Finite Group Databasebeta

← #27152134

Subgroups — #27152134

Order 12416.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261281C211yesmaximal, characteristic
2C2×#15690801C211yesmaximal, characteristic, complemented
3#20261102C212yesmaximal, complemented
4#20261098C212yesmaximal, complemented
5#20261091C212yesmaximal, complemented
6#20261088C212yesmaximal, complemented
7#20260737C211yesmaximal, characteristic, complemented
8#20260736C211yesmaximal, characteristic, complemented
9#20260734C211yesmaximal, characteristic, complemented
10M16×Dic97C211yesmaximal, characteristic, complemented
11#417×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C2×#13095495C2211yescharacteristic
14#15690808C2211yescharacteristic
15#15690806C2211yescharacteristic
16#15690801—21no—
17#15690801—21no—
18#15690801C2212yes—
19#15690801C2212yes—
20#15690799C2211yescharacteristic
21#15690797C2211yescharacteristic
22#15690782—21no—
23#15690782—21no—
24C4×C22⋊D97—21no—
25C4×C22⋊D97—21no—
26C2×C4⋊Dic97—21no—
27C2×C4⋊Dic97C2211yescharacteristic
28C2×#13095490—21no—
29C2×C4×Dic97C2211yescharacteristic
30#15690735C2212yes—
31#15690734C2212yes—
32#15690733C2212yes—
33#15690732C2212yes—
34#15690731C2212yes—
35#15690730C2212yes—
36#15690720C2212yes—
37C8×Dic97C2212yes—
38C2×#41×C97C2211yescharacteristic
39C2×M16×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#103×C97C2212yes—
42#102×C97C2212yes—
43#13095511—22no—
44#13095511—22no—
45C2×Dic194C2311yescharacteristic
46C2×D388C2311yescharacteristic
47C2×C22⋊D97—21no—
48C2×C22⋊D97—21no—
49C2×C22⋊D97—21no—
50C22×Dic97—21no—
51C22×Dic97D411yescharacteristic
52C22×Dic97D411yescharacteristic
53C2×C4×D97—21no—
54#13095495—22no—
55C2×#11715895C2312yes—
56C4⋊Dic97—21no—
57C4⋊Dic97—22no—
58C4⋊Dic97—22no—
59C4⋊Dic97C2312yes—
60#13095490—22no—
61#13095490—22no—
62#13095490—21no—
63#13095490—21no—
64C22⋊Dic97—21no—
65C22⋊Dic97—21no—
66#13095488—21no—
67#13095488—21no—
68C4×Dic97D411yescharacteristic
69C4×Dic97D411yescharacteristic
70C4×Dic97C2311yescharacteristic
71C4×Dic97C2311yescharacteristic
72#41×C97—22no—
73#41×C97—22no—
74C2×Q8×C97C2311yescharacteristic
75C2×D4×C97—21no—
76C2×D4×C97C2311yescharacteristic
77C22×C4×C97—21no—
78C22×C4×C97—21no—
79C22×C4×C97C2311yescharacteristic
80M16×C97—22no—
81C2×C8×C97C2312yes—
82C4⋊C4×C97—21no—
83C4⋊C4×C97C2312yes—
84C23×C97—21no—
85C23×C97C2×D411yescharacteristic
86C22×D97—21no—
87Q8×C97—21no—
88Q8×C97—21no—
89D4×C97—21no—
90D4×C97—21no—
91D4×C97—42no—
92C2×C4×C97—22no—
93C2×C4×C97—21no—
94C2×C4×C97—42no—
95C2×C4×C97—42no—
96C2×C4×C97C2×D411yescharacteristic
97C2×C4×C97#4112yes—
98C2×C4×C97#4112yes—
99C2×C4×C97C2411yescharacteristic
100C22⋊D97—42no—
101C22⋊D97—42no—
102C22⋊D97—42no—
103C2×Dic97—21no—
104C2×Dic97—22no—
105C2×Dic97—41no—
106C2×Dic97—41no—
107C2×Dic97—42no—
108C2×Dic97C2×D411yescharacteristic
109C2×Dic97C2×D411yescharacteristic
110C2×Dic97C2×D411yescharacteristic
111C2×Dic97C2×D411yescharacteristic
112D388—21no—
113D388—21no—
114C4×D97—42no—
115Dic194—21no—
116Dic194—21no—
117C8×C97—22no—
118#11715895—22no—
119C22×C97—21no—
120C22×C97—21no—
121C22×C97—41no—
122C22×C97—41no—
123C22×C97—41no—
124C22×C97#12412yes—
125C22×C97C22×D411yescharacteristic
126C2×D97—41no—
127C2×D97—41no—
128C2×D97—41no—
129C4×C97—41no—
130C4×C97—42no—
131C4×C97#12412yes—
132C4×C97C2×#4111yescharacteristic
133C4×C97C2×#4111yescharacteristic
134Dic97—21no—
135Dic97—21no—
136Dic97—21no—
137Dic97—21no—
138Dic97—41no—
139Dic97—42no—
140C2×C97—22no—
141C2×C97—81no—
142C2×C97C2×#12411yescharacteristic
143C2×C97#57511yescharacteristic
144C2×C97#57511yescharacteristic
145D97—81no—
146#3142—971nomaximal, Sylow
147C97#314211yescharacteristic, complemented, Sylow
148#532—971no—
149C2×#125—971no—
150#491—972no—
151#490—972no—
152#488—972no—
153#488—972no—
154#417—971no—
155#417—971no—
156#417—971no—
157#417D9711yescharacteristic, complemented
158C4×M16—971no—
159C2×#41—971no—
160C2×#41C2×D9711yescharacteristic
161C2×M16—971no—
162C2×M16C2×D9711yescharacteristic
163C4⋊Q8—971no—
164#127—971no—
165#125—972no—
166#125—972no—
167#125—1941no—
168#125—1941no—
169#124—971no—
170#124—971no—
171#121—1941no—
172#121—1941no—
173C4×D4—1941no—
174C4×D4—1941no—
175C2×C4⋊C4—971no—
176C2×C4⋊C4—1941no—
177C2×C4⋊C4—1941no—
178C2×C4⋊C4C2×D9711yescharacteristic
179C2×C42—971no—
180#103—972no—
181#103—972no—
182#103—972no—
183#103C2×D9712yes—
184#102—972no—
185#102—972no—
186#102—972no—
187#102C2×D9712yes—
188#97—972no—
189C4×C8—972no—
190#41—22no—
191#41—22no—
192#41—1942no—
193#41—1942no—
194C2×Q8—971no—
195C2×Q8C22×D9711yescharacteristic
196C2×D4—21no—
197C2×D4—971no—
198C2×D4—1941no—
199C2×D4—1941no—
200C2×D4—1941no—
201C2×D4C22×D9711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—971no—
205C22×C4—971no—
206C22×C4—1941no—
207C22×C4—1941no—
208C22×C4C22×D9711yescharacteristic
209M16—22no—
210M16—1942no—
211C2×C8—972no—
212C2×C8C22×D9712yes—
213C4⋊C4—21no—
214C4⋊C4—972no—
215C4⋊C4—1942no—
216C4⋊C4—1942no—
217C4⋊C4—1941no—
218C4⋊C4—1941no—
219C4⋊C4—1942no—
220C4⋊C4—1942no—
221C4⋊C4—1941no—
222C4⋊C4C22×D9712yes—
223C22⋊C4—1941no—
224C22⋊C4—1941no—
225C22⋊C4—1941no—
226C22⋊C4—1941no—
227C42—971no—
228C42—971no—
229C42—971no—
230C42—971no—
231C23—21no—
232C23—1941no—
233C23D4×D9711yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—1941no—
237Q8—1941no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—3882no—
242D4—3882no—
243D4—3882no—
244D4—1941no—
245D4—1941no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—971no—
251C2×C4—971no—
252C2×C4—971no—
253C2×C4—971no—
254C2×C4—3882no—
255C2×C4—3882no—
256C2×C4—3881no—
257C2×C4—3881no—
258C2×C4—1941no—
259C2×C4—1942no—
260C2×C4D4×D9711yescharacteristic
261C2×C4#1309551012yes—
262C2×C4#1309551112yes—
263C2×C4C23×D9711yescharacteristic
264C8—22no—
265C8—1942no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—3881no—
272C22—3881no—
273C22—3881no—
274C22#1569079612yes—
275C22C2×D4×D9711yescharacteristic
276C4—41no—
277C4—42no—
278C4—3881no—
279C4—3882no—
280C4—1941no—
281C4—1941no—
282C4—1941no—
283C4—1941no—
284C4#1569079612yes—
285C4C2×#1309551011yescharacteristic
286C4C2×#1309551111yescharacteristic
287C2—22no—
288C2—81no—
289C2—7761no—
290C2C2×#1569079611yescharacteristic
291C2#2026153111yescharacteristic
292C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#417×C97C2
socleC22×C97C22×D4
prime radical#20261281C2
hypercenter#417D97
hypocenterC97#3142
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#15690801C2semidirect
3. #20261102C2semidirect
4. #20261098C2semidirect
5. #20261091C2semidirect
6. #20261088C2semidirect
7. #20260737C2semidirect
8. #20260736C2semidirect
9. #20260734C2semidirect
10. M16×Dic97C2semidirect
11. #417×C97C2semidirect
147. C97#3142semidirect
157. #417D97semidirect

Maximal subgroups

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

subgroupindex
#202612812
C2×#156908012
#202611022
#202610982
#202610912
#202610882
#202607372
#202607362
#202607342
M16×Dic972
#417×C972
#314297