Finite Group Databasebeta

← #27140935

Subgroups — #27140935

Order 12416.

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

normal subgroups only

This group has 250 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#20261407C211yesmaximal, characteristic
2#20261083C211yesmaximal, characteristic, complemented
3#20260937C211yesmaximal, characteristic, complemented
4#20260478C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#475×C97C211yesmaximal, characteristic, complemented
8C2×#13095512—21no—
9C2×#13095498—21no—
10#15690807—21no—
11#15690807C2211yescharacteristic
12#15690785—22no—
13#15690785—21no—
14C4×D388—21no—
15C4×D388C2211yescharacteristic
16C4⋊#11715895C2211yescharacteristic
17#15690734—21no—
18#15690730—22no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×Q8×C97C2211yescharacteristic
22#106×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25#13095512—42no—
26#13095512—42no—
27C2×D388—22no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—22no—
31C2×D388D411yescharacteristic
32C2×D388D412yes—
33C2×C4×D97—22no—
34C2×C4×D97—21no—
35#13095498—42no—
36C2×#11715895—21no—
37C2×#11715895—21no—
38C4⋊Dic97D411yescharacteristic
39#13095488—22no—
40#13095488—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97D411yescharacteristic
43C2×C8×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C42×C97—21no—
49C42×C97C2311yescharacteristic
50C22×D97—21no—
51C22×D97—22no—
52Q8×C97—21no—
53Q8×C97—41no—
54Q8×C97—41no—
55Q8×C97SD1612yescomplemented
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—41no—
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×Dic97—21no—
63D388—21no—
64D388—21no—
65D388—42no—
66D388—44no—
67D388—41no—
68D388—41no—
69D388—41no—
70D388—41no—
71D388—82no—
72C4×D97—44no—
73C4×D97—41no—
74C4×D97—41no—
75C8×C97—41no—
76#11715895—41no—
77#11715895—41no—
78C22×C97#12011yescharacteristic
79C2×D97—41no—
80C2×D97—42no—
81C2×D97—42no—
82C2×D97—41no—
83C2×D97—42no—
84C2×D97—41no—
85C4×C97—22no—
86C4×C97—21no—
87C4×C97—41no—
88C4×C97—81no—
89C4×C97C2×SD1611yescharacteristic
90C4×C97#13511yescharacteristic
91C4×C97AGL(1,ℤ/8)11yescharacteristic
92C4×C97AGL(1,ℤ/8)11yescharacteristic
93Dic97—41no—
94C2×C97#44911yescharacteristic
95C2×C97#45011yescharacteristic
96C2×C97#45511yescharacteristic
97D97—81no—
98D97—82no—
99#1642—971nomaximal, Sylow
100C97#164211yescharacteristic, complemented, Sylow
101#550—971no—
102#486—971no—
103#475D9711yescharacteristic, complemented
104#461—971no—
105#331—971no—
106Q8⋊C8—971no—
107D4⋊C8—971no—
108C2×#41—1941no—
109C2×SD16—1941no—
110C4⋊Q8C2×D9711yescharacteristic
111#127—971no—
112#127—1941no—
113#121—1942no—
114#121—1941no—
115C4×Q8C2×D9711yescharacteristic
116C4×D4—971no—
117C4×D4—1941no—
118#106—21no—
119C4⋊C8—971no—
120C4⋊C8C2×D9711yescharacteristic
121#103—21no—
122#103—1941no—
123#102—1942no—
124C4×C8—971no—
125#41—3882no—
126#41—3882no—
127C2×Q8—21no—
128C2×Q8C22⋊D9711yescharacteristic
129C2×D4—972no—
130C2×D4—971no—
131C2×D4—1942no—
132C2×D4—1942no—
133C2×D4—1941no—
134C2×D4—1941no—
135C22×C4—1942no—
136C22×C4—1941no—
137SD16—3882no—
138C2×C8—21no—
139C2×C8—1941no—
140C2×C8—1941no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—971no—
145C4⋊C4C22⋊D9711yescharacteristic
146C22⋊C4—1942no—
147C22⋊C4—1941no—
148C42—21no—
149C42C22×D9711yescharacteristic
150C23—1942no—
151C23—1941no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1309549812yescomplemented
156D4—7762no—
157D4—3882no—
158D4—3884no—
159D4—3881no—
160D4—3881no—
161D4—3881no—
162D4—3881no—
163D4—1941no—
164D4—1941no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—3884no—
169C2×C4—3881no—
170C2×C4—3881no—
171C2×C4—1941no—
172C2×C4D4×D9711yescharacteristic
173C2×C4D4×D9711yescharacteristic
174C2×C4C2×C22⋊D9711yescharacteristic
175C8—41no—
176C8—3881no—
177C8—3881no—
178C22—3882no—
179C22—3881no—
180C22—3882no—
181C22—3881no—
182C22—3881no—
183C22—3882no—
184C22#1569077811yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—81no—
189C4—3881no—
190C4C2×#1309549811yescharacteristic
191C4#1569083111yescharacteristic
192C4#1569083511yescharacteristic
193C4#1569083711yescharacteristic
194C2—7762no—
195C2—7761no—
196C2#2026087911yescharacteristic
197C2#2026088511yescharacteristic
198C2#2026091111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#475×C97C2
socleC22×C97#120
prime radical#20261407C2
hypercenter#475D97
hypocenterC97#1642
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #20261083C2semidirect
3. #20260937C2semidirect
4. #20260478C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #475×C97C2semidirect
55. Q8×C97SD16semidirect
100. C97#1642semidirect
103. #475D97semidirect
155. Q8#13095498semidirect

Maximal subgroups

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

subgroupindex
#202614072
#202610832
#202609372
#202604782
Q8⋊#117158952
#202604622
#475×C972
#164297