Finite Group Databasebeta

← #27144749

Subgroups — #27144749

Order 12416.

788 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690783C211yesmaximal, characteristic
2C2×C4×D388C211yesmaximal, characteristic, complemented
3C2×#15690770C211yesmaximal, characteristic, complemented
4C2×#15690770C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
6#20260635C211yesmaximal, characteristic, complemented
7#20260632C211yesmaximal, characteristic, complemented
8#20260627C211yesmaximal, characteristic, complemented
9#20260613C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11#20260610C211yesmaximal, characteristic, complemented
12#20260606C211yesmaximal, characteristic, complemented
13C4×#13095490C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C23×Dic97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21C22×C4×D97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C22×C4×D97C2211yescharacteristic
24#15690783C418yescomplemented
25C4×D388—24no—
26#15690770—24no—
27#15690770—24no—
28C2×C4⋊Dic97C2211yescharacteristic
29C2×C4⋊Dic97C2211yescharacteristic
30C2×#13095490C2211yescharacteristic
31C2×#13095490C2211yescharacteristic
32C2×#13095490C2211yescharacteristic
33C2×#13095490C2211yescharacteristic
34C4⋊C4×D97—24no—
35C2×C22⋊Dic97C2211yescharacteristic
36C2×#13095488C2211yescharacteristic
37C2×#13095488C2211yescharacteristic
38C2×#13095488C2211yescharacteristic
39C2×#13095488C2211yescharacteristic
40C2×#13095488C2211yescharacteristic
41C2×#13095488C2211yescharacteristic
42C2×C4×Dic97C2211yescharacteristic
43C2×C4×Dic97C2211yescharacteristic
44C2×C4×Dic97C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47#15690717C2211yescharacteristic
48#15690717C2211yescharacteristic
49C2×C4⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97C2211yescharacteristic
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C42×C97C2211yescharacteristic
54#95×C97C2211yescharacteristic
55#95×C97C2211yescharacteristic
56C23×D97C2311yescharacteristic
57C23×D97C2311yescharacteristic
58C2×D388—22no—
59C2×D388—22no—
60C2×D388C2×C414yescomplemented
61C2×C22⋊D97—22no—
62C2×C22⋊D97—22no—
63C2×C22⋊D97—22no—
64C2×C22⋊D97—22no—
65C2×C22⋊D97C2×C414yescomplemented
66C2×C22⋊D97C2×C414yescomplemented
67C22×Dic97—22no—
68C22×Dic97—22no—
69C22×Dic97—41no—
70C22×Dic97C2311yescharacteristic
71C22×Dic97C2311yescharacteristic
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2311yescharacteristic
74C22×Dic97C2311yescharacteristic
75C22×Dic97C2311yescharacteristic
76C2×C4×D97—22no—
77C2×C4×D97—22no—
78C2×C4×D97—22no—
79C2×C4×D97—22no—
80C2×C4×D97—22no—
81C2×C4×D97—22no—
82C2×C4×D97—22no—
83C2×C4×D97—22no—
84C2×C4×D97—41no—
85C2×C4×D97—41no—
86C2×C4×D97—41no—
87C2×C4×D97C2×C414yescomplemented
88C4⋊Dic97—22no—
89C4⋊Dic97—22no—
90#13095490—22no—
91#13095490—22no—
92#13095490C2×C414yescomplemented
93#13095490D414yescomplemented
94C22⋊Dic97—41no—
95#13095488—22no—
96#13095488—22no—
97#13095488—22no—
98#13095488—41no—
99#13095488C2×C414yescomplemented
100#13095488D414yescomplemented
101C4×Dic97—22no—
102C4×Dic97—22no—
103C4×Dic97—22no—
104C24×C97C2311yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C22×C4×C97C2311yescharacteristic
107C22×C4×C97C2311yescharacteristic
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111C4⋊C4×C97—22no—
112C22⋊C4×C97—22no—
113C22⋊C4×C97—22no—
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×D97C22×C412yes—
135C22×D97C22×C412yes—
136C22×D97C2×D412yes—
137C22×D97#4112yes—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97—21no—
152C2×C4×C97C22×C412yes—
153C2×C4×C97C22×C412yes—
154C2×C4×C97C2×D412yes—
155C2×C4×C97C2×D412yes—
156C22⋊D97—28no—
157C22⋊D97—28no—
158C2×Dic97—21no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—21no—
163C2×Dic97—21no—
164C2×Dic97—21no—
165C2×Dic97—21no—
166C2×Dic97—21no—
167C2×Dic97—21no—
168C2×Dic97—21no—
169C2×Dic97—21no—
170C2×Dic97—44no—
171C2×Dic97C22×C412yes—
172C2×Dic97C22×C412yes—
173C2×Dic97C2×D412yes—
174C2×Dic97C2×D412yes—
175C2×Dic97C2×D412yes—
176C2×Dic97#4112yes—
177D388—28no—
178C4×D97—28no—
179C4×D97—44no—
180C4×D97—44no—
181C4×D97—44no—
182C22×C97—22no—
183C22×C97—22no—
184C22×C97—22no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97C23×C411yescharacteristic
190C22×C97C22×D411yescharacteristic
191C22×C97C22×D411yescharacteristic
192C22×C97C2×#4111yescharacteristic
193C22×C97C2×#4111yescharacteristic
194C22×C97H(5,𝔽2)11yescharacteristic
195C22×C972−1+411yescharacteristic
196C2×D97—22no—
197C2×D97—22no—
198C2×D97—22no—
199C2×D97—22no—
200C2×D97—22no—
201C2×D97—22no—
202C2×D97—22no—
203C2×D97—22no—
204C2×D97—22no—
205C2×D97—41no—
206C2×D97—41no—
207C2×D97—41no—
208C2×D97—41no—
209C2×D97C4×D414yescomplemented
210C4×C97—22no—
211C4×C97—22no—
212C4×C97—22no—
213C4×C97—22no—
214C4×C97—41no—
215C4×C97—41no—
216Dic97—22no—
217Dic97—22no—
218Dic97—22no—
219Dic97—41no—
220Dic97—41no—
221Dic97C4×D414yescomplemented
222C2×C97—42no—
223C2×C97C2×C4×D411yescharacteristic
224C2×C97C2×C4×D411yescharacteristic
225C2×C97#52011yescharacteristic
226C2×C97D4211yescharacteristic
227C2×C97#54711yescharacteristic
228C2×C97#54811yescharacteristic
229C2×C97#55311yescharacteristic
230D97—24no—
231D97—42no—
232#2361—971nomaximal, Sylow
233C97#236111yescharacteristic, complemented, Sylow
234C2×#121—971no—
235C2×C4×D4—971no—
236C2×C4×D4—971no—
237C2×C4×D4—971no—
238C22×C4⋊C4—971no—
239#390—971no—
240#389—971no—
241#388—971no—
242#388D9711yescharacteristic, complemented
243#386—971no—
244#386—971no—
245#385—971no—
246#385—971no—
247C4×C4⋊C4—971no—
248C4×C22⋊C4—971no—
249C22×D4—971no—
250C22×D4—971no—
251C22×D4—971no—
252C23×C4—971no—
253C23×C4—971no—
254C23×C4—971no—
255C23×C4—971no—
256C23×C4—971no—
257#121—978no—
258C4×D4—1944no—
259C4×D4—1944no—
260C4×D4—1944no—
261C2×C4⋊C4—971no—
262C2×C4⋊C4—971no—
263C2×C4⋊C4—971no—
264C2×C4⋊C4—971no—
265C2×C4⋊C4—971no—
266C2×C4⋊C4—971no—
267C2×C4⋊C4—1944no—
268C2×C4⋊C4C2×D9711yescharacteristic
269C2×C22⋊C4—971no—
270C2×C22⋊C4—971no—
271C2×C22⋊C4—971no—
272C2×C22⋊C4—971no—
273C2×C22⋊C4—971no—
274C2×C22⋊C4—971no—
275C2×C22⋊C4—971no—
276C2×C22⋊C4C2×D9711yescharacteristic
277C2×C22⋊C4C2×D9711yescharacteristic
278C2×C22⋊C4C2×D9711yescharacteristic
279C2×C42—971no—
280C2×C42—971no—
281C2×C42—971no—
282C2×C42C2×D9711yescharacteristic
283#95—971no—
284#95—971no—
285#95—971no—
286#95—971no—
287#95C2×D9711yescharacteristic
288#95C2×D9711yescharacteristic
289C24—971no—
290C24—971no—
291C24C22×D9711yescharacteristic
292C2×D4—974no—
293C2×D4—974no—
294C2×D4—974no—
295C2×D4—1942no—
296C2×D4—1942no—
297C2×D4—1942no—
298C2×D4—1942no—
299C2×D4—1942no—
300C2×D4—1942no—
301C22×C4—971no—
302C22×C4—971no—
303C22×C4—971no—
304C22×C4—971no—
305C22×C4—974no—
306C22×C4—971no—
307C22×C4—971no—
308C22×C4—3881no—
309C22×C4—3881no—
310C22×C4—3881no—
311C22×C4—3881no—
312C22×C4—1942no—
313C22×C4—1942no—
314C22×C4—1942no—
315C22×C4—1942no—
316C22×C4—1942no—
317C22×C4—1942no—
318C22×C4—1942no—
319C22×C4—1942no—
320C22×C4—1942no—
321C22×C4—1942no—
322C22×C4C22×D9711yescharacteristic
323C22×C4C22×D9711yescharacteristic
324C22×C4C22×D9711yescharacteristic
325C22×C4C22×D9711yescharacteristic
326C22×C4C22×D9711yescharacteristic
327C22×C4C22×D9711yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—974no—
330C4⋊C4—974no—
331C4⋊C4—1942no—
332C4⋊C4—1942no—
333C4⋊C4—1942no—
334C4⋊C4—1942no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—974no—
338C22⋊C4—974no—
339C22⋊C4—3881no—
340C22⋊C4—3881no—
341C22⋊C4—1942no—
342C22⋊C4—1942no—
343C22⋊C4—1942no—
344C22⋊C4C4×D9714yescomplemented
345C42—22no—
346C42—1942no—
347C42—1942no—
348C42—1942no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—972no—
356C23—972no—
357C23—972no—
358C23—972no—
359C23—1941no—
360C23—1941no—
361C23—1941no—
362C23—1941no—
363C23—1941no—
364C23—1941no—
365C23—1941no—
366C23—1941no—
367C23—1941no—
368C23—1941no—
369C23C2×C4×D9712yes—
370C23C23×D9711yescharacteristic
371D4—1948no—
372D4—1948no—
373D4—1948no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—972no—
389C2×C4—972no—
390C2×C4—972no—
391C2×C4—972no—
392C2×C4—972no—
393C2×C4—972no—
394C2×C4—3884no—
395C2×C4—3884no—
396C2×C4—3884no—
397C2×C4—3884no—
398C2×C4—1948no—
399C2×C4—1941no—
400C2×C4—1941no—
401C2×C4—1941no—
402C2×C4—1941no—
403C2×C4—1941no—
404C2×C4—1941no—
405C2×C4—1941no—
406C2×C4—1941no—
407C2×C4—1941no—
408C2×C4—1941no—
409C2×C4—1941no—
410C2×C4—1941no—
411C2×C4C2×C4×D9712yes—
412C2×C4C2×C4×D9712yes—
413C2×C4D4×D9712yes—
414C2×C4D4×D9712yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—974no—
423C22—3881no—
424C22—3881no—
425C22—3881no—
426C22—3881no—
427C22—1942no—
428C22—1942no—
429C22—1942no—
430C22—1942no—
431C22—1942no—
432C22—1942no—
433C22—1942no—
434C22—1942no—
435C22—1942no—
436C22C22×C4×D9711yescharacteristic
437C22C2×D4×D9711yescharacteristic
438C22C2×D4×D9711yescharacteristic
439C22#41×D9711yescharacteristic
440C22#41×D9711yescharacteristic
441C22#1569085611yescharacteristic
442C22#1569085711yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—41no—
448C4—41no—
449C4—974no—
450C4—3881no—
451C4—3881no—
452C4—1942no—
453C4—1942no—
454C4—1942no—
455C2—42no—
456C2—3882no—
457C2—1944no—
458C2C4×D4×D9711yescharacteristic
459C2C4×D4×D9711yescharacteristic
460C2#2026121511yescharacteristic
461C2#2026136911yescharacteristic
462C2#2026138311yescharacteristic
463C2#2026139311yescharacteristic
464C2#2026143611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690783C2
hypercenter#388D97
hypocenterC97#2361
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×#15690770C2semidirect
4. C2×#15690770C2semidirect
5. C2×C4⋊C4×D97C2semidirect
6. #20260635C2semidirect
7. #20260632C2semidirect
8. #20260627C2semidirect
9. #20260613C2semidirect
10. #20260611C2semidirect
11. #20260610C2semidirect
12. #20260606C2semidirect
13. C4×#13095490C2semidirect
14. #20260582C2semidirect
15. #388×C97C2semidirect
24. #15690783C4semidirect
60. C2×D388C2×C4semidirect
65. C2×C22⋊D97C2×C4semidirect
66. C2×C22⋊D97C2×C4semidirect
87. C2×C4×D97C2×C4semidirect
92. #13095490C2×C4semidirect
93. #13095490D4semidirect
99. #13095488C2×C4semidirect
100. #13095488D4semidirect
114. C22⋊C4×C97C2×C4semidirect
209. C2×D97C4×D4semidirect
221. Dic97C4×D4semidirect
233. C97#2361semidirect
242. #388D97semidirect
344. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907832
C2×C4×D3882
C2×#156907702
C2×C4⋊C4×D972
#202606352
#202606322
#202606272
#202606132
#202606112
#202606102
#202606062
C4×#130954902
#202605822
#388×C972
#236197