Finite Group Databasebeta

← #27140159

Subgroups — #27140159

Order 12416.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C4×D4×D97C211yesmaximal, characteristic
2#20260786C211yesmaximal, characteristic, complemented
3#20260757C211yesmaximal, characteristic, complemented
4#20260464C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×D4×D97C411yescharacteristic
9C22×C4×D97—21no—
10C2×#13095494—21no—
11C4×D388C2211yescharacteristic
12D4×Dic97C411yescharacteristic
13#15690772C411yescharacteristic
14C4×C22⋊D97—21no—
15#15690770—21no—
16C4⋊C4×D97C411yescharacteristic
17C22⋊C4×D97—21no—
18C42×D97C2211yescharacteristic
19C4⋊#11715895C2211yescharacteristic, complemented
20#15690739—21no—
21#15690739—21no—
22#15690722—21no—
23C4×#11715895C2211yescharacteristic, complemented
24C4×D4×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic, complemented
26C4×C8×C97C2211yescharacteristic, complemented
27C23×D97—21no—
28C2×D388C2×C411yescharacteristic
29C2×C22⋊D97—21no—
30D4×D97—22no—
31D4×D97—22no—
32C22×Dic97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97—42no—
37C2×C4×D97—42no—
38C2×C4×D97C2×C411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97D411yescharacteristic
41#13095494—42no—
42C2×#11715895—21no—
43C2×#11715895—21no—
44C4⋊Dic97C2×C411yescharacteristic
45#13095490—21no—
46C22⋊Dic97—21no—
47#13095488—21no—
48#13095488—21no—
49C4×Dic97C2×C411yescharacteristic
50C4×Dic97D411yescharacteristic
51C4×Dic97D411yescharacteristic
52C2×D4×C97C2×C411yescharacteristic
53C22×C4×C97—21no—
54C2×C8×C97—21no—
55C2×C8×C97—21no—
56C4⋊C4×C97C2×C411yescharacteristic
57C22⋊C4×C97—21no—
58C42×C97C2311yescharacteristic
59C23×C97—21no—
60C22×D97—24no—
61C22×D97—21no—
62C22×D97—42no—
63C22×D97—42no—
64C22×D97C22⋊C411yescharacteristic
65D4×C97—21no—
66D4×C97M1612yescomplemented
67C2×C4×C97—21no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97C22×C411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C22⋊D97—24no—
74C2×Dic97—21no—
75C2×Dic97—42no—
76C2×Dic97C22⋊C411yescharacteristic
77C2×Dic97C22⋊C411yescharacteristic
78C2×Dic97C22⋊C411yescharacteristic
79D388—21no—
80D388M1612yescomplemented
81C4×D97—21no—
82C4×D97—21no—
83C4×D97—41no—
84C4×D97—42no—
85C4×D97—41no—
86C4×D97—41no—
87C4×D97—42no—
88C4×D97D812yes—
89C4×D97SD1612yes—
90C8×C97—41no—
91C8×C97—41no—
92#11715895—41no—
93#11715895—41no—
94C22×C97—22no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97C2×C22⋊C411yescharacteristic
98C2×D97—21no—
99C2×D97—21no—
100C2×D97—22no—
101C2×D97—48no—
102C2×D97—41no—
103C2×D97—41no—
104C2×D97#10212yes—
105C4×C97—21no—
106C4×C97—41no—
107C4×C97C2×M1611yescharacteristic
108C4×C97C2×M1611yescharacteristic
109C4×C97C2×D811yescharacteristic
110C4×C97C2×SD1611yescharacteristic
111Dic97—21no—
112Dic97—21no—
113Dic97—41no—
114Dic97#10212yes—
115C2×C97—42no—
116C2×C97#40711yescharacteristic
117C2×C97C2×#10211yescharacteristic
118C2×C97#42111yescharacteristic
119D97—22no—
120D97—42no—
121#1486—971nomaximal, Sylow
122C97#148611yescharacteristic, complemented, Sylow
123C2×C4×D4—971no—
124#431—971no—
125#423—971no—
126D4⋊C8—971no—
127D4⋊C8—971no—
128D4⋊C8—971no—
129D4⋊C8D9711yescharacteristic, complemented
130C22×D4—971no—
131C23×C4—1941no—
132C2×M16—1941no—
133C4×D4—971no—
134C4×D4—971no—
135C4×D4—971no—
136C4×D4—1941no—
137C4×D4—1941no—
138C4×D4C2×D9711yescharacteristic
139C2×C4⋊C4—971no—
140C2×C22⋊C4—1941no—
141C2×C42—971no—
142C4⋊C8—971no—
143C4⋊C8—1941no—
144C4⋊C8—1941no—
145C4⋊C8C2×D9711yescharacteristic, complemented
146C22⋊C8—1941no—
147C4×C8—971no—
148C4×C8C2×D9711yescharacteristic, complemented
149C24—1941no—
150C2×D4—971no—
151C2×D4—1941no—
152C2×D4—1942no—
153C2×D4—1942no—
154C2×D4C4×D9711yescharacteristic
155C22×C4—21no—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—3882no—
160C22×C4—3882no—
161C22×C4—1941no—
162C22×C4—1941no—
163C22×C4—1941no—
164C22×C4—1941no—
165M16—3882no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—1941no—
169C2×C8—1941no—
170C4⋊C4—971no—
171C4⋊C4—1941no—
172C4⋊C4C4×D9711yescharacteristic
173C22⋊C4—21no—
174C22⋊C4—1941no—
175C22⋊C4—1941no—
176C22⋊C4—1941no—
177C42—971no—
178C42—971no—
179C42—971no—
180C42C22×D9711yescharacteristic
181C23—21no—
182C23—971no—
183C23—3882no—
184C23—3882no—
185C23—1944no—
186C23—1941no—
187D4—21no—
188D4—972no—
189D4—1944no—
190D4—1941no—
191D4#1309549412yescomplemented
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—971no—
196C2×C4—972no—
197C2×C4—972no—
198C2×C4—971no—
199C2×C4—971no—
200C2×C4—3881no—
201C2×C4—3882no—
202C2×C4—3882no—
203C2×C4—3882no—
204C2×C4—3881no—
205C2×C4—3881no—
206C2×C4—1941no—
207C2×C4—1941no—
208C2×C4—1941no—
209C2×C4C2×C4×D9711yescharacteristic
210C2×C4D4×D9711yescharacteristic
211C2×C4D4×D9711yescharacteristic
212C8—41no—
213C8—41no—
214C8—3881no—
215C8—3881no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—972no—
220C22—3888no—
221C22—3881no—
222C22—3881no—
223C22—1942no—
224C22—1941no—
225C22—1941no—
226C22C22⋊C4×D9711yescharacteristic
227C4—21no—
228C4—41no—
229C4—972no—
230C4—3881no—
231C4—1941no—
232C4—1941no—
233C4M16×D9711yescharacteristic
234C4C2×#1309549411yescharacteristic
235C4D8×D9711yescharacteristic
236C4SD16×D9711yescharacteristic
237C2—42no—
238C2—3882no—
239C2—1942no—
240C2#2026069711yescharacteristic
241C2#102×D9711yescharacteristic
242C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C4
cosocleC42×C97C23
hypercenterD4⋊C8D97
hypocenterC97#1486
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20260786C2semidirect
3. #20260757C2semidirect
4. #20260464C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. D4⋊C8×C97C2semidirect
19. C4⋊#11715895C22semidirect
23. C4×#11715895C22semidirect
25. C4⋊C8×C97C22semidirect
26. C4×C8×C97C22semidirect
66. D4×C97M16semidirect
80. D388M16semidirect
122. C97#1486semidirect
129. D4⋊C8D97semidirect
145. C4⋊C8C2×D97semidirect
148. C4×C8C2×D97semidirect
191. D4#13095494semidirect

Maximal subgroups

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

subgroupindex
C4×D4×D972
#202607862
#202607572
#202604642
D4⋊#117158952
#202604622
D4⋊C8×C972
#148697