Finite Group Databasebeta

← #27150129

Subgroups — #27150129

Order 12416.

418 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20260627C211yesmaximal, characteristic
2#20260627C211yesmaximal, characteristic
3#20260626C211yesmaximal, characteristic
4#20260625C211yesmaximal, characteristic
5#20260625C211yesmaximal, characteristic
6#20260624C211yesmaximal, characteristic
7#20260624C211yesmaximal, characteristic
8#20260622C211yesmaximal, characteristic, complemented
9#20260621C211yesmaximal, characteristic, complemented
10#20260620C211yesmaximal, characteristic, complemented
11#20260618C211yesmaximal, characteristic, complemented
12#20260617C211yesmaximal, characteristic, complemented
13#20260616C211yesmaximal, characteristic, complemented
14C42×Dic97C211yesmaximal, characteristic, complemented
15#387×C97C211yesmaximal, characteristic, complemented
16C2×C4⋊Dic97C2211yescharacteristic
17C2×C4⋊Dic97C2211yescharacteristic
18C2×#13095490C2211yescharacteristic
19C2×#13095490C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095488C2211yescharacteristic
22C2×#13095488C2211yescharacteristic
23C2×#13095488C2211yescharacteristic
24C2×#13095488C2211yescharacteristic
25C2×#13095488C2211yescharacteristic
26C2×#13095488C2211yescharacteristic
27C2×#13095488C2211yescharacteristic
28C2×C4×Dic97C2211yescharacteristic
29C2×C4×Dic97C2211yescharacteristic
30C2×C4×Dic97C2211yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34#15690717C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44C2×C4⋊C4×C97C2211yescharacteristic
45C2×C4⋊C4×C97C2211yescharacteristic
46C2×C42×C97C2211yescharacteristic
47#95×C97C2211yescharacteristic
48#95×C97C2211yescharacteristic
49#95×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51C23×D97C2311yescharacteristic
52C22×Dic97C2311yescharacteristic
53C22×Dic97C2311yescharacteristic
54C22×Dic97C2311yescharacteristic
55C22×Dic97C2311yescharacteristic
56C22×Dic97C2311yescharacteristic
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C4⋊Dic97—22no—
60C4⋊Dic97—22no—
61#13095490—22no—
62#13095490—22no—
63#13095490—22no—
64#13095488—22no—
65#13095488—22no—
66#13095488—22no—
67#13095488—22no—
68#13095488—22no—
69#13095488—22no—
70#13095488—22no—
71C4×Dic97—22no—
72C4×Dic97—22no—
73C4×Dic97—22no—
74C4×Dic97—22no—
75C4×Dic97—22no—
76C4×Dic97—22no—
77C22×C4×C97C2311yescharacteristic
78C22×C4×C97C2311yescharacteristic
79C22×C4×C97C2311yescharacteristic
80C22×C4×C97C2311yescharacteristic
81C22×C4×C97C2311yescharacteristic
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C4⋊C4×C97—22no—
85C4⋊C4×C97—22no—
86C42×C97—22no—
87C23×C97C2411yescharacteristic
88C22×D97—21no—
89C22×D97—21no—
90C22×D97—21no—
91C22×D97—21no—
92C22×D97—21no—
93C22×D97—21no—
94C22×D97—21no—
95C2×C4×C97—21no—
96C2×C4×C97—21no—
97C2×C4×C97—21no—
98C2×C4×C97—21no—
99C2×C4×C97—21no—
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#4112yes—
114C2×C4×C97#4112yes—
115C2×C4×C97#4112yes—
116C2×Dic97—21no—
117C2×Dic97—21no—
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—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97#4112yes—
134C2×Dic97#4112yes—
135C2×Dic97#4112yes—
136C2×Dic97#4112yes—
137C22×C97C2×#4111yescharacteristic
138C22×C97C2×#4111yescharacteristic
139C22×C97C2×#4111yescharacteristic
140C22×C97C2×#4111yescharacteristic
141C22×C97C2×#4111yescharacteristic
142C22×C97C2×#4111yescharacteristic
143C22×C97C2×#4111yescharacteristic
144C2×D97—41no—
145C2×D97—41no—
146C2×D97—41no—
147C2×D97—41no—
148C2×D97—41no—
149C2×D97—41no—
150C2×D97—41no—
151C4×C97—22no—
152C4×C97—22no—
153C4×C97—22no—
154C4×C97—41no—
155C4×C97—41no—
156C4×C97—41no—
157C4×C97—41no—
158Dic97—22no—
159Dic97—22no—
160Dic97—22no—
161Dic97—22no—
162Dic97—41no—
163Dic97—41no—
164Dic97—41no—
165C2×C97#52911yescharacteristic
166C2×C97#52911yescharacteristic
167C2×C97#52911yescharacteristic
168C2×C97#52911yescharacteristic
169C2×C97#52911yescharacteristic
170C2×C97#52911yescharacteristic
171C2×C97#52911yescharacteristic
172D97—81no—
173#2866—971nomaximal, Sylow
174C97#286611yescharacteristic, complemented, Sylow
175#388—971no—
176#388—971no—
177#388—971no—
178#388—971no—
179#388—971no—
180#388—971no—
181#388—971no—
182#387—971no—
183#387—971no—
184#387—971no—
185#387—971no—
186#387—971no—
187#387—971no—
188#387D9711yescharacteristic, complemented
189C43—971no—
190C2×C4⋊C4—971no—
191C2×C4⋊C4—971no—
192C2×C4⋊C4—971no—
193C2×C4⋊C4—971no—
194C2×C4⋊C4—971no—
195C2×C4⋊C4C2×D9711yescharacteristic
196C2×C4⋊C4C2×D9711yescharacteristic
197C2×C22⋊C4—971no—
198C2×C22⋊C4—971no—
199C2×C22⋊C4—971no—
200C2×C22⋊C4—971no—
201C2×C22⋊C4—971no—
202C2×C22⋊C4—971no—
203C2×C22⋊C4—971no—
204C2×C42—971no—
205C2×C42—971no—
206C2×C42—971no—
207C2×C42—971no—
208C2×C42—971no—
209C2×C42—971no—
210C2×C42C2×D9711yescharacteristic
211#95—971no—
212#95—971no—
213#95—971no—
214#95—971no—
215#95—971no—
216#95—971no—
217#95—971no—
218#95—971no—
219#95—971no—
220#95—971no—
221#95C2×D9711yescharacteristic
222#95C2×D9711yescharacteristic
223#95C2×D9711yescharacteristic
224#95C2×D9711yescharacteristic
225C24—971no—
226C22×C4—971no—
227C22×C4—971no—
228C22×C4—971no—
229C22×C4—971no—
230C22×C4—971no—
231C22×C4—971no—
232C22×C4—971no—
233C22×C4C22×D9711yescharacteristic
234C22×C4C22×D9711yescharacteristic
235C22×C4C22×D9711yescharacteristic
236C22×C4C22×D9711yescharacteristic
237C22×C4C22×D9711yescharacteristic
238C22×C4C22×D9711yescharacteristic
239C22×C4C22×D9711yescharacteristic
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—1942no—
243C4⋊C4—1942no—
244C4⋊C4—1942no—
245C4⋊C4—1942no—
246C4⋊C4—1942no—
247C22⋊C4—1942no—
248C22⋊C4—1942no—
249C22⋊C4—1942no—
250C22⋊C4—1942no—
251C22⋊C4—1942no—
252C22⋊C4—1942no—
253C22⋊C4—1942no—
254C42—22no—
255C42—1942no—
256C42—1942no—
257C42—1942no—
258C42—1942no—
259C42—1942no—
260C42—1942no—
261C23—1941no—
262C23—1941no—
263C23—1941no—
264C23—1941no—
265C23—1941no—
266C23—1941no—
267C23—1941no—
268C23C23×D9711yescharacteristic
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—972no—
288C2×C4—972no—
289C2×C4—972no—
290C2×C4—972no—
291C2×C4—1941no—
292C2×C4—1941no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4#1309551012yes—
309C2×C4#1309551112yes—
310C2×C4#1309551212yes—
311C22—3881no—
312C22—3881no—
313C22—3881no—
314C22—3881no—
315C22—3881no—
316C22—3881no—
317C22—3881no—
318C22#41×D9711yescharacteristic
319C22#41×D9711yescharacteristic
320C22#41×D9711yescharacteristic
321C22#41×D9711yescharacteristic
322C22C2×#1309551011yescharacteristic
323C22C2×#1309551111yescharacteristic
324C22C2×#1309551211yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—3881no—
333C4—3881no—
334C4—3881no—
335C4—1942no—
336C4—1942no—
337C4—1942no—
338C4—1942no—
339C2—7761no—
340C2#2026126211yescharacteristic
341C2#2026126311yescharacteristic
342C2#2026126511yescharacteristic
343C2#2026126811yescharacteristic
344C2#2026127011yescharacteristic
345C2#2026127111yescharacteristic
346C2#2026127211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#387×C97C2
prime radicalC23×D97C23
hypercenter#387D97
hypocenterC97#2866
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
8. #20260622C2semidirect
9. #20260621C2semidirect
10. #20260620C2semidirect
11. #20260618C2semidirect
12. #20260617C2semidirect
13. #20260616C2semidirect
14. C42×Dic97C2semidirect
15. #387×C97C2semidirect
174. C97#2866semidirect
188. #387D97semidirect

Maximal subgroups

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

subgroupindex
#202606272
#202606262
#202606252
#202606242
#202606222
#202606212
#202606202
#202606182
#202606172
#202606162
C42×Dic972
#387×C972
#286697