Finite Group Databasebeta

← #27143726

Subgroups — #27143726

Order 12416.

258 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 45 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#20260924C211yesmaximal, characteristic, complemented
2#20260921C211yesmaximal, characteristic
3#20260912C211yesmaximal, characteristic, complemented
4#20260530C211yesmaximal, characteristic, complemented
5#20260525C211yesmaximal, characteristic, complemented
6#20260522C211yesmaximal, characteristic, complemented
7#551×C97C211yesmaximal, characteristic, complemented
8#15690855C2211yescharacteristic
9#15690841—21no—
10#15690793—21no—
11#15690779—21no—
12#15690738—21no—
13#15690727C2211yescharacteristic
14#15690725C2211yescharacteristic
15#15690725C2211yescharacteristic
16#15690724—21no—
17#15690724—21no—
18#126×C97—21no—
19#124×C97C2211yescharacteristic
20#123×C97—21no—
21#123×C97C2211yescharacteristic
22C22⋊Q8×C97—21no—
23#120×C97C2211yescharacteristic
24C4×D4×C97—21no—
25#117×C97—21no—
26C2×C22⋊C4×C97—21no—
27#13095511—21no—
28#13095510—41no—
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C2×C22⋊D97—41no—
32D4×D97—41no—
33#13095501—41no—
34#13095498—41no—
35#13095495—21no—
36#13095490—41no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39C22⋊Dic97—41no—
40#13095488—41no—
41C24×C97D411yescharacteristic
42C2×Q8×C97D411yescharacteristic, complemented
43C2×D4×C97—21no—
44C2×D4×C97C2311yescharacteristic
45C22×C4×C97—21no—
46C22×C4×C97—21no—
47C22×C4×C97D411yescharacteristic, complemented
48C4⋊C4×C97—21no—
49C4⋊C4×C97—21no—
50C4⋊C4×C97—21no—
51C4⋊C4×C97—21no—
52C22⋊C4×C97—21no—
53C22⋊C4×C97—21no—
54C22⋊C4×C97—21no—
55C22⋊C4×C97—21no—
56C22⋊C4×C97—21no—
57C22⋊C4×C97—21no—
58C22⋊C4×C97D411yescharacteristic
59C22⋊C4×C97D411yescharacteristic, complemented
60C42×C97—21no—
61C42×C97D411yescharacteristic, complemented
62C23×C97—21no—
63C23×C97—42no—
64C23×C97—41no—
65C23×C97C2×D411yescharacteristic
66C23×C97C2×D411yescharacteristic
67C22×D97—21no—
68C22×D97—21no—
69Q8×C97—41no—
70D4×C97—41no—
71D4×C97—42no—
72C2×C4×C97—21no—
73C2×C4×C97—21no—
74C2×C4×C97—21no—
75C2×C4×C97—21no—
76C2×C4×C97—21no—
77C2×C4×C97—42no—
78C2×C4×C97—41no—
79C2×C4×C97—81no—
80C2×C4×C97C2×D411yescharacteristic
81C22⋊D97—41no—
82C22⋊D97—41no—
83C22⋊D97—41no—
84C22⋊D97—82no—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—41no—
88C2×Dic97—41no—
89D388—21no—
90C4×D97—41no—
91Dic194—21no—
92#11715895—41no—
93C22×C97—22no—
94C22×C97—21no—
95C22×C97—21no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97—41no—
99C22×C97—42no—
100C22×C97—41no—
101C22×C97—81no—
102C22×C97#12011yescharacteristic
103C2×D97—41no—
104C2×D97—81no—
105C2×D97—81no—
106C4×C97—21no—
107C4×C97—41no—
108C4×C97—41no—
109C4×C97—41no—
110C4×C97—41no—
111C4×C97—41no—
112Dic97—41no—
113Dic97—81no—
114Dic97—81no—
115C2×C97—21no—
116C2×C97—41no—
117C2×C97—41no—
118C2×C97—42no—
119C2×C97B(4,𝔽2)11yescharacteristic
120D97—81no—
121#2197—971nomaximal, Sylow
122C97#219711yescharacteristic, complemented, Sylow
123#551D9711yescharacteristic, complemented
124#458—971no—
125B(4,𝔽2)—971no—
126#455—971no—
127#354—971no—
128#352—971no—
129AΣL(1,𝔽16)—971no—
130H(5,𝔽2)—971no—
131#137—1941no—
132#126—21no—
133#124C2×D9711yescharacteristic
134#123—21no—
135#123—1941no—
136#123C2×D9711yescharacteristic
137C22⋊Q8—21no—
138#120—1941no—
139#120C2×D9711yescharacteristic
140C4×D4—21no—
141#117—21no—
142C2×C22⋊C4—21no—
143#104—1941no—
144#100—971no—
145#99—971no—
146#99—971no—
147#99—1941no—
148#99—1941no—
149C24C22⋊D9711yescharacteristic
150#41—3881no—
151#41—1941no—
152C2×Q8C22⋊D9711yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—3881no—
155C2×D4—3881no—
156C2×D4—1941no—
157C2×D4—1941no—
158C2×D4C22×D9711yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C22⋊D9711yescharacteristic, complemented
162Q16—3881no—
163SD16—3881no—
164M16—1941no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—3881no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—3881no—
177C22⋊C4—3881no—
178C22⋊C4—1941no—
179C22⋊C4—1941no—
180C22⋊C4C22⋊D9711yescharacteristic
181C22⋊C4C22⋊D9711yescharacteristic, complemented
182C42—21no—
183C42C22⋊D9711yescharacteristic, complemented
184C23—21no—
185C23—41no—
186C23—42no—
187C23—1941no—
188C23—1941no—
189C23C2×C22⋊D9711yescharacteristic
190C23C2×C22⋊D9711yescharacteristic
191Q8—41no—
192Q8—1941no—
193D4—41no—
194D4—42no—
195D4—7762no—
196D4—3881no—
197D4—3881no—
198D4—3881no—
199D4—1941no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—3881no—
209C2×C4—3881no—
210C2×C4—3881no—
211C2×C4—1941no—
212C2×C4—1941no—
213C2×C4C2×C22⋊D9711yescharacteristic
214C8—3881no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—7761no—
225C22—7761no—
226C22—3881no—
227C22#1569077911yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—7761no—
235C4—7761no—
236C4—3881no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—7761no—
242C2#2026092211yescharacteristic
subgroupG/H
centerC2#20260922
commutator, cosocleC2×D4×C97C23
FrattiniC2×D4C22×D97
Fitting#551×C97C2
socleC2×C97B(4,𝔽2)
prime radical#20260921C2
hypercenter#551D97
hypocenterC97#2197
LCSG ▷ C2×D4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #551

Complements

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

NH ≅ G/Ntype
1. #20260924C2semidirect
3. #20260912C2semidirect
4. #20260530C2semidirect
5. #20260525C2semidirect
6. #20260522C2semidirect
7. #551×C97C2semidirect
42. C2×Q8×C97D4semidirect
47. C22×C4×C97D4semidirect
59. C22⋊C4×C97D4semidirect
61. C42×C97D4semidirect
122. C97#2197semidirect
123. #551D97semidirect
152. C2×Q8C22⋊D97semidirect
161. C22×C4C22⋊D97semidirect
181. C22⋊C4C22⋊D97semidirect
183. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202609242
#202609212
#202609122
#202605302
#202605252
#202605222
#551×C972
#219797