Finite Group Databasebeta

← #27140808

Subgroups — #27140808

Order 12416.

236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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#20261109C211yesmaximal, characteristic, complemented
2#20260935C211yesmaximal, characteristic, complemented
3#20260930C211yesmaximal, characteristic
4#20260484C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#554×C97C211yesmaximal, characteristic, complemented
8C2×#13095498—21no—
9C2×#13095498—22no—
10C2×#13095496—21no—
11#15690807C2211yescharacteristic
12C4⋊#11715895C2211yescharacteristic
13C4⋊#11715895C2211yescharacteristic
14#15690730—21no—
15#15690730—21no—
16C4×#11715895C2211yescharacteristic
17C4⋊Q8×C97C2211yescharacteristic
18#125×C97—21no—
19C22⋊Q8×C97—21no—
20C22⋊Q8×C97—22no—
21C4×Q8×C97C2211yescharacteristic
22C4×D4×C97—21no—
23C4×D4×C97C2211yescharacteristic
24C2×C4⋊C4×C97—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28#13095498—44no—
29#13095498—42no—
30#13095496—42no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C2×#11715895—21no—
34C2×Q8×C97D411yescharacteristic
35C2×Q8×C97D412yes—
36C2×D4×C97D411yescharacteristic
37C22×C4×C97—22no—
38C22×C4×C97—21no—
39C4⋊C4×C97—22no—
40C4⋊C4×C97—22no—
41C4⋊C4×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97D411yescharacteristic
45C4⋊C4×C97D411yescharacteristic
46C22⋊C4×C97—22no—
47C22⋊C4×C97—21no—
48C42×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51C22×D97—41no—
52Q8×C97—21no—
53Q8×C97—21no—
54Q8×C97—42no—
55D4×C97—21no—
56D4×C97SD1612yescomplemented
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—22no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—82no—
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67D388—42no—
68D388—41no—
69D388—41no—
70D388—41no—
71D388—41no—
72#11715895—22no—
73#11715895—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×C97—41no—
85C4×C97—41no—
86C4×C97—42no—
87C4×C97C2×SD1611yescharacteristic
88C4×C97#13511yescharacteristic
89C4×C97AGL(1,ℤ/8)11yescharacteristic
90C4×C97#13711yescharacteristic
91C2×C97—42no—
92C2×C97#44811yescharacteristic
93C2×C97#44911yescharacteristic
94C2×C97#45311yescharacteristic
95D97—161no—
96#1622—971nomaximal, Sylow
97C97#162211yescharacteristic, complemented, Sylow
98#554D9711yescharacteristic, complemented
99#492—971no—
100#460—971no—
101#459—971no—
102#333—971no—
103Q8⋊C8—971no—
104D4⋊C8—971no—
105C2×SD16—1941no—
106C2×SD16—1942no—
107C2×D8—1941no—
108C4⋊Q8C2×D9711yescharacteristic
109#127—971no—
110#125—21no—
111C22⋊Q8—21no—
112C22⋊Q8—22no—
113C4×Q8C2×D9711yescharacteristic
114C4×D4—21no—
115C4×D4C2×D9711yescharacteristic
116C2×C4⋊C4—21no—
117C4⋊C8—971no—
118C4⋊C8—971no—
119#102—1941no—
120#102—1941no—
121C4×C8—971no—
122C2×Q8C22⋊D9711yescharacteristic
123C2×Q8C22⋊D9712yes—
124C2×D4—1941no—
125C2×D4—1941no—
126C2×D4—1941no—
127C2×D4C22⋊D9711yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130SD16—3882no—
131SD16—3884no—
132D8—3882no—
133C2×C8—1941no—
134C2×C8—1941no—
135C2×C8—1941no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4C22⋊D9711yescharacteristic
142C4⋊C4C22⋊D9711yescharacteristic
143C22⋊C4—22no—
144C22⋊C4—21no—
145C42—21no—
146C42C22×D9711yescharacteristic
147C23—21no—
148C23—3881no—
149Q8—21no—
150Q8—21no—
151Q8—42no—
152D4—21no—
153D4—3882no—
154D4—3881no—
155D4—3881no—
156D4—3881no—
157D4—3881no—
158D4#1309549812yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4C2×C22⋊D9711yescharacteristic
167C2×C4C2×C22⋊D9711yescharacteristic
168C2×C4C2×C22⋊D9711yescharacteristic
169C8—3881no—
170C8—3881no—
171C8—1942no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—7761no—
176C22—7761no—
177C22—7761no—
178C22#1569077911yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—42no—
184C4C2×#1309549811yescharacteristic
185C4#1569083011yescharacteristic
186C4#1569083511yescharacteristic
187C4#1569084111yescharacteristic
188C2—42no—
189C2—15521no—
190C2#2026087411yescharacteristic
191C2#2026088011yescharacteristic
192C2#2026090311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#554×C97C2
socleC22×C97#120
prime radical#20260930C2
hypercenter#554D97
hypocenterC97#1622
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #20261109C2semidirect
2. #20260935C2semidirect
4. #20260484C2semidirect
5. Q8⋊#11715895C2semidirect
6. D4⋊#11715895C2semidirect
7. #554×C97C2semidirect
56. D4×C97SD16semidirect
97. C97#1622semidirect
98. #554D97semidirect
158. D4#13095498semidirect

Maximal subgroups

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

subgroupindex
#202611092
#202609352
#202609302
#202604842
Q8⋊#117158952
D4⋊#117158952
#554×C972
#162297