Finite Group Databasebeta

← #27140810

Subgroups — #27140810

Order 12416.

236 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261442C211yesmaximal, characteristic, complemented
2#20261108C211yesmaximal, characteristic, complemented
3#20260936C211yesmaximal, characteristic, complemented
4#20260930C211yesmaximal, characteristic
5#20260483C211yesmaximal, characteristic, complemented
6#20260467C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095500—21no—
9C2×#13095500—22no—
10C2×#13095496—21no—
11#15690810C2211yescharacteristic
12#15690807C2211yescharacteristic
13#15690801—21no—
14#15690789—21no—
15#15690787—22no—
16C4×Dic194C2211yescharacteristic
17D4×Dic97—21no—
18C2×C4⋊Dic97—21no—
19C4⋊#11715895C2211yescharacteristic
20#15690732—21no—
21#15690730—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194D411yescharacteristic
26C2×Dic194D412yes—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C22×Dic97—22no—
31#13095500—44no—
32#13095500—42no—
33#13095496—42no—
34C2×#11715895—21no—
35C4⋊Dic97—22no—
36C4⋊Dic97—21no—
37C4⋊Dic97—21no—
38C4⋊Dic97D411yescharacteristic
39#13095490—22no—
40#13095490—21no—
41C22⋊Dic97—22no—
42C4×Dic97—21no—
43C2×D4×C97D411yescharacteristic
44C22×C4×C97—21no—
45C2×C8×C97—21no—
46C2×C8×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51C22×D97—41no—
52D4×C97—21no—
53D4×C97SD1612yescomplemented
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97—22no—
63C2×Dic97—82no—
64D388—42no—
65D388—41no—
66D388—41no—
67D388—41no—
68D388—41no—
69Dic194—21no—
70Dic194—21no—
71Dic194—42no—
72C8×C97—22no—
73C8×C97—41no—
74#11715895—41no—
75C22×C97—22no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97#12011yescharacteristic
79C2×D97—81no—
80C2×D97—81no—
81C2×D97—81no—
82C4×C97—21no—
83C4×C97—41no—
84C4×C97C2×SD1611yescharacteristic
85C4×C97#13511yescharacteristic
86C4×C97AGL(1,ℤ/8)11yescharacteristic
87C4×C97#13711yescharacteristic
88Dic97—41no—
89Dic97—42no—
90Dic97—41no—
91C2×C97—42no—
92C2×C97#44811yescharacteristic
93C2×C97#44911yescharacteristic
94C2×C97#45311yescharacteristic
95D97—161no—
96#1622—971nomaximal, Sylow
97C97#162211yescharacteristic, complemented, Sylow
98#554—971no—
99#492—971no—
100#460—971no—
101#459—971no—
102#333—971no—
103Q8⋊C8—971no—
104D4⋊C8D9711yescharacteristic, complemented
105C2×SD16—1941no—
106C2×SD16—1942no—
107C2×D8—1941no—
108C4⋊Q8—971no—
109#127—971no—
110#125—1941no—
111C22⋊Q8—1942no—
112C22⋊Q8—1941no—
113C4×Q8—971no—
114C4×D4—1941no—
115C4×D4C2×D9711yescharacteristic
116C2×C4⋊C4—1941no—
117C4⋊C8—971no—
118C4⋊C8C2×D9711yescharacteristic
119#102—1941no—
120#102—1941no—
121C4×C8C2×D9711yescharacteristic
122C2×Q8—972no—
123C2×Q8—971no—
124C2×D4—1941no—
125C2×D4—1941no—
126C2×D4—1941no—
127C2×D4D38811yescharacteristic
128C22×C4—21no—
129C22×C4—1942no—
130SD16—3884no—
131SD16—3882no—
132D8—3882no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—1941no—
136C4⋊C4—971no—
137C4⋊C4—1942no—
138C4⋊C4—1942no—
139C4⋊C4—1941no—
140C4⋊C4—1941no—
141C4⋊C4—1941no—
142C4⋊C4D38811yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—1942no—
145C42—1941no—
146C42C22×D9711yescharacteristic
147C23—21no—
148C23—3881no—
149Q8—3882no—
150Q8—1941no—
151Q8—1941no—
152D4—21no—
153D4—3882no—
154D4—3881no—
155D4—3881no—
156D4—3881no—
157D4—3881no—
158D4#1309550012yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—7762no—
163C2×C4—1941no—
164C2×C4—1941no—
165C2×C4—1942no—
166C2×C4D4×D9711yescharacteristic
167C2×C4D4×D9711yescharacteristic
168C2×C4C2×D38811yescharacteristic
169C8—22no—
170C8—41no—
171C8—3881no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—7761no—
176C22—7761no—
177C22—7761no—
178C22#1569078011yescharacteristic
179C4—21no—
180C4—41no—
181C4—3881no—
182C4—3881no—
183C4—3882no—
184C4C2×#1309550011yescharacteristic
185C4#1569083111yescharacteristic
186C4#1569083411yescharacteristic
187C4#1569084311yescharacteristic
188C2—42no—
189C2—15521no—
190C2#2026087611yescharacteristic
191C2#2026088111yescharacteristic
192C2#2026090411yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radical#20260930C2
hypercenterD4⋊C8D97
hypocenterC97#1622
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. #20261442C2semidirect
2. #20261108C2semidirect
3. #20260936C2semidirect
5. #20260483C2semidirect
6. #20260467C2semidirect
7. D4⋊C8×C97C2semidirect
53. D4×C97SD16semidirect
97. C97#1622semidirect
104. D4⋊C8D97semidirect
158. D4#13095500semidirect

Maximal subgroups

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

subgroupindex
#202614422
#202611082
#202609362
#202609302
#202604832
#202604672
D4⋊C8×C972
#162297