Finite Group Databasebeta

← #27144894

Subgroups — #27144894

Order 12416.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690782C211yesmaximal, characteristic
2#20260638C211yesmaximal, characteristic, complemented
3#20260635C211yesmaximal, characteristic, complemented
4#20260624C211yesmaximal, characteristic, complemented
5#20260623C211yesmaximal, characteristic, complemented
6#20260622C211yesmaximal, characteristic, complemented
7#20260617C211yesmaximal, characteristic, complemented
8#20260614C211yesmaximal, characteristic, complemented
9#20260611C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11#20260609C211yesmaximal, characteristic, complemented
12#20260608C211yesmaximal, characteristic, complemented
13#20260596C211yesmaximal, characteristic, complemented
14#20260593C211yesmaximal, characteristic, complemented
15#386×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C23×Dic97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20#15690782C418yescomplemented
21C2×C4⋊Dic97C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45C22×D4×C97C2211yescharacteristic
46C23×C4×C97C2211yescharacteristic
47C2×C22⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52C23×D97C2311yescharacteristic
53C2×C22⋊D97—41no—
54C2×C22⋊D97—41no—
55C2×C22⋊D97—41no—
56C2×C22⋊D97—41no—
57C2×C22⋊D97C2×C414yes—
58C2×C22⋊D97C2×C414yescomplemented
59C22×Dic97—41no—
60C22×Dic97—41no—
61C22×Dic97C2×C414yescomplemented
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C2×C4×D97—41no—
70C2×C4×D97—41no—
71C2×C4×D97—41no—
72C4⋊Dic97—41no—
73#13095490—41no—
74#13095490—41no—
75#13095490—41no—
76#13095490C2×C414yescomplemented
77C22⋊Dic97—41no—
78C22⋊Dic97—41no—
79C22⋊Dic97—41no—
80C22⋊Dic97—41no—
81C22⋊Dic97C2×C414yescomplemented
82#13095488—41no—
83#13095488—41no—
84#13095488—41no—
85#13095488C2×C414yescomplemented
86C4×Dic97—41no—
87C4×Dic97—41no—
88C24×C97C2311yescharacteristic
89C24×C97C2311yescharacteristic
90C2×D4×C97—41no—
91C2×D4×C97—41no—
92C2×D4×C97C2×C414yescomplemented
93C22×C4×C97—41no—
94C22×C4×C97—41no—
95C22×C4×C97—41no—
96C22×C4×C97C2311yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C22⋊C4×C97—41no—
102C22⋊C4×C97—41no—
103C22⋊C4×C97—41no—
104C23×C97—21no—
105C23×C97—21no—
106C23×C97—21no—
107C23×C97—21no—
108C23×C97—21no—
109C23×C97—21no—
110C23×C97—21no—
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97—21no—
115C23×C97—21no—
116C23×C97C22×C412yes—
117C23×C97C22×C412yes—
118C23×C97C2411yescharacteristic
119C22×D97—21no—
120C22×D97—21no—
121C22×D97—21no—
122C22×D97—21no—
123C22×D97—21no—
124C22×D97—21no—
125C22×D97C22×C412yes—
126D4×C97—44no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—44no—
142C2×C4×C97C22×C412yes—
143C22⋊D97—44no—
144C22⋊D97—44no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—21no—
149C2×Dic97—21no—
150C2×Dic97—21no—
151C2×Dic97—21no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—21no—
157C2×Dic97—21no—
158C2×Dic97—21no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—21no—
163C2×Dic97—44no—
164C2×Dic97C22×C412yes—
165C2×Dic97C22×C412yes—
166C2×Dic97C22×C412yes—
167C4×D97—44no—
168C22×C97—22no—
169C22×C97—22no—
170C22×C97—22no—
171C22×C97—22no—
172C22×C97—22no—
173C22×C97—22no—
174C22×C97—41no—
175C22×C97—41no—
176C22×C97—41no—
177C22×C97—41no—
178C22×C97—41no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97—41no—
182C22×C97C23×C411yescharacteristic
183C22×C97H(5,𝔽2)11yescharacteristic
184C22×C97H(5,𝔽2)11yescharacteristic
185C22×C97H(5,𝔽2)11yescharacteristic
186C22×C97H(5,𝔽2)11yescharacteristic
187C22×C97H(5,𝔽2)11yescharacteristic
188C22×C972−1+411yescharacteristic
189C2×D97—22no—
190C2×D97—22no—
191C2×D97—22no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—41no—
196C4×C97—22no—
197C4×C97—41no—
198C4×C97—41no—
199C4×C97—41no—
200C4×C97—41no—
201Dic97—22no—
202Dic97—22no—
203Dic97—22no—
204Dic97—41no—
205Dic97—41no—
206Dic97—41no—
207Dic97—41no—
208C2×C97—42no—
209C2×C97—42no—
210C2×C97#51811yescharacteristic
211C2×C97#51811yescharacteristic
212C2×C97#52011yescharacteristic
213C2×C97#56011yescharacteristic
214C2×C97#56011yescharacteristic
215C2×C97#56211yescharacteristic
216C2×C97#56211yescharacteristic
217D97—42no—
218#2377—971nomaximal, Sylow
219C97#237711yescharacteristic, complemented, Sylow
220C2×#121—971no—
221#390—971no—
222#390—971no—
223#388—971no—
224#388—971no—
225#387—971no—
226#387—971no—
227#386—971no—
228#386—971no—
229#386—971no—
230#386D9711yescharacteristic, complemented
231#385—971no—
232#385—971no—
233#381—971no—
234#380—971no—
235C22×D4—971no—
236C22×D4—971no—
237C22×D4C2×D9711yescharacteristic
238C23×C4—971no—
239C23×C4—971no—
240C23×C4C2×D9711yescharacteristic
241#121—978no—
242C2×C4⋊C4—971no—
243C2×C4⋊C4—971no—
244C2×C4⋊C4—971no—
245C2×C4⋊C4—971no—
246C2×C4⋊C4—971no—
247C2×C22⋊C4—971no—
248C2×C22⋊C4—971no—
249C2×C22⋊C4—971no—
250C2×C22⋊C4—971no—
251C2×C22⋊C4—971no—
252C2×C22⋊C4—971no—
253C2×C22⋊C4—971no—
254C2×C22⋊C4—971no—
255C2×C22⋊C4—971no—
256C2×C22⋊C4C2×D9711yescharacteristic
257C2×C22⋊C4C2×D9711yescharacteristic
258C2×C22⋊C4C2×D9711yescharacteristic
259C2×C42—971no—
260C2×C42—971no—
261#95—971no—
262#95—971no—
263#95—971no—
264#95—971no—
265#95—971no—
266#95—971no—
267#95—971no—
268#95—971no—
269#95C2×D9711yescharacteristic
270#95C2×D9711yescharacteristic
271C24—971no—
272C24C22×D9711yescharacteristic
273C24C22×D9711yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—974no—
277C2×D4—974no—
278C2×D4—3881no—
279C2×D4—3881no—
280C2×D4—3881no—
281C2×D4—3881no—
282C2×D4C4×D9714yescomplemented
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—974no—
287C22×C4—971no—
288C22×C4—971no—
289C22×C4—971no—
290C22×C4—971no—
291C22×C4—971no—
292C22×C4—971no—
293C22×C4—971no—
294C22×C4—3881no—
295C22×C4—3881no—
296C22×C4—3881no—
297C22×C4—3881no—
298C22×C4—3881no—
299C22×C4C22×D9711yescharacteristic
300C22×C4C22×D9711yescharacteristic
301C22×C4C22×D9711yescharacteristic
302C22×C4C22×D9711yescharacteristic
303C22×C4C22×D9711yescharacteristic
304C4⋊C4—974no—
305C4⋊C4—3881no—
306C4⋊C4—3881no—
307C4⋊C4—3881no—
308C4⋊C4—3881no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—974no—
313C22⋊C4—974no—
314C22⋊C4—3881no—
315C22⋊C4—3881no—
316C22⋊C4—3881no—
317C22⋊C4—3881no—
318C22⋊C4—3881no—
319C22⋊C4—3881no—
320C22⋊C4—3881no—
321C42—3881no—
322C42—3881no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—972no—
336C23—1941no—
337C23—1941no—
338C23—1941no—
339C23—1941no—
340C23—1941no—
341C23—1941no—
342C23C2×C4×D9712yes—
343C23C2×C4×D9712yes—
344C23C23×D9711yescharacteristic
345D4—44no—
346D4—3884no—
347D4—3884no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—44no—
363C2×C4—972no—
364C2×C4—972no—
365C2×C4—972no—
366C2×C4—3884no—
367C2×C4—3884no—
368C2×C4—1941no—
369C2×C4—1941no—
370C2×C4—1941no—
371C2×C4—1941no—
372C2×C4—1941no—
373C2×C4—1941no—
374C2×C4—1941no—
375C2×C4—1941no—
376C2×C4—1941no—
377C2×C4—1941no—
378C2×C4—1941no—
379C2×C4—1941no—
380C2×C4—1941no—
381C2×C4—1941no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4—1941no—
385C2×C4—1941no—
386C2×C4C2×C4×D9712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—3881no—
402C22—3881no—
403C22—3881no—
404C22—3881no—
405C22—1942no—
406C22—1942no—
407C22—1942no—
408C22C22×C4×D9711yescharacteristic
409C22#1569085511yescharacteristic
410C22#1569085511yescharacteristic
411C22#1569085511yescharacteristic
412C22#1569085511yescharacteristic
413C22#1569085611yescharacteristic
414C22#1569085711yescharacteristic
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—3881no—
421C4—3881no—
422C4—3881no—
423C4—3881no—
424C4—1942no—
425C4—1942no—
426C4—1942no—
427C2—42no—
428C2—42no—
429C2—3882no—
430C2#2026120711yescharacteristic
431C2#2026120811yescharacteristic
432C2#2026121411yescharacteristic
433C2#2026147311yescharacteristic
434C2#2026147411yescharacteristic
435C2#2026148011yescharacteristic
436C2#2026148211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#386×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690782C2
hypercenter#386D97
hypocenterC97#2377
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #20260638C2semidirect
3. #20260635C2semidirect
4. #20260624C2semidirect
5. #20260623C2semidirect
6. #20260622C2semidirect
7. #20260617C2semidirect
8. #20260614C2semidirect
9. #20260611C2semidirect
10. #20260611C2semidirect
11. #20260609C2semidirect
12. #20260608C2semidirect
13. #20260596C2semidirect
14. #20260593C2semidirect
15. #386×C97C2semidirect
20. #15690782C4semidirect
58. C2×C22⋊D97C2×C4semidirect
61. C22×Dic97C2×C4semidirect
76. #13095490C2×C4semidirect
81. C22⋊Dic97C2×C4semidirect
85. #13095488C2×C4semidirect
92. C2×D4×C97C2×C4semidirect
219. C97#2377semidirect
230. #386D97semidirect
282. C2×D4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907822
#202606382
#202606352
#202606242
#202606232
#202606222
#202606172
#202606142
#202606112
#202606092
#202606082
#202605962
#202605932
#386×C972
#237797