Finite Group Databasebeta

← #27140198

Subgroups — #27140198

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#20261267C211yesmaximal, characteristic
2#20260786C211yesmaximal, characteristic, complemented
3#20260757C211yesmaximal, characteristic, complemented
4#20260467C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095494—21no—
9#15690803—21no—
10#15690801C411yescharacteristic
11#15690797C411yescharacteristic
12#15690792—21no—
13#15690790C411yescharacteristic
14#15690781C411yescharacteristic
15C4×Dic194C2211yescharacteristic
16C4×C22⋊D97—21no—
17#15690769—21no—
18C42×D97C2211yescharacteristic
19C4⋊#11715895C2211yescharacteristic
20#15690739—21no—
21#15690739—21no—
22#15690722—21no—
23C4×#11715895C2211yescharacteristic
24C4×D4×C97C2211yescharacteristic
25C4⋊C8×C97C2211yescharacteristic
26C4×C8×C97C2211yescharacteristic
27C2×Dic194C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32#13095494—42no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97C2×C411yescharacteristic
38#13095490—21no—
39#13095490—21no—
40C22⋊Dic97—21no—
41C22⋊Dic97—21no—
42C22⋊Dic97—21no—
43#13095488—21no—
44C4×Dic97—21no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97D411yescharacteristic
47C4×Dic97D411yescharacteristic
48C2×D4×C97C2×C411yescharacteristic
49C22×C4×C97—21no—
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C4⋊C4×C97C2×C411yescharacteristic
53C22⋊C4×C97—21no—
54C42×C97C2311yescharacteristic
55C23×C97—21no—
56C22×D97C22⋊C411yescharacteristic
57D4×C97—21no—
58D4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97C22×C411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C22⋊D97—42no—
66C2×Dic97—21no—
67C2×Dic97—21no—
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C22⋊C411yescharacteristic
71C4×D97—21no—
72C4×D97—21no—
73C4×D97—21no—
74C4×D97—21no—
75C4×D97—41no—
76Dic194—21no—
77Dic194—21no—
78C8×C97—41no—
79C8×C97—41no—
80#11715895—41no—
81#11715895—41no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97C2×C22⋊C411yescharacteristic
86C2×D97—21no—
87C2×D97—21no—
88C2×D97—21no—
89C4×C97—21no—
90C4×C97—41no—
91C4×C97#13111yescharacteristic
92C4×C97#13111yescharacteristic
93C4×C97AGL(1,ℤ/8)11yescharacteristic
94C4×C97#13711yescharacteristic
95Dic97—21no—
96Dic97—21no—
97Dic97—21no—
98Dic97—41no—
99Dic97—41no—
100C2×C97—81no—
101C2×C97#40811yescharacteristic
102C2×C97#41711yescharacteristic
103C2×C97#42111yescharacteristic
104D97—41no—
105#1495—971nomaximal, Sylow
106C97#149511yescharacteristic, complemented, Sylow
107#529—971no—
108#431—971no—
109#423—971no—
110Q8⋊C8—971no—
111Q8⋊C8—971no—
112D4⋊C8—971no—
113D4⋊C8D9711yescharacteristic, complemented
114C2×M16—1941no—
115#126—1941no—
116#125—971no—
117#124—971no—
118#123—1941no—
119C22⋊Q8—971no—
120#121—971no—
121C4×Q8—971no—
122C4×D4—1941no—
123C4×D4C2×D9711yescharacteristic
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×Q8—971no—
134C2×D4—1941no—
135C2×D4C4×D9711yescharacteristic
136C22×C4—21no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—971no—
140M16—3882no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—1941no—
144C2×C8—1941no—
145C4⋊C4—971no—
146C4⋊C4—1941no—
147C4⋊C4—1941no—
148C4⋊C4—1941no—
149C4⋊C4—1941no—
150C4⋊C4C4×D9711yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—1941no—
153C22⋊C4—1941no—
154C22⋊C4—1941no—
155C22⋊C4—1941no—
156C42—971no—
157C42—971no—
158C42—971no—
159C42—1941no—
160C42C22×D9711yescharacteristic
161C23—21no—
162C23—971no—
163Q8—1941no—
164Q8—1941no—
165D4—21no—
166D4—21no—
167D4—3882no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—971no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—3881no—
175C2×C4—1941no—
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—41no—
189C22—41no—
190C22—41no—
191C22—1941no—
192C22—1941no—
193C22—1941no—
194C22C22⋊C4×D9711yescharacteristic
195C4—21no—
196C4—41no—
197C4—3881no—
198C4—3881no—
199C4—1941no—
200C4—1941no—
201C4—1941no—
202C4#1569081611yescharacteristic
203C4#1569081811yescharacteristic
204C4#1569083411yescharacteristic
205C4#1569083911yescharacteristic
206C2—81no—
207C2—3881no—
208C2#2026070011yescharacteristic
209C2#2026073311yescharacteristic
210C2#2026075011yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radical#15690781C4
cosocleC42×C97C23
hypercenterD4⋊C8D97
hypocenterC97#1495
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. #20260467C2semidirect
5. #20260465C2semidirect
6. D4⋊#11715895C2semidirect
7. D4⋊C8×C97C2semidirect
106. C97#1495semidirect
113. D4⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612672
#202607862
#202607572
#202604672
#202604652
D4⋊#117158952
D4⋊C8×C972
#149597