Finite Group Databasebeta

← #27140200

Subgroups — #27140200

Order 12416.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261272C211yesmaximal, characteristic
2#20260786C211yesmaximal, characteristic, complemented
3#20260757C211yesmaximal, characteristic, complemented
4Q8⋊#11715895C211yesmaximal, characteristic, complemented
5#20260464C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095494—21no—
9#15690805—21no—
10#15690801C411yescharacteristic
11#15690799C411yescharacteristic
12#15690795—21no—
13#15690786C411yescharacteristic
14#15690785C411yescharacteristic
15C4×D388—21no—
16C4×D388C2211yescharacteristic
17#15690767—21no—
18C42×D97C2211yescharacteristic
19C4⋊#11715895C2211yescharacteristic
20#15690739—21no—
21#15690739—21no—
22#15690722—21no—
23C4×#11715895C2211yescharacteristic
24C4×Q8×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27C2×D388—21no—
28C2×D388C2×C411yescharacteristic
29C2×C4×D97—21no—
30C2×C4×D97C2×C411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C2×C4×D97D411yescharacteristic
33#13095494—42no—
34C2×#11715895—21no—
35C2×#11715895—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97C2×C411yescharacteristic
38#13095490—21no—
39#13095490—21no—
40#13095488—21no—
41#13095488—21no—
42#13095488—21no—
43#13095488—21no—
44#13095488—21no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97D411yescharacteristic
47C4×Dic97D411yescharacteristic
48C2×Q8×C97C2×C411yescharacteristic
49C2×C8×C97—21no—
50C2×C8×C97—21no—
51C4⋊C4×C97—21no—
52C4⋊C4×C97C2×C411yescharacteristic
53C42×C97—21no—
54C42×C97C2311yescharacteristic
55C22×D97—21no—
56C22×D97C22⋊C411yescharacteristic
57Q8×C97—21no—
58Q8×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97C22×C411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×Dic97—21no—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C22⋊C411yescharacteristic
68D388—21no—
69D388—21no—
70D388—42no—
71C4×D97—21no—
72C4×D97—21no—
73C4×D97—21no—
74C4×D97—21no—
75C4×D97—41no—
76C4×D97—41no—
77C4×D97—41no—
78C8×C97—41no—
79C8×C97—41no—
80#11715895—41no—
81#11715895—41no—
82C22×C97C2×C22⋊C411yescharacteristic
83C2×D97—21no—
84C2×D97—21no—
85C2×D97—21no—
86C2×D97—41no—
87C2×D97—41no—
88C2×D97—41no—
89C4×C97—21no—
90C4×C97—41no—
91C4×C97—41no—
92C4×C97#13111yescharacteristic
93C4×C97#13111yescharacteristic
94C4×C97AGL(1,ℤ/8)11yescharacteristic
95C4×C97#13711yescharacteristic
96Dic97—21no—
97Dic97—21no—
98Dic97—21no—
99Dic97—41no—
100C2×C97#40811yescharacteristic
101C2×C97#41711yescharacteristic
102C2×C97#42111yescharacteristic
103D97—41no—
104D97—81no—
105#1495—971nomaximal, Sylow
106C97#149511yescharacteristic, complemented, Sylow
107#529—971no—
108#431—971no—
109#423—971no—
110Q8⋊C8—971no—
111Q8⋊C8D9711yescharacteristic, complemented
112D4⋊C8—971no—
113D4⋊C8—971no—
114C2×M16—1941no—
115#126—1941no—
116#125—971no—
117#124—971no—
118#123—1941no—
119C22⋊Q8—971no—
120#121—971no—
121C4×Q8C2×D9711yescharacteristic
122C4×D4—971no—
123C4×D4—1941no—
124#117—1941no—
125C2×C42—971no—
126C4⋊C8—971no—
127C4⋊C8—1941no—
128C4⋊C8—1941no—
129C4⋊C8C2×D9711yescharacteristic
130C22⋊C8—1941no—
131C4×C8—971no—
132C4×C8C2×D9711yescharacteristic
133C2×Q8C4×D9711yescharacteristic
134C2×D4—971no—
135C2×D4—1941no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—1941no—
140M16—3882no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—1941no—
144C2×C8—1941no—
145C4⋊C4—21no—
146C4⋊C4—971no—
147C4⋊C4—1941no—
148C4⋊C4—1941no—
149C4⋊C4—1941no—
150C4⋊C4C4×D9711yescharacteristic
151C22⋊C4—1941no—
152C22⋊C4—1941no—
153C22⋊C4—1941no—
154C22⋊C4—1941no—
155C22⋊C4—1941no—
156C42—21no—
157C42—971no—
158C42—971no—
159C42—971no—
160C42C22×D9711yescharacteristic
161C23—971no—
162C23—1941no—
163Q8—21no—
164Q8—21no—
165D4—3882no—
166D4—1941no—
167D4—1941no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—971no—
171C2×C4—971no—
172C2×C4—971no—
173C2×C4—3881no—
174C2×C4—3881no—
175C2×C4—3881no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4—1941no—
180C2×C4—1941no—
181C2×C4C2×C4×D9711yescharacteristic
182C2×C4D4×D9711yescharacteristic
183C2×C4D4×D9711yescharacteristic
184C8—41no—
185C8—41no—
186C8—3881no—
187C8—3881no—
188C22—3881no—
189C22—3881no—
190C22—3881no—
191C22—1941no—
192C22—1941no—
193C22—1941no—
194C22C22⋊C4×D9711yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—3881no—
199C4—1941no—
200C4—1941no—
201C4—1941no—
202C4#1569081611yescharacteristic
203C4#1569081811yescharacteristic
204C4#1569083711yescharacteristic
205C4#1569084211yescharacteristic
206C2—7761no—
207C2—3881no—
208C2#2026070011yescharacteristic
209C2#2026073511yescharacteristic
210C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingQ8⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radical#15690785C4
cosocleC42×C97C23
hypercenterQ8⋊C8D97
hypocenterC97#1495
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
2. #20260786C2semidirect
3. #20260757C2semidirect
4. Q8⋊#11715895C2semidirect
5. #20260464C2semidirect
6. #20260462C2semidirect
7. Q8⋊C8×C97C2semidirect
106. C97#1495semidirect
111. Q8⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612722
#202607862
#202607572
Q8⋊#117158952
#202604642
#202604622
Q8⋊C8×C972
#149597