Finite Group Databasebeta

← #27143388

Subgroups — #27143388

Order 12416.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261307C211yesmaximal, characteristic
2#20260716C211yesmaximal, characteristic
3#20260707C211yesmaximal, characteristic
4#20260526C211yesmaximal, characteristic, complemented
5#20260525C211yesmaximal, characteristic, complemented
6#20260524C211yesmaximal, characteristic, complemented
7#352×C97C211yesmaximal, characteristic, complemented
8C2×#13095510C2211yescharacteristic
9C22×Dic194C411yescharacteristic, complemented
10M16×D97—21no—
11#15690809—21no—
12#15690796—21no—
13#15690794C411yescharacteristic, complemented
14#15690793C411yescharacteristic, complemented
15#15690791C2211yescharacteristic
16#15690788—21no—
17#15690787—21no—
18#15690766—21no—
19#15690766C411yescharacteristic, complemented
20#15690728—21no—
21#15690727C2211yescharacteristic
22#15690725C2211yescharacteristic
23#15690724—21no—
24#123×C97C2211yescharacteristic
25#100×C97C2211yescharacteristic
26#99×C97C2211yescharacteristic
27#13095510—41no—
28#13095510—41no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194—41no—
32C2×Dic194—41no—
33C2×Dic194D411yescharacteristic
34C2×C22⋊D97C2×C411yescharacteristic
35C2×C22⋊D97D411yescharacteristic
36C22×Dic97C2×C411yescharacteristic
37C22×Dic97C2×C411yescharacteristic
38C22×Dic97D411yescharacteristic
39C2×C4×D97D411yescharacteristic
40#13095495—21no—
41#13095494—41no—
42C8×D97—41no—
43C4⋊Dic97—21no—
44#13095490—21no—
45#13095490—21no—
46#13095490—21no—
47#13095490—41no—
48C22⋊Dic97—21no—
49C22⋊Dic97—21no—
50C22⋊Dic97C2×C411yescharacteristic
51#13095488—21no—
52#13095488—21no—
53C4×Dic97—21no—
54C4×Dic97—41no—
55C2×D4×C97C2311yescharacteristic
56C22×C4×C97C2×C411yescharacteristic
57M16×C97—21no—
58C4⋊C4×C97—21no—
59C22⋊C4×C97—21no—
60C22⋊C4×C97—21no—
61C22⋊C4×C97C2×C411yescharacteristic
62C23×C97C22×C411yescharacteristic
63C23×C97C2×D411yescharacteristic
64C22×D97C22⋊C411yescharacteristic
65D4×C97—41no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97C2×D411yescharacteristic
71C22⋊D97—41no—
72C22⋊D97—41no—
73C2×Dic97—21no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97C22⋊C411yescharacteristic
82C2×Dic97C22⋊C411yescharacteristic
83C2×Dic97C22⋊C411yescharacteristic
84C4×D97—21no—
85C4×D97—21no—
86Dic194—41no—
87Dic194—42no—
88Dic194—41no—
89Dic194—41no—
90C8×C97—41no—
91#11715895—41no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97—41no—
95C22×C97—41no—
96C22×C97C2×C22⋊C411yescharacteristic
97C2×D97—21no—
98C2×D97—41no—
99C4×C97—21no—
100C4×C97—41no—
101C4×C97—41no—
102C4×C97—81no—
103Dic97—21no—
104Dic97—21no—
105Dic97—41no—
106Dic97—41no—
107Dic97—81no—
108Dic97#9911yescharacteristic
109Dic97#9911yescharacteristic
110C2×C97—21no—
111C2×C97—41no—
112C2×C97—41no—
113C2×C97C2×#9911yescharacteristic
114D97—41no—
115#2123—971nomaximal, Sylow
116C97#212311yescharacteristic, complemented, Sylow
117#536—971no—
118#413—971no—
119#410—971no—
120#352—971no—
121#352—971no—
122#352—971no—
123#352D9711yescharacteristic, complemented
124C2×#41—971no—
125C22×Q8—971no—
126C2×M16—1941no—
127C4⋊Q8—1941no—
128#124—1941no—
129#123—971no—
130#123—971no—
131#123—971no—
132#123C2×D9711yescharacteristic
133C22⋊Q8—1941no—
134C22⋊Q8—1941no—
135#117—971no—
136#117—1941no—
137#101—1941no—
138#100—971no—
139#100C2×D9711yescharacteristic
140#99—971no—
141#99—1941no—
142#99C2×D9711yescharacteristic
143#41—3881no—
144#41—3881no—
145C2×Q8—971no—
146C2×Q8—3881no—
147C2×Q8—3881no—
148C2×Q8—1941no—
149C2×Q8—1941no—
150C2×D4—971no—
151C2×D4—971no—
152C2×D4C22×D9711yescharacteristic
153C22×C4—971no—
154C22×C4—971no—
155C22×C4—971no—
156C22×C4—971no—
157C22×C4C4×D9711yescharacteristic
158M16—21no—
159M16—3881no—
160M16—1941no—
161C2×C8—3881no—
162C4⋊C4—21no—
163C4⋊C4—3881no—
164C4⋊C4—1941no—
165C4⋊C4—1941no—
166C4⋊C4—1941no—
167C4⋊C4—1941no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—971no—
171C22⋊C4—1941no—
172C22⋊C4—1941no—
173C22⋊C4—1941no—
174C22⋊C4—1941no—
175C22⋊C4C4×D9711yescharacteristic
176C42—3881no—
177C42—1941no—
178C23—971no—
179C23C2×C4×D9711yescharacteristic
180C23D4×D9711yescharacteristic
181Q8—3881no—
182Q8—3881no—
183Q8—3881no—
184Q8—3882no—
185D4—41no—
186D4—3881no—
187D4—3881no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—971no—
193C2×C4—971no—
194C2×C4—971no—
195C2×C4—3881no—
196C2×C4—3881no—
197C2×C4—1941no—
198C2×C4—1941no—
199C2×C4—1941no—
200C2×C4—1941no—
201C2×C4—1941no—
202C2×C4—1941no—
203C2×C4—1941no—
204C2×C4—1941no—
205C2×C4D4×D9711yescharacteristic
206C8—41no—
207C8—3881no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—3881no—
213C22—1941no—
214C22C22⋊C4×D9711yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—7761no—
220C4—971no—
221C4—971no—
222C4—3881no—
223C4—3881no—
224C4—1941no—
225C4—1941no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—3881no—
230C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC23×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#352×C97C2
socleC2×C97C2×#99
prime radicalC2×#13095510C22
cosocleC2×D4×C97C23
hypercenter#352D97
hypocenterC97#2123
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #20260526C2semidirect
5. #20260525C2semidirect
6. #20260524C2semidirect
7. #352×C97C2semidirect
9. C22×Dic194C4semidirect
13. #15690794C4semidirect
14. #15690793C4semidirect
19. #15690766C4semidirect
116. C97#2123semidirect
123. #352D97semidirect

Maximal subgroups

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

subgroupindex
#202613072
#202607162
#202607072
#202605262
#202605252
#202605242
#352×C972
#212397