Finite Group Databasebeta

← #27140267

Subgroups — #27140267

Order 12416.

274 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C22⋊Q8×D97C211yesmaximal, characteristic
2#20260697C211yesmaximal, characteristic
3#20260597C211yesmaximal, characteristic
4#20260473C211yesmaximal, characteristic, complemented
5#20260472C211yesmaximal, characteristic, complemented
6#20260471C211yesmaximal, characteristic, complemented
7#328×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97C411yescharacteristic, complemented
9C22×C4×D97C2211yescharacteristic
10C2×#13095494—21no—
11#15690790C411yescharacteristic, complemented
12#15690789C2211yescharacteristic
13#15690788—21no—
14#15690787—21no—
15#15690786C411yescharacteristic, complemented
16C4⋊C4×D97—21no—
17C4⋊C4×D97C411yescharacteristic, complemented
18C2×#13095488—21no—
19C22⋊C4×D97—21no—
20C22⋊#11715895C2211yescharacteristic
21#15690722—21no—
22#15690717—21no—
23#15690717C2211yescharacteristic
24C22⋊Q8×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27C23×D97D411yescharacteristic
28C2×Dic194—21no—
29C2×Dic194C2×C411yescharacteristic
30Q8×D97—42no—
31C22×Dic97—21no—
32C22×Dic97D411yescharacteristic
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97—42no—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97C2×C411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C2×C4×D97D411yescharacteristic
40#13095494—42no—
41C2×#11715895—21no—
42C4⋊Dic97—21no—
43C4⋊Dic97C2×C411yescharacteristic
44#13095490—21no—
45#13095490—21no—
46#13095490—21no—
47C22⋊Dic97—21no—
48#13095488—21no—
49#13095488—21no—
50#13095488—42no—
51C2×Q8×C97C2×C411yescharacteristic
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54C2×C8×C97—21no—
55C4⋊C4×C97—21no—
56C4⋊C4×C97C2×C411yescharacteristic
57C22⋊C4×C97—21no—
58C23×C97C2×D411yescharacteristic
59C22×D97—21no—
60C22×D97—21no—
61C22×D97—22no—
62C22×D97—41no—
63C22×D97C22⋊C411yescharacteristic
64C22×D97C22⋊C411yescharacteristic
65Q8×C97—41no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97C22×C411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97C22⋊C411yescharacteristic
81C2×Dic97C22⋊C411yescharacteristic
82C4×D97—22no—
83C4×D97—22no—
84C4×D97—41no—
85C4×D97—41no—
86C4×D97—42no—
87C4×D97—42no—
88Dic194—42no—
89Dic194—41no—
90C8×C97—41no—
91#11715895—41no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97C2×C22⋊C411yescharacteristic
95C22×C97#13711yescharacteristic
96C22×C97#13711yescharacteristic
97C2×D97—21no—
98C2×D97—21no—
99C2×D97—21no—
100C2×D97—21no—
101C2×D97—21no—
102C2×D97—42no—
103C2×D97—42no—
104C2×D97#9912yes—
105C4×C97—21no—
106C4×C97—21no—
107C4×C97—41no—
108C4×C97—41no—
109C4×C97—81no—
110Dic97—21no—
111Dic97—21no—
112Dic97—41no—
113Dic97—41no—
114Dic97—81no—
115C2×C97—21no—
116C2×C97—21no—
117C2×C97C2×#9911yescharacteristic
118C2×C97#41911yescharacteristic
119C2×C97#42111yescharacteristic
120D97—22no—
121D97—41no—
122#1511—971nomaximal, Sylow
123C97#151111yescharacteristic, complemented, Sylow
124C2×C22⋊Q8—971no—
125#407—971no—
126#381—971no—
127#328—971no—
128#328—971no—
129#328—971no—
130#328D9711yescharacteristic, complemented
131C22×Q8—971no—
132C23×C4—971no—
133C2×M16—1941no—
134C22⋊Q8—971no—
135C22⋊Q8—971no—
136C22⋊Q8—971no—
137C22⋊Q8—1941no—
138C22⋊Q8—1941no—
139C22⋊Q8C2×D9711yescharacteristic
140C2×C4⋊C4—971no—
141C2×C4⋊C4—1941no—
142C2×C22⋊C4—1941no—
143C2×C22⋊C4—1941no—
144C22⋊C8—971no—
145C22⋊C8—1941no—
146C22⋊C8C2×D9711yescharacteristic
147#95—971no—
148#95—1941no—
149#95C2×D9711yescharacteristic
150C24—971no—
151C2×Q8—971no—
152C2×Q8—3882no—
153C2×Q8—1941no—
154C2×Q8C4×D9711yescharacteristic
155C22×C4—21no—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—971no—
160C22×C4—971no—
161C22×C4—3882no—
162C22×C4—1941no—
163C22×C4—1941no—
164C22×C4—1941no—
165C22×C4C22×D9711yescharacteristic
166M16—3882no—
167C2×C8—21no—
168C2×C8—1941no—
169C4⋊C4—21no—
170C4⋊C4—971no—
171C4⋊C4—1941no—
172C4⋊C4—1941no—
173C4⋊C4—1941no—
174C4⋊C4—1941no—
175C4⋊C4C4×D9711yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—3882no—
178C22⋊C4—1941no—
179C22⋊C4—1941no—
180C22⋊C4—1941no—
181C23—971no—
182C23—971no—
183C23—3881no—
184C23—1942no—
185C23—1941no—
186C23—1941no—
187C23D4×D9711yescharacteristic
188Q8—41no—
189Q8—3881no—
190Q8—3882no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—971no—
198C2×C4—971no—
199C2×C4—3881no—
200C2×C4—3881no—
201C2×C4—3882no—
202C2×C4—3882no—
203C2×C4—3881no—
204C2×C4—3881no—
205C2×C4—3881no—
206C2×C4—3881no—
207C2×C4—1942no—
208C2×C4—1942no—
209C2×C4—1941no—
210C2×C4—1941no—
211C2×C4C2×C4×D9711yescharacteristic
212C2×C4D4×D9711yescharacteristic
213C8—41no—
214C8—3881no—
215C22—21no—
216C22—21no—
217C22—972no—
218C22—3882no—
219C22—3882no—
220C22—1941no—
221C22—1941no—
222C22—1941no—
223C22—1941no—
224C22—1941no—
225C22C22⋊C4×D9711yescharacteristic
226C22#1569083911yescharacteristic
227C22#1569084211yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—7761no—
234C4—3881no—
235C4—3881no—
236C4—1941no—
237C4—1941no—
238C2—21no—
239C2—21no—
240C2—3881no—
241C2—1942no—
242C2#99×D9711yescharacteristic
243C2#2026074211yescharacteristic
244C2#2026075011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#328×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC22×C4×C97C23
hypercenter#328D97
hypocenterC97#1511
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #20260473C2semidirect
5. #20260472C2semidirect
6. #20260471C2semidirect
7. #328×C97C2semidirect
8. C2×Q8×D97C4semidirect
11. #15690790C4semidirect
15. #15690786C4semidirect
17. C4⋊C4×D97C4semidirect
123. C97#1511semidirect
130. #328D97semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D972
#202606972
#202605972
#202604732
#202604722
#202604712
#328×C972
#151197