Finite Group Databasebeta

← #27140825

Subgroups — #27140825

Order 12416.

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

normal subgroups only

This group has 256 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#20261083C211yesmaximal, characteristic, complemented
2#20260930C211yesmaximal, characteristic, complemented
3#20260930C211yesmaximal, characteristic, complemented
4#20260481C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#547×C97C211yesmaximal, characteristic, complemented
8C2×#13095496—21no—
9C2×#13095496—21no—
10#15690807C2211yescharacteristic
11C4⋊#11715895C2211yescharacteristic
12C4⋊#11715895C2211yescharacteristic
13#15690730—21no—
14#15690730—21no—
15#15690730—22no—
16C4×#11715895C2211yescharacteristic
17C2×#41×C97—21no—
18C4⋊Q8×C97C2211yescharacteristic
19#123×C97—22no—
20C22⋊Q8×C97—21no—
21#121×C97—21no—
22C4×D4×C97C2211yescharacteristic
23C4×D4×C97C2211yescharacteristic
24C2×C4⋊C4×C97—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28#13095496—42no—
29#13095496—42no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33#41×C97—42no—
34#41×C97—42no—
35C2×Q8×C97—21no—
36C2×D4×C97—22no—
37C2×D4×C97D411yescharacteristic
38C2×D4×C97D411yescharacteristic
39C22×C4×C97—22no—
40C22×C4×C97—21no—
41C22×C4×C97—21no—
42C4⋊C4×C97—22no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97D411yescharacteristic
45C4⋊C4×C97D411yescharacteristic
46C4⋊C4×C97D412yes—
47C22⋊C4×C97—22no—
48C22⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C42×C97C2311yescharacteristic
51C23×C97—21no—
52C23×C97—21no—
53C22×D97—41no—
54Q8×C97—41no—
55Q8×C97—41no—
56D4×C97—21no—
57D4×C97—21no—
58D4×C97—21no—
59D4×C97—44no—
60D4×C97D812yescomplemented
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—22no—
64C2×C4×C97—44no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72D388—42no—
73D388—41no—
74D388—41no—
75D388—41no—
76D388—41no—
77#11715895—41no—
78#11715895—41no—
79#11715895—41no—
80C22×C97—22no—
81C22×C97—41no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97—41no—
86C22×C97#12011yescharacteristic
87C2×D97—81no—
88C2×D97—81no—
89C2×D97—81no—
90C4×C97—21no—
91C4×C97—41no—
92C4×C97—42no—
93C4×C97—41no—
94C4×C97C2×D811yescharacteristic
95C4×C97#13511yescharacteristic
96C4×C97AGL(1,ℤ/8)11yescharacteristic
97C4×C97AGL(1,ℤ/8)11yescharacteristic
98C2×C97—42no—
99C2×C97—81no—
100C2×C97#44711yescharacteristic
101C2×C97#44911yescharacteristic
102C2×C97#45411yescharacteristic
103D97—161no—
104#1625—971nomaximal, Sylow
105C97#162511yescharacteristic, complemented, Sylow
106#547D9711yescharacteristic, complemented
107#486—971no—
108#459—971no—
109#459—971no—
110#332—971no—
111D4⋊C8—971no—
112D4⋊C8—971no—
113C2×#41—21no—
114C2×D8—1941no—
115C2×D8—1941no—
116C4⋊Q8C2×D9711yescharacteristic
117#127—971no—
118#123—22no—
119C22⋊Q8—21no—
120#121—21no—
121C4×D4C2×D9711yescharacteristic
122C4×D4C2×D9711yescharacteristic
123C2×C4⋊C4—21no—
124C4⋊C8—971no—
125C4⋊C8—971no—
126#102—1941no—
127#102—1941no—
128#102—1942no—
129C4×C8—971no—
130#41—42no—
131#41—42no—
132C2×Q8—21no—
133C2×D4—22no—
134C2×D4—1941no—
135C2×D4—1941no—
136C2×D4—1941no—
137C2×D4C22⋊D9711yescharacteristic
138C2×D4C22⋊D9711yescharacteristic
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142D8—3882no—
143D8—3882no—
144C2×C8—1941no—
145C2×C8—1941no—
146C2×C8—1941no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4C22⋊D9711yescharacteristic
150C4⋊C4C22⋊D9711yescharacteristic
151C4⋊C4C22⋊D9712yes—
152C22⋊C4—22no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D9711yescharacteristic
156C23—21no—
157C23—21no—
158C23—3881no—
159Q8—41no—
160Q8—41no—
161D4—21no—
162D4—21no—
163D4—21no—
164D4—44no—
165D4—3882no—
166D4—3881no—
167D4—3881no—
168D4—3881no—
169D4—3881no—
170D4#1309549612yescomplemented
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4C2×C22⋊D9711yescharacteristic
180C2×C4C2×C22⋊D9711yescharacteristic
181C2×C4C2×C22⋊D9711yescharacteristic
182C8—3881no—
183C8—3881no—
184C8—3881no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—7761no—
192C22—7761no—
193C22—7761no—
194C22#1569077911yescharacteristic
195C4—21no—
196C4—41no—
197C4—42no—
198C4—41no—
199C4C2×#1309549611yescharacteristic
200C4#1569083011yescharacteristic
201C4#1569083511yescharacteristic
202C4#1569083611yescharacteristic
203C2—42no—
204C2—81no—
205C2—15521no—
206C2#2026086911yescharacteristic
207C2#2026088011yescharacteristic
208C2#2026090811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#547×C97C2
socleC22×C97#120
hypercenter#547D97
hypocenterC97#1625
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20261083C2semidirect
2. #20260930C2semidirect
3. #20260930C2semidirect
4. #20260481C2semidirect
5. D4⋊#11715895C2semidirect
6. D4⋊#11715895C2semidirect
7. #547×C97C2semidirect
60. D4×C97D8semidirect
105. C97#1625semidirect
106. #547D97semidirect
170. D4#13095496semidirect

Maximal subgroups

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

subgroupindex
#202610832
#202609302
#202604812
D4⋊#117158952
#547×C972
#162597