Finite Group Databasebeta

← #27145035

Subgroups — #27145035

Order 12416.

432 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C97⋊#401C211yesmaximal, characteristic, complemented
2#20260677C211yesmaximal, characteristic, complemented
3#20260675C211yesmaximal, characteristic, complemented
4#20260664C211yesmaximal, characteristic
5#20260662C211yesmaximal, characteristic
6#20260661C211yesmaximal, characteristic
7#20260657C211yesmaximal, characteristic
8C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
10#20260586C211yesmaximal, characteristic, complemented
11#20260582C211yesmaximal, characteristic
12#20260582C211yesmaximal, characteristic
13#20260582C211yesmaximal, characteristic
14#20260580C211yesmaximal, characteristic, complemented
15#400×C97C211yesmaximal, characteristic, complemented
16C22×C4×D97C2211yescharacteristic
17C2×C4⋊Dic97C2211yescharacteristic
18C2×C4⋊Dic97C2211yescharacteristic
19C2×#13095490C2211yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×#13095488C2211yescharacteristic
22C2×#13095488C2211yescharacteristic
23C2×#13095488C2211yescharacteristic
24C2×#13095488C2211yescharacteristic
25C2×#13095488C2211yescharacteristic
26C2×#13095488C2211yescharacteristic
27C2×C4×Dic97C2211yescharacteristic
28C2×C4×Dic97C2211yescharacteristic
29C2×C4×Dic97C2211yescharacteristic
30C2×C4×Dic97C2211yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33#15690717C2211yescharacteristic
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
46#95×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
59C2×C4×D97—22no—
60C2×C4×D97—22no—
61C2×C4×D97—22no—
62C4⋊Dic97—22no—
63C4⋊Dic97—22no—
64#13095490—22no—
65#13095490—22no—
66#13095488—22no—
67#13095488—22no—
68#13095488—22no—
69#13095488—22no—
70#13095488—22no—
71#13095488—22no—
72C4×Dic97—22no—
73C4×Dic97—22no—
74C4×Dic97—22no—
75C4×Dic97—22no—
76C4×Dic97—22no—
77C4×Dic97—22no—
78C22×C4×C97C2311yescharacteristic
79C22×C4×C97C2311yescharacteristic
80C22×C4×C97C2311yescharacteristic
81C22×C4×C97C2311yescharacteristic
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C22×C4×C97C2311yescharacteristic
85C4⋊C4×C97—22no—
86C4⋊C4×C97—22no—
87C23×C97C2411yescharacteristic
88C22×D97—21no—
89C22×D97—21no—
90C22×D97—21no—
91C22×D97—21no—
92C22×D97—21no—
93C22×D97—21no—
94C22×D97#4112yes—
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—21no—
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—
137C4×D97—44no—
138C22×C97C2×#4111yescharacteristic
139C22×C97C2×#4111yescharacteristic
140C22×C97C2×#4111yescharacteristic
141C22×C97C2×#4111yescharacteristic
142C22×C97C2×#4111yescharacteristic
143C22×C97C2×#4111yescharacteristic
144C22×C97C2×#4111yescharacteristic
145C2×D97—22no—
146C2×D97—22no—
147C2×D97—22no—
148C2×D97—41no—
149C2×D97—41no—
150C2×D97—41no—
151C2×D97—41no—
152C4×C97—22no—
153C4×C97—22no—
154C4×C97—41no—
155C4×C97—41no—
156C4×C97—41no—
157C4×C97—41no—
158C4×C97—41no—
159Dic97—22no—
160Dic97—41no—
161Dic97—41no—
162Dic97—41no—
163Dic97—41no—
164Dic97—41no—
165Dic97#12614yes—
166C2×C97C2×#12611yescharacteristic
167C2×C97#52911yescharacteristic
168C2×C97#52911yescharacteristic
169C2×C97#52911yescharacteristic
170C2×C97#52911yescharacteristic
171C2×C97#52911yescharacteristic
172C2×C97#52911yescharacteristic
173D97—42no—
174#2401—971nomaximal, Sylow
175C97#240111yescharacteristic, complemented, Sylow
176#401—971no—
177#400—971no—
178#400—971no—
179#400D9711yescharacteristic, complemented
180#397—971no—
181#397—971no—
182#397—971no—
183#396—971no—
184C4×C4⋊C4—971no—
185C4×C4⋊C4—971no—
186C4×C4⋊C4—971no—
187C4×C22⋊C4—971no—
188C4×C22⋊C4—971no—
189C4×C22⋊C4—971no—
190#376—971no—
191C23×C4—971no—
192C2×C4⋊C4—971no—
193C2×C4⋊C4—971no—
194C2×C4⋊C4—971no—
195C2×C4⋊C4—971no—
196C2×C4⋊C4C2×D9711yescharacteristic
197C2×C4⋊C4C2×D9711yescharacteristic
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—
210#95—971no—
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
225#95C2×D9711yescharacteristic
226C24—971no—
227C22×C4—971no—
228C22×C4—971no—
229C22×C4—971no—
230C22×C4—971no—
231C22×C4—971no—
232C22×C4—971no—
233C22×C4—971no—
234C22×C4—1942no—
235C22×C4—1942no—
236C22×C4—1942no—
237C22×C4C22×D9711yescharacteristic
238C22×C4C22×D9711yescharacteristic
239C22×C4C22×D9711yescharacteristic
240C22×C4C22×D9711yescharacteristic
241C22×C4C22×D9711yescharacteristic
242C22×C4C22×D9711yescharacteristic
243C22×C4C22×D9711yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—1942no—
247C4⋊C4—1942no—
248C4⋊C4—1942no—
249C4⋊C4—1942no—
250C22⋊C4—1942no—
251C22⋊C4—1942no—
252C22⋊C4—1942no—
253C22⋊C4—1942no—
254C22⋊C4—1942no—
255C22⋊C4—1942no—
256C42—1942no—
257C42—1942no—
258C42—1942no—
259C42—1942no—
260C42—1942no—
261C42—1942no—
262C23—972no—
263C23—1941no—
264C23—1941no—
265C23—1941no—
266C23—1941no—
267C23—1941no—
268C23—1941no—
269C23C23×D9711yescharacteristic
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—21no—
288C2×C4—21no—
289C2×C4—972no—
290C2×C4—972no—
291C2×C4—972no—
292C2×C4—972no—
293C2×C4—3884no—
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—1941no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4#1309551012yes—
312C2×C4#1309551212yes—
313C22—3881no—
314C22—3881no—
315C22—3881no—
316C22—3881no—
317C22—1942no—
318C22—1942no—
319C22—1942no—
320C22#41×D9711yescharacteristic
321C22#41×D9711yescharacteristic
322C22#41×D9711yescharacteristic
323C22#41×D9711yescharacteristic
324C22#41×D9711yescharacteristic
325C22C2×#1309551011yescharacteristic
326C22C2×#1309551211yescharacteristic
327C4—22no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—974no—
335C4—3881no—
336C4—3881no—
337C4—3881no—
338C4—3881no—
339C4—3881no—
340C4—1942no—
341C2—3882no—
342C2#126×D9711yescharacteristic
343C2#2026126211yescharacteristic
344C2#2026126411yescharacteristic
345C2#2026126811yescharacteristic
346C2#2026127011yescharacteristic
347C2#2026127011yescharacteristic
348C2#2026127111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#400×C97C2
prime radicalC23×D97C23
hypercenter#400D97
hypocenterC97#2401
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C97⋊#401C2semidirect
2. #20260677C2semidirect
3. #20260675C2semidirect
8. C4⋊C4×Dic97C2semidirect
9. C4⋊C4×Dic97C2semidirect
10. #20260586C2semidirect
14. #20260580C2semidirect
15. #400×C97C2semidirect
175. C97#2401semidirect
179. #400D97semidirect

Maximal subgroups

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

subgroupindex
C97⋊#4012
#202606772
#202606752
#202606642
#202606622
#202606612
#202606572
C4⋊C4×Dic972
#202605862
#202605822
#202605802
#400×C972
#240197