Finite Group Databasebeta

← #27140181

Subgroups — #27140181

Order 12416.

252 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 73 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×Q8×D97C211yesmaximal, characteristic
2#20260791C211yesmaximal, characteristic
3C4×#13095494C211yesmaximal, characteristic
4#20260467C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260465C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97C411yescharacteristic
9C2×#13095494—21no—
10Q8×Dic97C411yescharacteristic
11C4×Dic194—21no—
12C4×Dic194C2211yescharacteristic
13#15690775—21no—
14#15690775C411yescharacteristic
15C4⋊C4×D97—21no—
16C4⋊C4×D97C411yescharacteristic
17C42×D97—21no—
18C42×D97C2211yescharacteristic
19C4⋊#11715895C2211yescharacteristic
20#15690722—21no—
21#15690720—21no—
22#15690720—21no—
23C4×#11715895C2211yescharacteristic
24C4×Q8×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27C2×Dic194—21no—
28C2×Dic194C2×C411yescharacteristic
29Q8×D97—22no—
30Q8×D97—22no—
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36#13095494—24no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C4⋊Dic97—21no—
40C4⋊Dic97C2×C411yescharacteristic
41#13095490—21no—
42#13095490—21no—
43#13095490—21no—
44C4×Dic97—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C4×Dic97C2×C411yescharacteristic
48C4×Dic97D411yescharacteristic
49C4×Dic97D411yescharacteristic
50C2×Q8×C97C2×C411yescharacteristic
51C2×C8×C97—21no—
52C2×C8×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97C2×C411yescharacteristic
55C42×C97—21no—
56C42×C97C2311yescharacteristic
57C22×D97C22⋊C411yescharacteristic
58Q8×C97—21no—
59Q8×C97M1612yescomplemented
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97C22×C411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×Dic97—21no—
66C2×Dic97—21no—
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97C22⋊C411yescharacteristic
70C4×D97—24no—
71C4×D97—22no—
72C4×D97—21no—
73C4×D97—21no—
74C4×D97—21no—
75C4×D97—21no—
76C4×D97—42no—
77Dic194—24no—
78Dic194—21no—
79Dic194M1612yescomplemented
80C8×C97—22no—
81C8×C97—41no—
82#11715895—22no—
83#11715895—41no—
84C22×C97C2×C22⋊C411yescharacteristic
85C2×D97—21no—
86C2×D97—21no—
87C2×D97#10412yes—
88C4×C97—22no—
89C4×C97—21no—
90C4×C97—41no—
91C4×C97C2×M1611yescharacteristic
92C4×C97C2×M1611yescharacteristic
93C4×C97#13711yescharacteristic
94C4×C97#13711yescharacteristic
95Dic97—21no—
96Dic97—22no—
97Dic97—21no—
98Dic97—21no—
99Dic97—41no—
100C2×C97#40711yescharacteristic
101C2×C97#41911yescharacteristic
102C2×C97C2×#10411yescharacteristic
103D97—22no—
104#1491—971nomaximal, Sylow
105C97#149111yescharacteristic, complemented, Sylow
106C2×C4×Q8—971no—
107#432—971no—
108C4×M16—971no—
109Q8⋊C8—971no—
110Q8⋊C8—971no—
111Q8⋊C8—971no—
112Q8⋊C8D9711yescharacteristic, complemented
113C22×Q8—971no—
114C2×M16—1941no—
115C4×Q8—971no—
116C4×Q8—971no—
117C4×Q8—971no—
118C4×Q8—1941no—
119C4×Q8—1941no—
120C4×Q8C2×D9711yescharacteristic
121C2×C4⋊C4—971no—
122C2×C4⋊C4—1941no—
123C2×C42—971no—
124C2×C42—1941no—
125C4⋊C8—971no—
126C4⋊C8C2×D9711yescharacteristic
127C22⋊C8—1941no—
128#97—1941no—
129#97—1941no—
130C4×C8—971no—
131C4×C8C2×D9711yescharacteristic
132C2×Q8—971no—
133C2×Q8—1941no—
134C2×Q8—1942no—
135C2×Q8—1942no—
136C2×Q8C4×D9711yescharacteristic
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—1941no—
141C22×C4—1941no—
142M16—1944no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—1941no—
146C2×C8—1941no—
147C4⋊C4—21no—
148C4⋊C4—971no—
149C4⋊C4—1941no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4C4×D9711yescharacteristic
154C42—21no—
155C42—971no—
156C42—971no—
157C42—971no—
158C42—1941no—
159C42—1941no—
160C42—1941no—
161C42C22×D9711yescharacteristic
162C23—971no—
163Q8—21no—
164Q8—972no—
165Q8—1944no—
166Q8—1941no—
167Q8#1309549412yescomplemented
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—971no—
171C2×C4—971no—
172C2×C4—971no—
173C2×C4—3882no—
174C2×C4—1942no—
175C2×C4—1944no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4—1941no—
180C2×C4—1941no—
181C2×C4—1941no—
182C2×C4C2×C4×D9711yescharacteristic
183C2×C4D4×D9711yescharacteristic
184C2×C4D4×D9711yescharacteristic
185C8—22no—
186C8—41no—
187C8—3881no—
188C8—1942no—
189C22—972no—
190C22—1941no—
191C22—1941no—
192C22C22⋊C4×D9711yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—3881no—
197C4—1941no—
198C4—1942no—
199C4—1941no—
200C4—1941no—
201C4M16×D9711yescharacteristic
202C4C2×#1309549411yescharacteristic
203C4#1569083911yescharacteristic
204C4#1569084211yescharacteristic
205C2—1942no—
206C2#2026069711yescharacteristic
207C2#2026074211yescharacteristic
208C2#104×D9711yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingQ8⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenterQ8⋊C8D97
hypocenterC97#1491
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #20260467C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260465C2semidirect
7. Q8⋊C8×C97C2semidirect
59. Q8×C97M16semidirect
79. Dic194M16semidirect
105. C97#1491semidirect
112. Q8⋊C8D97semidirect
167. Q8#13095494semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×D972
#202607912
C4×#130954942
#202604672
Q8⋊#117158952
#202604652
Q8⋊C8×C972
#149197