Finite Group Databasebeta

← #27140143

Subgroups — #27140143

Order 12416.

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

normal subgroups only

This group has 212 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#20260787C211yesmaximal, characteristic, complemented
3#20260786C211yesmaximal, characteristic, complemented
4Q8⋊#11715895C211yesmaximal, characteristic, complemented
5#20260464C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8#15690805—21no—
9#15690801C411yescharacteristic
10#15690799C411yescharacteristic
11#15690795—21no—
12#15690786C411yescharacteristic
13#15690785C411yescharacteristic
14C4×D388—21no—
15C4×D388C2211yescharacteristic
16#15690767—21no—
17C42×D97C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690722—21no—
22C4×#11715895C2211yescharacteristic
23C8×Dic97—21no—
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
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—21no—
38#13095490—21no—
39#13095488—21no—
40#13095488—21no—
41#13095488—21no—
42#13095488—21no—
43#13095488—21no—
44C4×Dic97C2×C411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97D411yescharacteristic
47C2×Q8×C97C2×C411yescharacteristic
48C2×C8×C97—21no—
49C2×C8×C97—21no—
50C4⋊C4×C97—21no—
51C4⋊C4×C97C2×C411yescharacteristic
52C42×C97—21no—
53C42×C97C2311yescharacteristic
54C22×D97—21no—
55C22×D97C22⋊C411yescharacteristic
56Q8×C97—21no—
57Q8×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97C22×C411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×Dic97—21no—
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C22⋊C411yescharacteristic
67D388—21no—
68D388—21no—
69D388—42no—
70C4×D97—21no—
71C4×D97—21no—
72C4×D97—21no—
73C4×D97—21no—
74C4×D97—41no—
75C4×D97—41no—
76C4×D97—41no—
77C8×C97—41no—
78C8×C97—41no—
79#11715895—41no—
80#11715895—41no—
81C22×C97C2×C22⋊C411yescharacteristic
82C2×D97—21no—
83C2×D97—21no—
84C2×D97—21no—
85C2×D97—41no—
86C2×D97—41no—
87C2×D97—41no—
88C4×C97—21no—
89C4×C97—41no—
90C4×C97—41no—
91C4×C97#13111yescharacteristic
92C4×C97#13111yescharacteristic
93C4×C97#13511yescharacteristic
94C4×C97#13511yescharacteristic
95Dic97—21no—
96Dic97—21no—
97Dic97—21no—
98Dic97—41no—
99C2×C97#40811yescharacteristic
100C2×C97#41611yescharacteristic
101C2×C97#42111yescharacteristic
102D97—41no—
103D97—81no—
104#1482—971nomaximal, Sylow
105C97#148211yescharacteristic, complemented, Sylow
106#529—971no—
107#431—971no—
108#431—971no—
109Q8⋊C8—971no—
110Q8⋊C8D9711yescharacteristic, complemented
111D4⋊C8—971no—
112D4⋊C8—971no—
113#126—1941no—
114#125—971no—
115#124—971no—
116#123—1941no—
117C22⋊Q8—971no—
118#121—971no—
119C4×Q8C2×D9711yescharacteristic
120C4×D4—971no—
121C4×D4—1941no—
122#117—1941no—
123C2×C42—971no—
124C4⋊C8—971no—
125C4⋊C8—1941no—
126C4⋊C8C2×D9711yescharacteristic
127C22⋊C8—1941no—
128C22⋊C8—1941no—
129C4×C8—971no—
130C4×C8—1941no—
131C4×C8C2×D9711yescharacteristic
132C2×Q8C4×D9711yescharacteristic
133C2×D4—971no—
134C2×D4—1941no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—1941no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—1941no—
142C2×C8—1941no—
143C4⋊C4—21no—
144C4⋊C4—971no—
145C4⋊C4—1941no—
146C4⋊C4—1941no—
147C4⋊C4—1941no—
148C4⋊C4C4×D9711yescharacteristic
149C22⋊C4—1941no—
150C22⋊C4—1941no—
151C22⋊C4—1941no—
152C22⋊C4—1941no—
153C22⋊C4—1941no—
154C42—21no—
155C42—971no—
156C42—971no—
157C42—971no—
158C42C22×D9711yescharacteristic
159C23—971no—
160C23—1941no—
161Q8—21no—
162Q8—21no—
163D4—3882no—
164D4—1941no—
165D4—1941no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—971no—
169C2×C4—971no—
170C2×C4—971no—
171C2×C4—3881no—
172C2×C4—3881no—
173C2×C4—3881no—
174C2×C4—1941no—
175C2×C4—1941no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4C2×C4×D9711yescharacteristic
180C2×C4D4×D9711yescharacteristic
181C2×C4D4×D9711yescharacteristic
182C8—41no—
183C8—41no—
184C8—3881no—
185C8—3881no—
186C22—3881no—
187C22—3881no—
188C22—3881no—
189C22—1941no—
190C22—1941no—
191C22—1941no—
192C22C22⋊C4×D9711yescharacteristic
193C4—21no—
194C4—41no—
195C4—41no—
196C4—3881no—
197C4—1941no—
198C4—1941no—
199C4—1941no—
200C4#1569081611yescharacteristic
201C4#1569081811yescharacteristic
202C4#1569083111yescharacteristic
203C4#1569083311yescharacteristic
204C2—7761no—
205C2—3881no—
206C2#2026070011yescharacteristic
207C2#2026073011yescharacteristic
208C2#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#1482
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. #20260787C2semidirect
3. #20260786C2semidirect
4. Q8⋊#11715895C2semidirect
5. #20260464C2semidirect
6. #20260462C2semidirect
7. Q8⋊C8×C97C2semidirect
105. C97#1482semidirect
110. Q8⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612722
#202607872
#202607862
Q8⋊#117158952
#202604642
#202604622
Q8⋊C8×C972
#148297