2211yescharacteristic, complemented
42C2×C4×Dic97C2211yescharacteristic, complemented
43#15690717C2211yescharacteristic, complemented
44#15690717C2211yescharacteristic, complemented
45#15690717C2211yescharacteristic, complemented
46#15690717C2211yescharacteristic, complemented
47C2×C4⋊C4×C97C2211yescharacteristic, complemented
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C42×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
63C23×D97C2311yescharacteristic
64C2×D388—22no—
65C2×D388—22no—
66C2×D388—41no—
67C2×D388—41no—
68C2×D388C2×C414yes—
69C2×D388C2×C414yescomplemented
70C2×C22⋊D97—41no—
71C2×C22⋊D97—41no—
72C2×C22⋊D97C2×C414yescomplemented
73C22×Dic97C2311yescharacteristic
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2311yescharacteristic
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C2×C4×D97—22no—
79C2×C4×D97—22no—
80C2×C4×D97—41no—
81C2×C4×D97—41no—
82C2×C4×D97—41no—
83C2×C4×D97—41no—
84C2×C4×D97—41no—
85C2×C4×D97—41no—
86C2×C4×D97D414yescomplemented
87C4⋊Dic97—22no—
88#13095490—41no—
89C22⋊Dic97—22no—
90C22⋊Dic97—41no—
91#13095488—22no—
92#13095488—22no—
93#13095488—22no—
94#13095488—22no—
95#13095488—22no—
96#13095488—41no—
97#13095488—41no—
98#13095488—41no—
99#13095488—41no—
100#13095488C2×C414yescomplemented
101#13095488C2×C414yescomplemented
102C4×Dic97—41no—
103C4×Dic97—41no—
104C24×C97C2311yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C22×C4×C97C2311yescharacteristic
107C22×C4×C97C2311yescharacteristic
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111C4⋊C4×C97—41no—
112C22⋊C4×C97—22no—
113C22⋊C4×C97—41no—
114C22⋊C4×C97C2×C414yescomplemented
115C42×C97—22no—
116C23×C97—21no—
117C23×C97—21no—
118C23×C97—21no—
119C23×C97—21no—
120C23×C97—21no—
121C23×C97—21no—
122C23×C97C22×C412yes—
123C23×C97C2411yescharacteristic
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
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—28no—
141C22×D97—44no—
142C22×D97—44no—
143C22×D97—44no—
144C22×D97—44no—
145C22×D97—44no—
146C22×D97—44no—
147C22×D97C22×C412yes—
148C22×D97C22×C412yes—
149C22×D97C22×C412yes—
150C22×D97C2×D412yes—
151C22×D97#4112yes—
152C2×C4×C97—21no—
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—21no—
167C2×C4×C97C22×C412yes—
168C2×C4×C97C22×C412yes—
169C2×C4×C97C2×D412yes—
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×Dic97C22×C412yes—
185C2×Dic97C2×D412yes—
186D388—28no—
187D388—44no—
188C4×D97—44no—
189C4×D97—44no—
190C4×D97—44no—
191C22×C97—22no—
192C22×C97—22no—
193C22×C97—22no—
194C22×C97—41no—
195C22×C97—41no—
196C22×C97—41no—
197C22×C97—41no—
198C22×C97C23×C411yescharacteristic
199C22×C97C22×D411yescharacteristic
200C22×C97C2×#4111yescharacteristic
201C22×C97H(5,𝔽2)11yescharacteristic
202C22×C97H(5,𝔽2)11yescharacteristic
203C22×C97H(5,𝔽2)11yescharacteristic
204C22×C97H(5,𝔽2)11yescharacteristic
205C2×D97—22no—
206C2×D97—22no—
207C2×D97—22no—
208C2×D97—22no—
209C2×D97—22no—
210C2×D97—22no—
211C2×D97—22no—
212C2×D97—22no—
213C2×D97—22no—
214C2×D97—22no—
215C2×D97—22no—
216C2×D97—22no—
217C2×D97—416no—
218C2×D97—41no—
219C2×D97—41no—
220C2×D97—41no—
221C2×D97—41no—
222C2×D97—41no—
223C2×D97—41no—
224C2×D97—41no—
225C2×D97—41no—
226C2×D97C4×D414yes—
227C4×C97—22no—
228C4×C97—22no—
229C4×C97—22no—
230C4×C97—41no—
231C4×C97—41no—
232C4×C97—41no—
233Dic97—22no—
234Dic97—22no—
235Dic97—41no—
236Dic97—41no—
237Dic97—41no—
238C2×C97—42no—
239C2×C97C2×C4×D411yescharacteristic
240C2×C97#51811yescharacteristic
241C2×C97#51811yescharacteristic
242C2×C97#53411yescharacteristic
243C2×C97#53511yescharacteristic
244C2×C97#53811yescharacteristic
245C2×C97#53811yescharacteristic
246D97—24no—
247D97—42no—
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#388D9711yescharacteristic, complemented
260#386—971no—
261#386—971no—
262#386—971no—
263#385—971no—
264#379—971no—
265C4×C22⋊C4—971no—
266C25—971no—
267C22×D4—971no—
268C22×D4—971no—
269C22×D4—971no—
270C23×C4—971no—
271C23×C4—971no—
272C23×C4—971no—
273#120—978no—
274C4×D4—1944no—
275C2×C4⋊C4—971no—
276C2×C4⋊C4—971no—
277C2×C4⋊C4C2×D9711yescharacteristic, complemented
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
295C2×C42—971no—
296C2×C42—971no—
297C2×C42C2×D9711yescharacteristic, complemented
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—971no—
308C24—3881no—
309C24—3881no—
310C24—3881no—
311C24—3881no—
312C24—1942no—
313C24—1942no—
314C24C22×D9711yescharacteristic
315C2×D4—974no—
316C2×D4—974no—
317C2×D4—974no—
318C2×D4—3881no—
319C2×D4—3881no—
320C2×D4—3881no—
321C2×D4—3881no—
322C2×D4—1942no—
323C2×D4—1942no—
324C22×C4—971no—
325C22×C4—971no—
326C22×C4—974no—
327C22×C4—971no—
328C22×C4—971no—
329C22×C4—971no—
330C22×C4—3881no—
331C22×C4—3881no—
332C22×C4—3881no—
333C22×C4—3881no—
334C22×C4—3881no—
335C22×C4—3881no—
336C22×C4—1942no—
337C22×C4—1942no—
338C22×C4C22×D9711yescharacteristic
339C22×C4C22×D9711yescharacteristic
340C22×C4C22×D9711yescharacteristic
341C22×C4C22×D9711yescharacteristic
342C22×C4C22×D9711yescharacteristic
343C22×C4C22×D9711yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—3881no—
346C4⋊C4—1942no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—974no—
350C22⋊C4—974no—
351C22⋊C4—3881no—
352C22⋊C4—3881no—
353C22⋊C4—3881no—
354C22⋊C4—3881no—
355C22⋊C4—3881no—
356C22⋊C4—1942no—
357C22⋊C4—1942no—
358C22⋊C4—1942no—
359C22⋊C4—1942no—
360C22⋊C4—1942no—
361C22⋊C4—1942no—
362C22⋊C4C4×D9714yescomplemented
363C42—22no—
364C42—3881no—
365C42—3881no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—972no—
373C23—972no—
374C23—972no—
375C23—972no—
376C23—972no—
377C23—3884no—
378C23—3884no—
379C23—3884no—
380C23—3884no—
381C23—3884no—
382C23—3884no—
383C23—1948no—
384C23—1941no—
385C23—1941no—
386C23—1941no—
387C23—1941no—
388C23—1941no—
389C23—1941no—
390C23—1941no—
391C23—1941no—
392C23—1941no—
393C23—1941no—
394C23—1941no—
395C23—1941no—
396C23—1941no—
397C23—1941no—
398C23—1941no—
399C23—1941no—
400C23C2×C4×D9712yes—
401C23C23×D9711yescharacteristic
402D4—3884no—
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—21no—
419C2×C4—21no—
420C2×C4—972no—
421C2×C4—972no—
422C2×C4—3884no—
423C2×C4—3884no—
424C2×C4—3884no—
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×C4C2×C4×D9712yes—
439C2×C4C2×C4×D9712yes—
440C2×C4D4×D9712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—974no—
449C22—38816no—
450C22—3881no—
451C22—3881no—
452C22—3881no—
453C22—3881no—
454C22—3881no—
455C22—3881no—
456C22—3881no—
457C22—3881no—
458C22—1942no—
459C22—1942no—
460C22—1942no—
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#1569085611yescharacteristic
475C22#1569085611yescharacteristic
476C22#1569085611yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—3881no—
484C4—3881no—
485C4—3881no—
486C4—1942no—
487C4—1942no—
488C2—42no—
489C2—3882no—
490C2—3882no—
491C2—1944no—
492C2C4×D4×D9711yescharacteristic
493C2#2026120711yescharacteristic
494C2#2026120911yescharacteristic
495C2#2026129611yescharacteristic
496C2#2026130111yescharacteristic
497C2#2026131911yescharacteristic
498C2#2026132411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690780C2
hypercenter#388D97
hypocenterC97#2321
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×C22⋊C4×D97C2semidirect
4. #20260636C2semidirect
5. #20260635C2semidirect
6. #20260628C2semidirect
7. #20260627C2semidirect
8. #20260626C2semidirect
9. #20260613C2semidirect
10. #20260613C2semidirect
11. #20260611C2semidirect
12. #20260606C2semidirect
13. #20260590C2semidirect
14. #20260582C2semidirect
15. #388×C97C2semidirect
22. C22×C4×D97C22semidirect
23. #15690780C4semidirect
25. C2×C4⋊Dic97C22semidirect
26. C2×#13095490C22semidirect
37. C2×#13095488C22semidirect
38. C2×#13095488C22semidirect
39. C2×#13095488C22semidirect
41. C2×C4×Dic97C22semidirect
42. C2×C4×Dic97C22semidirect
43. #15690717C22semidirect
44. #15690717C22semidirect
45. #15690717C22semidirect
46. #15690717C22semidirect
47. C2×C4⋊C4×C97C22semidirect
51. C2×C42×C97C22semidirect
52. #95×C97C22semidirect
53. #95×C97C22semidirect
60. C23×D97C2×C4semidirect
69. C2×D388C2×C4semidirect
72. C2×C22⋊D97C2×C4semidirect
86. C2×C4×D97D4semidirect
100. #13095488C2×C4semidirect
101. #13095488C2×C4semidirect
114. C22⋊C4×C97C2×C4semidirect
250. C97#2321semidirect
259. #388D97semidirect
277. C2×C4⋊C4C2×D97semidirect
297. C2×C42C2×D97semidirect
302. #95C2×D97semidirect
303. #95C2×D97semidirect
362. 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×C22⋊C4×D972
#202606362
#202606352
#202606282
#202606272
#202606262
#202606132
#202606112
#202606062
#202605902
#202605822
#388×C972
#232197