Finite Group Databasebeta

← #37571764

Subgroups — #37571764

Order 24448.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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#27133391C211yesmaximal, characteristic
2#27133068C211yesmaximal, characteristic
3#27132949C211yesmaximal, characteristic
4#27132873C211yesmaximal, characteristic
5#27132408C211yesmaximal, characteristic
6#27132395C211yesmaximal, characteristic
7Q8⋊C8×C191C211yesmaximal, characteristic
8C2×#15690277—21no—
9#20259322C2211yescharacteristic
10#20259322C2211yescharacteristic
11#20259320—21no—
12#20259314—21no—
13#20259312—22no—
14C4×Dic382—21no—
15C4×Dic382C2211yescharacteristic
16#20259287—21no—
17#20259256—22no—
18#20259254—21no—
19C4⋊(C191⋊C8)C2211yescharacteristic
20#20259247—21no—
21#20259245—21no—
22C4×Q8×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4×C8×C191C2211yescharacteristic
25C2×Dic382—21no—
26C2×Dic382—21no—
27C2×Dic382D411yescharacteristic
28#15690277—42no—
29C2×C191⋊C8—21no—
30C4⋊Dic191—21no—
31C4⋊Dic191—21no—
32C4⋊Dic191—21no—
33C4⋊Dic191D411yescharacteristic
34C4⋊Dic191D412yes—
35Dic191⋊C4—22no—
36Dic191⋊C4—22no—
37Dic191⋊C4—22no—
38Dic191⋊C4—21no—
39C4×Dic191—22no—
40C4×Dic191—21no—
41C2×Q8×C191D411yescharacteristic
42C2×C8×C191—21no—
43C2×C8×C191—21no—
44C4⋊C4×C191—21no—
45C4⋊C4×C191D411yescharacteristic
46C42×C191—21no—
47C42×C191C2311yescharacteristic
48Dic382—21no—
49Dic382—41no—
50Dic382—41no—
51Dic382—41no—
52Dic382—41no—
53Dic382Q1612yescomplemented
54C2×Dic191—21no—
55C2×Dic191—21no—
56C2×Dic191—22no—
57C2×Dic191—41no—
58C191⋊C8—41no—
59Q8×C191—21no—
60Q8×C191—21no—
61C2×C4×C191—21no—
62C2×C4×C191—21no—
63C2×C4×C191C2×D411yescharacteristic
64C2×C4×C191C2×D411yescharacteristic
65C2×C4×C191C2×D411yescharacteristic
66C8×C191—22no—
67C8×C191—41no—
68C22×C191#12011yescharacteristic
69C4×C191—21no—
70C4×C191—41no—
71C4×C191—41no—
72C4×C191C2×Q1611yescharacteristic
73C4×C191#13511yescharacteristic
74C4×C191AGL(1,ℤ/8)11yescharacteristic
75C4×C191#13711yescharacteristic
76Dic191—22no—
77Dic191—42no—
78Dic191—41no—
79Dic191—81no—
80C2×C191#44911yescharacteristic
81C2×C191#45111yescharacteristic
82C2×C191#45611yescharacteristic
83C191#164811yescharacteristic, complemented, Sylow
84#1648—1911nomaximal, Sylow
85#557—1911no—
86#500—1911no—
87#475—1911no—
88#462—1911no—
89#332—1911no—
90Q8⋊C8—1911no—
91Q8⋊C8D19111yescharacteristic
92C2×Q16—3821no—
93C4⋊Q8—3821no—
94C4⋊Q8—1911no—
95C4⋊Q8—1911no—
96#125—3821no—
97#125—3822no—
98C4×Q8—3821no—
99C4×Q8—3821no—
100C4×Q8—1911no—
101C4×Q8C2×D19111yescharacteristic
102#107—3822no—
103#106—3821no—
104C4⋊C8—1911no—
105C4⋊C8C2×D19111yescharacteristic
106#103—3821no—
107#103—3821no—
108C4×C8C2×D19111yescharacteristic
109C2×Q8—3821no—
110C2×Q8—3821no—
111C2×Q8—1911no—
112C2×Q8D76411yescharacteristic
113Q16—7642no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—3821no—
117C4⋊C4—21no—
118C4⋊C4—3822no—
119C4⋊C4—3822no—
120C4⋊C4—3822no—
121C4⋊C4—3821no—
122C4⋊C4—3821no—
123C4⋊C4—3821no—
124C4⋊C4—3821no—
125C4⋊C4—1911no—
126C4⋊C4—1912no—
127C4⋊C4D76411yescharacteristic
128C42—21no—
129C42—3822no—
130C42—3821no—
131C42C22×D19111yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—3821no—
135Q8—1912no—
136Q8—7641no—
137Q8—7641no—
138Q8—7641no—
139Q8—7641no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—3821no—
143C2×C4—3821no—
144C2×C4—3822no—
145C2×C4—7641no—
146C2×C4D4×D19111yescharacteristic
147C2×C4D4×D19111yescharacteristic
148C2×C4C2×D76411yescharacteristic
149C8—22no—
150C8—41no—
151C8—7641no—
152C22#2025929211yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—3822no—
157C4—15281no—
158C4—7642no—
159C4—7641no—
160C4Q16×D19111yescharacteristic
161C4#2025934411yescharacteristic
162C4#2025935011yescharacteristic
163C4#2025935111yescharacteristic
164C2#2713281011yescharacteristic
165C2#2713282011yescharacteristic
166C2#2713284711yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
FittingQ8⋊C8×C191C2
socle, prime radicalC22×C191#120
hypercenterQ8⋊C8D191
hypocenterC191#1648
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
53. Dic382Q16semidirect
83. C191#1648semidirect

Maximal subgroups

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

subgroupindex
#271333912
#271330682
#271329492
#271328732
#271324082
#271323952
Q8⋊C8×C1912
#1648191