Finite Group Databasebeta

← #27144590

Subgroups — #27144590

Order 12416.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690766C211yesmaximal, characteristic
2#20260619C211yesmaximal, characteristic, complemented
3#20260619C211yesmaximal, characteristic, complemented
4#20260617C211yesmaximal, characteristic, complemented
5#20260617C211yesmaximal, characteristic, complemented
6#20260608C211yesmaximal, characteristic
7#20260607C211yesmaximal, characteristic
8#20260602C211yesmaximal, characteristic, complemented
9#20260599C211yesmaximal, characteristic, complemented
10#20260596C211yesmaximal, characteristic
11C4×#13095490C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic97C211yesmaximal, characteristic
13#20260581C211yesmaximal, characteristic, complemented
14C2×#15690717C211yesmaximal, characteristic
15C4×C22⋊C4×C97C211yesmaximal, characteristic
16C23×Dic97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18#15690766C418yescomplemented
19C2×#13095490C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×C22⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C4×Dic97C2211yescharacteristic
28C2×C4×Dic97C2211yescharacteristic
29C2×C4×Dic97C2211yescharacteristic
30C2×C4×Dic97C2211yescharacteristic
31#15690717C2211yescharacteristic
32#15690717C2211yescharacteristic
33#15690717C2211yescharacteristic
34#15690717C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45C23×C4×C97C2211yescharacteristic
46C2×C22⋊C4×C97C2211yescharacteristic
47C2×C22⋊C4×C97C2211yescharacteristic
48C2×C42×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52C22×Dic97—22no—
53C22×Dic97—22no—
54C22×Dic97—22no—
55C22×Dic97—22no—
56C22×Dic97—41no—
57C22×Dic97C2×C414yes—
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66#13095490—22no—
67#13095490—22no—
68#13095490—41no—
69#13095490—41no—
70#13095490C2×C414yescomplemented
71#13095490C2×C414yescomplemented
72C22⋊Dic97—22no—
73C22⋊Dic97C2×C414yes—
74C4×Dic97—22no—
75C4×Dic97—41no—
76C4×Dic97C2×C414yescomplemented
77C4×Dic97C2×C414yescomplemented
78C24×C97C2311yescharacteristic
79C22×C4×C97—22no—
80C22×C4×C97—22no—
81C22×C4×C97—41no—
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C22×C4×C97C2311yescharacteristic
85C22×C4×C97C2311yescharacteristic
86C22×C4×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C22⋊C4×C97—22no—
89C22⋊C4×C97C2×C414yes—
90C42×C97—22no—
91C42×C97—41no—
92C23×C97—21no—
93C23×C97—21no—
94C23×C97—21no—
95C23×C97—21no—
96C23×C97C22×C412yes—
97C23×C97#4112yes—
98C23×C97#4112yes—
99C23×C97C2411yescharacteristic
100C2×C4×C97—21no—
101C2×C4×C97—21no—
102C2×C4×C97—21no—
103C2×C4×C97—21no—
104C2×C4×C97—21no—
105C2×C4×C97—21no—
106C2×C4×C97—21no—
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—44no—
116C2×C4×C97C22×C412yes—
117C2×C4×C97C22×C412yes—
118C2×C4×C97#4112yes—
119C2×Dic97—28no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—44no—
140C2×Dic97C22×C412yes—
141C2×Dic97C22×C412yes—
142C2×Dic97C22×C412yes—
143C2×Dic97C22×C412yes—
144C2×Dic97#4112yes—
145C22×C97—22no—
146C22×C97—22no—
147C22×C97—22no—
148C22×C97—22no—
149C22×C97—22no—
150C22×C97—22no—
151C22×C97#11714yes—
152C22×C97C23×C411yescharacteristic
153C22×C97C2×#4111yescharacteristic
154C22×C97C2×#4111yescharacteristic
155C22×C97C2×#4111yescharacteristic
156C22×C97C2×#4111yescharacteristic
157C22×C97H(5,𝔽2)11yescharacteristic
158C22×C972−1+411yescharacteristic
159C4×C97—22no—
160C4×C97—22no—
161C4×C97—22no—
162C4×C97—41no—
163C4×C97—41no—
164C4×C97—41no—
165Dic97—22no—
166Dic97—22no—
167Dic97—22no—
168Dic97—22no—
169Dic97—22no—
170Dic97—41no—
171Dic97—41no—
172Dic97—41no—
173C2×C97—24no—
174C2×C97C2×#11711yescharacteristic
175C2×C97C4×#4111yescharacteristic
176C2×C97#52011yescharacteristic
177C2×C97#55111yescharacteristic
178C2×C97#55111yescharacteristic
179C2×C97#55211yescharacteristic
180C2×C97#55211yescharacteristic
181#2343—971nomaximal, Sylow
182C97#234311yescharacteristic, complemented, Sylow
183C2×#117—971no—
184#387—971no—
185#387—971no—
186#387—971no—
187#387—971no—
188#385—971no—
189#385—971no—
190#383—971no—
191#382—971no—
192#381—971no—
193C4×C4⋊C4—971no—
194C4×C22⋊C4—971no—
195C4×C22⋊C4D9711yescharacteristic
196#376—971no—
197C2×#95—971no—
198C23×C4—971no—
199C23×C4—971no—
200C23×C4C2×D9711yescharacteristic
201#117—978no—
202C2×C4⋊C4—971no—
203C2×C4⋊C4—971no—
204C2×C4⋊C4—971no—
205C2×C4⋊C4—971no—
206C2×C4⋊C4—971no—
207C2×C4⋊C4—971no—
208C2×C22⋊C4—971no—
209C2×C22⋊C4—971no—
210C2×C22⋊C4C2×D9711yescharacteristic
211C2×C22⋊C4C2×D9711yescharacteristic
212C2×C42—971no—
213C2×C42—971no—
214C2×C42—971no—
215C2×C42—971no—
216C2×C42C2×D9711yescharacteristic
217C2×C42C2×D9711yescharacteristic
218#95—971no—
219#95—971no—
220#95—971no—
221#95—971no—
222#95—971no—
223#95—971no—
224#95—971no—
225#95—971no—
226#95—971no—
227#95—971no—
228#95—971no—
229#95—971no—
230#95—971no—
231#95—971no—
232#95C2×D9711yescharacteristic
233#95C2×D9711yescharacteristic
234C24C22×D9711yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—974no—
239C22×C4—971no—
240C22×C4—971no—
241C22×C4—971no—
242C22×C4—971no—
243C22×C4—971no—
244C22×C4—971no—
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—3881no—
248C22×C4—1942no—
249C22×C4—1942no—
250C22×C4—1942no—
251C22×C4—1942no—
252C22×C4C22×D9711yescharacteristic
253C22×C4C22×D9711yescharacteristic
254C22×C4C22×D9711yescharacteristic
255C22×C4C22×D9711yescharacteristic
256C22×C4C22×D9711yescharacteristic
257C22×C4C22×D9711yescharacteristic
258C4⋊C4—974no—
259C4⋊C4—974no—
260C4⋊C4—3881no—
261C4⋊C4—3881no—
262C4⋊C4—1942no—
263C4⋊C4—1942no—
264C22⋊C4—22no—
265C22⋊C4—974no—
266C22⋊C4—1942no—
267C22⋊C4C4×D9714yes—
268C42—22no—
269C42—41no—
270C42—974no—
271C42—974no—
272C42—3881no—
273C42—1942no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D9712yes—
279C23#1309551112yes—
280C23#1309551112yes—
281C23C23×D9711yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—972no—
299C2×C4—972no—
300C2×C4—972no—
301C2×C4—972no—
302C2×C4—972no—
303C2×C4—3884no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4—1941no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4—1941no—
312C2×C4—1941no—
313C2×C4—1948no—
314C2×C4—1941no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4C2×C4×D9712yes—
325C2×C4C2×C4×D9712yes—
326C2×C4#1309551012yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#1569076714yes—
334C22C22×C4×D9711yescharacteristic
335C22#41×D9711yescharacteristic
336C22C2×#1309551011yescharacteristic
337C22C2×#1309551111yescharacteristic
338C22C2×#1309551111yescharacteristic
339C22#1569085511yescharacteristic
340C22#1569085711yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3881no—
348C4—3881no—
349C4—3881no—
350C4—1942no—
351C4—1942no—
352C4—1942no—
353C4—1942no—
354C4—1942no—
355C2—24no—
356C2C2×#1569076711yescharacteristic
357C2C4×#1309551011yescharacteristic
358C2#2026121411yescharacteristic
359C2#2026141111yescharacteristic
360C2#2026141411yescharacteristic
361C2#2026141811yescharacteristic
362C2#2026142511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC4×C22⋊C4×C97C2
socle, cosocleC23×C97C24
prime radicalC24×C97C23
hypercenterC4×C22⋊C4D97
hypocenterC97#2343
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20260619C2semidirect
3. #20260619C2semidirect
4. #20260617C2semidirect
5. #20260617C2semidirect
8. #20260602C2semidirect
9. #20260599C2semidirect
11. C4×#13095490C2semidirect
13. #20260581C2semidirect
18. #15690766C4semidirect
70. #13095490C2×C4semidirect
71. #13095490C2×C4semidirect
76. C4×Dic97C2×C4semidirect
77. C4×Dic97C2×C4semidirect
182. C97#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#156907662
#202606192
#202606172
#202606082
#202606072
#202606022
#202605992
#202605962
C4×#130954902
C22⋊C4×Dic972
#202605812
C2×#156907172
C4×C22⋊C4×C972
#234397