Finite Group Databasebeta

← #27157171

Subgroups — #27157171

Order 12416.

340 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#15690867C211yesmaximal, characteristic, complemented
2#20261677C211yesmaximal, characteristic, complemented
3#20261587C212yesmaximal, complemented
4#20261586C212yesmaximal, complemented
5#121×D97C211yesmaximal, characteristic
6C22×#11715897—21no—
7C2×C4⋊#87604—21no—
8C2×C22⋊#87604—21no—
9#15690867—21no—
10#15690867C2212yescomplemented
11(C2×#87604)⋊C4C2212yescomplemented
12C2×D4×D97—21no—
13C2×D4×D97C411yescharacteristic, complemented
14C22×C4×D97C2211yescharacteristic
15#15690785C2211yescharacteristic
16#15690784C411yescharacteristic, complemented
17#15690783—21no—
18#15690782—21no—
19#15690781C2211yescharacteristic
20C4⋊C4×D97C411yescharacteristic, complemented
21C22⋊C4×D97—21no—
22#121×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C2×#11715897—42no—
26C4⋊#87604—42no—
27C22⋊#87604—42no—
28C23×D97—21no—
29C23×D97D411yescharacteristic
30C2×D388—21no—
31C2×D388C2×C411yescharacteristic
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C2×C22⋊D97—21no—
35D4×D97—42no—
36D4×D97—42no—
37D4×D97—42no—
38C22×Dic97D411yescharacteristic
39C2×C4×D97—21no—
40C2×C4×D97—42no—
41C2×C4×D97C2×C411yescharacteristic
42C2×C4×D97D411yescharacteristic
43C2×C4×D97D411yescharacteristic
44C2×C4×D97C2311yescharacteristic
45C4⋊Dic97C2×C411yescharacteristic
46#13095490—21no—
47C22⋊Dic97—21no—
48#13095488—21no—
49#13095488—21no—
50C2×D4×C97—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97C2×C411yescharacteristic
53C4⋊C4×C97C2×C411yescharacteristic
54C22⋊C4×C97—21no—
55C23×C97—21no—
56C23×C97C22⋊C411yescharacteristic
57C22×D97—21no—
58C22×D97—41no—
59C22×D97—41no—
60C22×D97—42no—
61C22×D97—42no—
62C22×D97—42no—
63C22×D97C22⋊C411yescharacteristic
64C22×D97D812yes—
65C22×D97SD1612yes—
66C22×D97C2×D411yescharacteristic
67C2×#87604—42no—
68C2×#87604—42no—
69C2×#87604—42no—
70D4×C97—41no—
71D4×C97—42no—
72C2×C4×C97—21no—
73C2×C4×C97—41no—
74C2×C4×C97C22⋊C411yescharacteristic
75C2×C4×C97C22×C411yescharacteristic
76C22⋊D97—42no—
77C22⋊D97—41no—
78C22⋊D97—41no—
79C22⋊D97—42no—
80C2×Dic97—22no—
81C2×Dic97—21no—
82C2×Dic97C22⋊C411yescharacteristic
83C2×Dic97C2×D411yescharacteristic
84D388—41no—
85D388—41no—
86D388—41no—
87C4×D97—22no—
88C4×D97—22no—
89C4×D97—41no—
90C4×D97—41no—
91C4×D97—42no—
92#11715897—42no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97#10212yes—
99C22×C97C2×C22⋊C411yescharacteristic
100C2×D97—21no—
101C2×D97—21no—
102C2×D97—21no—
103C2×D97—21no—
104C2×D97—42no—
105C2×D97—42no—
106C2×D97—41no—
107C2×D97—41no—
108C2×D97—41no—
109C2×D97—84no—
110C2×D97#10212yes—
111C2×D97C2×D811yescharacteristic
112C2×D97C2×SD1611yescharacteristic
113#87604—82no—
114C4×C97—21no—
115C4×C97—21no—
116C4×C97—41no—
117Dic97—21no—
118Dic97—41no—
119Dic97#9912yes—
120C2×C97—22no—
121C2×C97—81no—
122C2×C97C2×#9911yescharacteristic
123C2×C97C2×#10211yescharacteristic
124C2×C97#42111yescharacteristic
125D97—21no—
126D97—22no—
127D97—21no—
128D97—81no—
129#1519—971nomaximal, Sylow
130C97#151911yescharacteristic, complemented, Sylow
131C2×#121—971no—
132C2×C22⋊C8—971no—
133#380—971no—
134#327—972no—
135#327—972no—
136C22×D4—971no—
137C22×D4—1941no—
138C23×C4—971no—
139C22×C8—1941no—
140#121—971no—
141#121—971no—
142#121—971no—
143#121—1941no—
144#121—1941no—
145#121#8760411yescharacteristic, complemented
146C2×C4⋊C4—971no—
147C2×C4⋊C4—1941no—
148C2×C22⋊C4—1941no—
149C2×C22⋊C4—1941no—
150C22⋊C8—972no—
151C22⋊C8—1941no—
152#95—972no—
153C24—971no—
154C24—1941no—
155C2×D4—21no—
156C2×D4—971no—
157C2×D4—3882no—
158C2×D4—3882no—
159C2×D4—3882no—
160C2×D4—1941no—
161C2×D4—1941no—
162C2×D4—1941no—
163C2×D4—1941no—
164C2×D4C2×#8760411yescharacteristic
165C22×C4—971no—
166C22×C4—971no—
167C22×C4—971no—
168C22×C4—971no—
169C22×C4—971no—
170C22×C4—3882no—
171C22×C4—1941no—
172C22×C4—1942no—
173C22×C4C2×#8760411yescharacteristic
174C2×C8—3882no—
175C2×C8—1942no—
176C4⋊C4—971no—
177C4⋊C4—3882no—
178C4⋊C4—1941no—
179C4⋊C4C2×#8760411yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—3882no—
182C22⋊C4—1941no—
183C22⋊C4—1941no—
184C22⋊C4—1941no—
185C23—21no—
186C23—972no—
187C23—972no—
188C23—971no—
189C23—971no—
190C23—3881no—
191C23—3881no—
192C23—3882no—
193C23—3882no—
194C23—3882no—
195C23—1941no—
196C23C22⋊#8760411yescharacteristic
197D4—41no—
198D4—42no—
199D4—3881no—
200D4—3881no—
201D4—3881no—
202D4—3882no—
203D4—3881no—
204D4—3881no—
205D4—3882no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—971no—
209C2×C4—971no—
210C2×C4—3881no—
211C2×C4—3881no—
212C2×C4—3882no—
213C2×C4—3882no—
214C2×C4—3882no—
215C2×C4—3882no—
216C2×C4—1942no—
217C2×C4—1942no—
218C2×C4—1942no—
219C2×C4—1941no—
220C2×C4C22⋊#8760411yescharacteristic
221C2×C4C22×#8760411yescharacteristic
222C8—3882no—
223C22—21no—
224C22—21no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—7764no—
229C22—971no—
230C22—972no—
231C22—971no—
232C22—3882no—
233C22—3882no—
234C22—3881no—
235C22—3881no—
236C22—3881no—
237C22—1941no—
238C22—1941no—
239C22—1941no—
240C22—1941no—
241C22#1569087212yes—
242C22C2×C22⋊#8760411yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—7762no—
247C4—972no—
248C4—3881no—
249C4—1941no—
250C2—22no—
251C2—81no—
252C2—7761no—
253C2—1941no—
254C2—1942no—
255C2—1941no—
256C2D97⋊#9911yescharacteristic
257C2C2×#1569087211yescharacteristic
258C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#121×C97C4
socleC22×C97C2×C22⋊C4
prime radical#121×D97C2
cosocleC2×C4×D97C23
hypercenter#121#87604
hypocenterC97#1519
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15690867C2semidirect
2. #20261677C2semidirect
3. #20261587C2semidirect
4. #20261586C2semidirect
10. #15690867C22semidirect
11. (C2×#87604)⋊C4C22semidirect
13. C2×D4×D97C4semidirect
16. #15690784C4semidirect
20. C4⋊C4×D97C4semidirect
22. #121×C97C4semidirect
130. C97#1519semidirect
145. #121#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908672
#202616772
#202615872
#202615862
#121×D972
#151997