Finite Group Databasebeta

← #27141754

Subgroups — #27141754

Order 12416.

312 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 95 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#41×Dic97C211yesmaximal, characteristic
2#20260758C211yesmaximal, characteristic, complemented
3C2×#15690737C211yesmaximal, characteristic
4C4×#13095490C211yesmaximal, characteristic, complemented
5#20260496C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7C2×#104×C97C211yesmaximal, characteristic
8C2×#13095495C2211yescharacteristic
9Q8×Dic97C411yescharacteristic, complemented
10Q8×Dic97C411yescharacteristic, complemented
11D4×Dic97—21no—
12D4×Dic97—21no—
13D4×Dic97C411yescharacteristic, complemented
14D4×Dic97C411yescharacteristic, complemented
15#15690769—21no—
16#15690769C2211yescharacteristic
17C2×#13095490—21no—
18C2×C4×Dic97—21no—
19C2×C4×Dic97C2211yescharacteristic
20C2×C4×Dic97C2211yescharacteristic
21#15690739—21no—
22#15690739—21no—
23#15690737—22no—
24#15690717—21no—
25C2×#41×C97C2211yescharacteristic
26C2×M16×C97C2211yescharacteristic
27C2×C42×C97C2211yescharacteristic
28#104×C97—22no—
29C22×Dic97—21no—
30C22×Dic97—21no—
31C22×Dic97—21no—
32C22×Dic97D411yescharacteristic
33C22×Dic97D411yescharacteristic
34#13095495—21no—
35#13095495—21no—
36C2×#11715895—21no—
37C4⋊Dic97—21no—
38C4⋊Dic97—21no—
39C4⋊Dic97C2×C411yescharacteristic
40C4⋊Dic97C2×C411yescharacteristic
41#13095490—24no—
42C22⋊Dic97—21no—
43C22⋊Dic97—21no—
44C22⋊Dic97—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C4×Dic97—21no—
48C4×Dic97—21no—
49C4×Dic97—21no—
50C4×Dic97C2×C411yescharacteristic
51C4×Dic97C2×C411yescharacteristic
52C4×Dic97D411yescharacteristic
53C4×Dic97D411yescharacteristic
54#41×C97—21no—
55#41×C97—21no—
56#41×C97D412yes—
57#41×C97Q812yes—
58C2×Q8×C97C2×C411yescharacteristic
59C2×D4×C97—21no—
60C2×D4×C97C2×C411yescharacteristic
61C22×C4×C97—21no—
62C22×C4×C97—21no—
63C22×C4×C97C2311yescharacteristic
64M16×C97—21no—
65M16×C97—21no—
66C2×C8×C97—21no—
67C42×C97—21no—
68C42×C97—21no—
69C23×C97—21no—
70C23×C97C2×D411yescharacteristic
71Q8×C97—21no—
72Q8×C97C4⋊C412yescomplemented
73D4×C97—22no—
74D4×C97—22no—
75D4×C97—21no—
76D4×C97C4⋊C412yescomplemented
77C2×C4×C97—21no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—41no—
82C2×C4×C97—41no—
83C2×C4×C97C22×C411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D411yescharacteristic
86C2×C4×C97C2×Q811yescharacteristic
87C2×C4×C97#4111yescharacteristic
88C2×C4×C97#4111yescharacteristic
89C2×Dic97—22no—
90C2×Dic97—22no—
91C2×Dic97—21no—
92C2×Dic97—21no—
93C2×Dic97—41no—
94C2×Dic97—42no—
95C2×Dic97—41no—
96C2×Dic97—41no—
97C2×Dic97—42no—
98C2×Dic97C22⋊C411yescharacteristic
99C2×Dic97C22⋊C411yescharacteristic
100C2×Dic97C22⋊C411yescharacteristic
101C2×Dic97C22⋊C411yescharacteristic
102C8×C97—41no—
103#11715895—41no—
104C22×C97—21no—
105C22×C97—21no—
106C22×C97—22no—
107C22×C97—41no—
108C22×C97—41no—
109C22×C97C2×C22⋊C411yescharacteristic
110C22×C97#12111yescharacteristic
111C22×C97C22⋊Q811yescharacteristic
112C4×C97—22no—
113C4×C97—42no—
114C4×C97C2×C4⋊C411yescharacteristic
115C4×C97#11711yescharacteristic
116C4×C97#12111yescharacteristic
117C4×C97C22⋊Q811yescharacteristic
118Dic97—21no—
119Dic97—21no—
120Dic97—21no—
121Dic97—42no—
122Dic97—41no—
123Dic97—41no—
124Dic97#10412yescomplemented
125C2×C97—21no—
126C2×C97—21no—
127C2×C97—42no—
128C2×C97#38011yescharacteristic
129C2×C97C2×#10411yescharacteristic
130C2×C97#42111yescharacteristic
131#1806—971nomaximal, Sylow
132C97#180611yescharacteristic, complemented, Sylow
133C4×#41—971no—
134#423—971no—
135C2×#104—971no—
136C2×#104D9711yescharacteristic
137C4×C4⋊C4—971no—
138#339—971no—
139#339—971no—
140C2×#41C2×D9711yescharacteristic
141C2×M16—971no—
142C2×M16C2×D9711yescharacteristic
143C4×Q8—971no—
144C4×Q8—971no—
145C4×D4—971no—
146C4×D4—971no—
147C4×D4—1941no—
148C4×D4—1941no—
149#117—971no—
150#117—1941no—
151C2×C4⋊C4—1941no—
152C2×C42—971no—
153C2×C42—971no—
154C2×C42—1941no—
155C2×C42C2×D9711yescharacteristic
156C4⋊C8—1941no—
157C4⋊C8—1941no—
158#104—22no—
159#104—1942no—
160#95—1941no—
161#41—21no—
162#41—21no—
163#41Dic19412yes—
164#41C22⋊D9712yes—
165C2×Q8C4×D9711yescharacteristic
166C2×D4—21no—
167C2×D4C4×D9711yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—971no—
171C22×C4—971no—
172C22×C4—1941no—
173C22×C4—1941no—
174C22×C4—1941no—
175C22×C4C22×D9711yescharacteristic
176M16—21no—
177M16—21no—
178M16—1941no—
179M16—1941no—
180C2×C8—21no—
181C2×C8—1941no—
182C4⋊C4—971no—
183C4⋊C4—971no—
184C4⋊C4—1941no—
185C4⋊C4—1941no—
186C4⋊C4—1944no—
187C22⋊C4—1941no—
188C22⋊C4—1941no—
189C22⋊C4—1941no—
190C42—21no—
191C42—21no—
192C42—971no—
193C42—971no—
194C42—971no—
195C42—971no—
196C42—1941no—
197C42—1941no—
198C42—1941no—
199C42—1941no—
200C42—1941no—
201C23—21no—
202C23D4×D9711yescharacteristic
203Q8—21no—
204Q8#1309549012yescomplemented
205D4—22no—
206D4—22no—
207D4—21no—
208D4#1309549012yescomplemented
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—971no—
216C2×C4—971no—
217C2×C4—971no—
218C2×C4—971no—
219C2×C4—3882no—
220C2×C4—3881no—
221C2×C4—3881no—
222C2×C4—3882no—
223C2×C4—3881no—
224C2×C4—1942no—
225C2×C4—1942no—
226C2×C4—1941no—
227C2×C4—1941no—
228C2×C4C2×C4×D9711yescharacteristic
229C2×C4D4×D9711yescharacteristic
230C2×C4C2×C22⋊D9711yescharacteristic
231C2×C4C2×Dic19411yescharacteristic
232C2×C4#1309551011yescharacteristic
233C2×C4#1309551011yescharacteristic
234C8—41no—
235C8—3881no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22C22⋊C4×D9711yescharacteristic
242C22#1569078211yescharacteristic
243C22#1569078711yescharacteristic
244C4—22no—
245C4—42no—
246C4—972no—
247C4—3882no—
248C4—3881no—
249C4—3881no—
250C4—1941no—
251C4—1941no—
252C4—1941no—
253C4C2×#1309549011yescharacteristic
254C4#1569076611yescharacteristic
255C4#1569078211yescharacteristic
256C4#1569078711yescharacteristic
257C2—21no—
258C2—21no—
259C2—42no—
260C2#2026059311yescharacteristic
261C2#104×D9711yescharacteristic
262C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#104×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenterC2×#104D97
hypocenterC97#1806
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #20260758C2semidirect
4. C4×#13095490C2semidirect
5. #20260496C2semidirect
6. #20260495C2semidirect
9. Q8×Dic97C4semidirect
10. Q8×Dic97C4semidirect
13. D4×Dic97C4semidirect
14. D4×Dic97C4semidirect
72. Q8×C97C4⋊C4semidirect
76. D4×C97C4⋊C4semidirect
124. Dic97#104semidirect
132. C97#1806semidirect
204. Q8#13095490semidirect
208. D4#13095490semidirect

Maximal subgroups

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

subgroupindex
#41×Dic972
#202607582
C2×#156907372
C4×#130954902
#202604962
#202604952
C2×#104×C972
#180697