Finite Group Databasebeta

← #27140148

Subgroups — #27140148

Order 12416.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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
2C4⋊C8×D97C211yesmaximal, characteristic, complemented
3C4×#13095494C211yesmaximal, characteristic, complemented
4Q8⋊#11715895C211yesmaximal, characteristic, complemented
5#20260464C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095494—21no—
9C2×C8×D97—21no—
10#15690805—21no—
11#15690801C411yescharacteristic
12#15690799C411yescharacteristic
13#15690795—21no—
14#15690786C411yescharacteristic
15#15690785C411yescharacteristic
16C4×D388—21no—
17C4×D388C2211yescharacteristic
18#15690767—21no—
19C42×D97C2211yescharacteristic
20C4⋊#11715895C2211yescharacteristic
21#15690739—21no—
22#15690720—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—24no—
34C2×#11715895—21no—
35C2×#11715895—21no—
36C8×D97—42no—
37C4⋊Dic97—21no—
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—21no—
40#13095490—21no—
41#13095488—21no—
42#13095488—21no—
43#13095488—21no—
44#13095488—21no—
45#13095488—21no—
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97D411yescharacteristic
49C2×Q8×C97C2×C411yescharacteristic
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97C2×C411yescharacteristic
54C42×C97—21no—
55C42×C97C2311yescharacteristic
56C22×D97—21no—
57C22×D97C22⋊C411yescharacteristic
58Q8×C97—21no—
59Q8×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97C22×C411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×Dic97—21no—
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C22⋊C411yescharacteristic
69D388—21no—
70D388—21no—
71D388—42no—
72C4×D97—22no—
73C4×D97—21no—
74C4×D97—21no—
75C4×D97—21no—
76C4×D97—21no—
77C4×D97—41no—
78C4×D97—41no—
79C8×C97—22no—
80C8×C97—41no—
81#11715895—22no—
82#11715895—41no—
83C22×C97C2×C22⋊C411yescharacteristic
84C2×D97—21no—
85C2×D97—21no—
86C2×D97—41no—
87C2×D97—41no—
88C2×D97—41no—
89C2×D97#10412yes—
90C4×C97—21no—
91C4×C97—41no—
92C4×C97—41no—
93C4×C97#13111yescharacteristic
94C4×C97#13111yescharacteristic
95C4×C97AGL(1,ℤ/8)11yescharacteristic
96C4×C97#13711yescharacteristic
97Dic97—21no—
98Dic97—21no—
99Dic97—21no—
100Dic97—41no—
101C2×C97#40811yescharacteristic
102C2×C97#41711yescharacteristic
103C2×C97C2×#10411yescharacteristic
104D97—22no—
105D97—81no—
106#1483—971nomaximal, Sylow
107C97#148311yescharacteristic, complemented, Sylow
108#529—971no—
109C2×C4⋊C8—971no—
110C4×M16—971no—
111Q8⋊C8—971no—
112Q8⋊C8D9711yescharacteristic, complemented
113D4⋊C8—971no—
114D4⋊C8—971no—
115C2×M16—1941no—
116C22×C8—1941no—
117#126—1941no—
118#125—971no—
119#124—971no—
120#123—1941no—
121C22⋊Q8—971no—
122#121—971no—
123C4×Q8C2×D9711yescharacteristic
124C4×D4—971no—
125C4×D4—1941no—
126#117—1941no—
127C2×C42—971no—
128C4⋊C8—971no—
129C4⋊C8—1941no—
130C4⋊C8C2×D9711yescharacteristic
131#97—1941no—
132C4×C8—971no—
133C4×C8C2×D9711yescharacteristic
134C2×Q8C4×D9711yescharacteristic
135C2×D4—971no—
136C2×D4—1941no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—1941no—
141M16—1944no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—3882no—
145C2×C8—1941no—
146C2×C8—1941no—
147C4⋊C4—21no—
148C4⋊C4—971no—
149C4⋊C4—1941no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4C4×D9711yescharacteristic
153C22⋊C4—1941no—
154C22⋊C4—1941no—
155C22⋊C4—1941no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C42—21no—
159C42—971no—
160C42—971no—
161C42—971no—
162C42C22×D9711yescharacteristic
163C23—971no—
164C23—1941no—
165Q8—21no—
166Q8—21no—
167D4—3882no—
168D4—1941no—
169D4—1941no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—3881no—
176C2×C4—3881no—
177C2×C4—1942no—
178C2×C4—1941no—
179C2×C4—1941no—
180C2×C4—1941no—
181C2×C4—1941no—
182C2×C4—1941no—
183C2×C4C2×C4×D9711yescharacteristic
184C2×C4D4×D9711yescharacteristic
185C2×C4D4×D9711yescharacteristic
186C8—22no—
187C8—41no—
188C8—3881no—
189C8—1942no—
190C22—972no—
191C22—3881no—
192C22—3881no—
193C22—3881no—
194C22—1941no—
195C22—1941no—
196C22C22⋊C4×D9711yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—3881no—
201C4—1941no—
202C4—1941no—
203C4—1941no—
204C4#1569081611yescharacteristic
205C4#1569081811yescharacteristic
206C4#1569083711yescharacteristic
207C4#1569084211yescharacteristic
208C2—7761no—
209C2—1942no—
210C2#2026070011yescharacteristic
211C2#2026073511yescharacteristic
212C2#104×D9711yescharacteristic
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#1483
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. C4⋊C8×D97C2semidirect
3. C4×#13095494C2semidirect
4. Q8⋊#11715895C2semidirect
5. #20260464C2semidirect
6. #20260462C2semidirect
7. Q8⋊C8×C97C2semidirect
107. C97#1483semidirect
112. Q8⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612722
C4⋊C8×D972
C4×#130954942
Q8⋊#117158952
#202604642
#202604622
Q8⋊C8×C972
#148397