Finite Group Databasebeta

← #27144626

Subgroups — #27144626

Order 12416.

504 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690766C211yesmaximal, characteristic
2#20260632C211yesmaximal, characteristic, complemented
3#20260630C211yesmaximal, characteristic, complemented
4#20260629C211yesmaximal, characteristic
5#20260628C211yesmaximal, characteristic
6#20260623C211yesmaximal, characteristic
7#20260602C211yesmaximal, characteristic, complemented
8#20260602C211yesmaximal, characteristic, complemented
9#20260593C211yesmaximal, characteristic
10C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
11C4×#13095490C211yesmaximal, characteristic, complemented
12C4×#13095490C211yesmaximal, characteristic, complemented
13#20260586C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic97C211yesmaximal, characteristic
15#388×C97C211yesmaximal, characteristic
16C23×Dic97C2211yescharacteristic
17#15690766C418yescomplemented
18C2×C4⋊Dic97C2211yescharacteristic
19C2×#13095490C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×C22⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C4×Dic97C2211yescharacteristic
29C2×C4×Dic97C2211yescharacteristic
30C2×C4×Dic97C2211yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45C2×C4⋊C4×C97C2211yescharacteristic
46C2×C22⋊C4×C97C2211yescharacteristic
47C2×C22⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52C22×Dic97—22no—
53C22×Dic97—22no—
54C22×Dic97C2×C414yes—
55C22×Dic97C2311yescharacteristic
56C22×Dic97C2311yescharacteristic
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C4⋊Dic97—22no—
64#13095490—22no—
65#13095490—22no—
66#13095490—22no—
67#13095490—22no—
68#13095490C2×C414yes—
69#13095490C2×C414yes—
70C22⋊Dic97—22no—
71C22⋊Dic97—41no—
72C22⋊Dic97C2×C414yes—
73C4×Dic97—22no—
74C4×Dic97—22no—
75C4×Dic97—41no—
76C4×Dic97—41no—
77C4×Dic97—41no—
78C4×Dic97C2×C414yes—
79C4×Dic97C2×C414yes—
80C24×C97C2311yescharacteristic
81C22×C4×C97C2311yescharacteristic
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C22×C4×C97C2311yescharacteristic
85C22×C4×C97C2311yescharacteristic
86C22×C4×C97C2311yescharacteristic
87C4⋊C4×C97—22no—
88C22⋊C4×C97—22no—
89C22⋊C4×C97—41no—
90C22⋊C4×C97C2×C414yes—
91C42×C97—41no—
92C23×C97—21no—
93C23×C97—21no—
94C23×C97—21no—
95C23×C97—21no—
96C23×C97—21no—
97C23×C97—21no—
98C23×C97C22×C412yes—
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×C97C22×C412yes—
116C2×C4×C97C22×C412yes—
117C2×C4×C97#4112yes—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—28no—
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×Dic97C22×C412yes—
137C2×Dic97C22×C412yes—
138C2×Dic97C22×C412yes—
139C2×Dic97C22×C412yes—
140C2×Dic97#4112yes—
141C2×Dic97#4112yes—
142C2×Dic97#4112yes—
143C22×C97—22no—
144C22×C97—22no—
145C22×C97—22no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97—41no—
149C22×C97—41no—
150C22×C97C23×C411yescharacteristic
151C22×C97C2×#4111yescharacteristic
152C22×C97C2×#4111yescharacteristic
153C22×C97C2×#4111yescharacteristic
154C22×C97C2×#4111yescharacteristic
155C22×C97H(5,𝔽2)11yescharacteristic
156C22×C972−1+411yescharacteristic
157C4×C97—22no—
158C4×C97—22no—
159C4×C97—22no—
160C4×C97—41no—
161C4×C97—41no—
162C4×C97—41no—
163Dic97—22no—
164Dic97—22no—
165Dic97—22no—
166Dic97—22no—
167Dic97—41no—
168Dic97—41no—
169Dic97—41no—
170Dic97#11714yes—
171C2×C97—42no—
172C2×C97C2×#11711yescharacteristic
173C2×C97C4×#4111yescharacteristic
174C2×C97#52011yescharacteristic
175C2×C97#55311yescharacteristic
176C2×C97#55311yescharacteristic
177C2×C97#55611yescharacteristic
178C2×C97#55611yescharacteristic
179#2347—971nomaximal, Sylow
180C97#234711yescharacteristic, complemented, Sylow
181C2×#117—971no—
182#389—971no—
183#389—971no—
184#388—971no—
185#388—971no—
186#388—971no—
187#388D9711yescharacteristic
188#383—971no—
189#383—971no—
190#380—971no—
191C4×C4⋊C4—971no—
192C4×C4⋊C4—971no—
193C4×C4⋊C4—971no—
194C4×C4⋊C4—971no—
195C4×C22⋊C4—971no—
196C23×C4—971no—
197#117—978no—
198C2×C4⋊C4—971no—
199C2×C4⋊C4—971no—
200C2×C4⋊C4—971no—
201C2×C4⋊C4—971no—
202C2×C4⋊C4—971no—
203C2×C4⋊C4—971no—
204C2×C4⋊C4—971no—
205C2×C4⋊C4C2×D9711yescharacteristic
206C2×C22⋊C4—971no—
207C2×C22⋊C4—971no—
208C2×C22⋊C4—971no—
209C2×C22⋊C4C2×D9711yescharacteristic
210C2×C22⋊C4C2×D9711yescharacteristic
211C2×C22⋊C4C2×D9711yescharacteristic
212C2×C42—971no—
213C2×C42—971no—
214C2×C42—971no—
215C2×C42—971no—
216C2×C42—971no—
217C2×C42—971no—
218C2×C42—971no—
219C2×C42C2×D9711yescharacteristic
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#95C2×D9711yescharacteristic
231#95C2×D9711yescharacteristic
232C24C22×D9711yescharacteristic
233C22×C4—971no—
234C22×C4—974no—
235C22×C4—971no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4—971no—
241C22×C4—971no—
242C22×C4—1942no—
243C22×C4—1942no—
244C22×C4C22×D9711yescharacteristic
245C22×C4C22×D9711yescharacteristic
246C22×C4C22×D9711yescharacteristic
247C22×C4C22×D9711yescharacteristic
248C22×C4C22×D9711yescharacteristic
249C22×C4C22×D9711yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—974no—
252C4⋊C4—974no—
253C4⋊C4—1942no—
254C4⋊C4—1942no—
255C4⋊C4—1942no—
256C4⋊C4—1942no—
257C4⋊C4—1942no—
258C22⋊C4—22no—
259C22⋊C4—41no—
260C22⋊C4—974no—
261C22⋊C4—3881no—
262C22⋊C4—1942no—
263C22⋊C4C4×D9714yes—
264C42—41no—
265C42—974no—
266C42—974no—
267C42—3881no—
268C42—3881no—
269C42—3881no—
270C42—1942no—
271C42—1942no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D9712yes—
279C23C23×D9711yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
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—972no—
296C2×C4—972no—
297C2×C4—972no—
298C2×C4—972no—
299C2×C4—972no—
300C2×C4—972no—
301C2×C4—972no—
302C2×C4—1948no—
303C2×C4—1941no—
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—1941no—
314C2×C4—1941no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4C2×C4×D9712yes—
321C2×C4C2×C4×D9712yes—
322C2×C4#1309551012yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D9711yescharacteristic
331C22#41×D9711yescharacteristic
332C22#41×D9711yescharacteristic
333C22#41×D9711yescharacteristic
334C22C2×#1309551011yescharacteristic
335C22#1569085511yescharacteristic
336C22#1569085711yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—974no—
344C4—3881no—
345C4—3881no—
346C4—3881no—
347C4—1942no—
348C4—1942no—
349C4—1942no—
350C4—1942no—
351C2—42no—
352C2#117×D9711yescharacteristic
353C2C4×#1309551011yescharacteristic
354C2#2026121411yescharacteristic
355C2#2026142811yescharacteristic
356C2#2026143011yescharacteristic
357C2#2026145211yescharacteristic
358C2#2026145411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC24×C97C23
hypercenter#388D97
hypocenterC97#2347
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20260632C2semidirect
3. #20260630C2semidirect
7. #20260602C2semidirect
8. #20260602C2semidirect
10. C4⋊C4×Dic97C2semidirect
11. C4×#13095490C2semidirect
12. C4×#13095490C2semidirect
13. #20260586C2semidirect
17. #15690766C4semidirect
180. C97#2347semidirect

Maximal subgroups

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

subgroupindex
C2×#156907662
#202606322
#202606302
#202606292
#202606282
#202606232
#202606022
#202605932
C4⋊C4×Dic972
C4×#130954902
#202605862
C22⋊C4×Dic972
#388×C972
#234797