Finite Group Databasebeta

← #37571694

Subgroups — #37571694

Order 24448.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#27133383C211yesmaximal, characteristic
2#27133014C211yesmaximal, characteristic, complemented
3#27132959C211yesmaximal, characteristic, complemented
4#27132953C211yesmaximal, characteristic
5#27132408C211yesmaximal, characteristic, complemented
6#27132395C211yesmaximal, characteristic, complemented
7D4⋊C8×C191C211yesmaximal, characteristic
8#20259322C2211yescharacteristic
9#20259322C2211yescharacteristic
10#20259315—22no—
11#20259309—21no—
12#20259301—21no—
13C4×Dic382C2211yescharacteristic
14#20259287—21no—
15#20259281—21no—
16#20259256—21no—
17#20259253—21no—
18C4⋊(C191⋊C8)C2211yescharacteristic
19#20259247—21no—
20#20259247—22no—
21#20259243—21no—
22C4×D4×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4×C8×C191C2211yescharacteristic
25C2×Dic382—21no—
26C2×Dic382—21no—
27C2×Dic382D411yescharacteristic
28C2×C191⋊C8—21no—
29C4⋊Dic191—21no—
30C4⋊Dic191—21no—
31C4⋊Dic191—21no—
32C4⋊Dic191D412yes—
33C4⋊Dic191D411yescharacteristic
34Dic191⋊C4—22no—
35Dic191⋊C4—21no—
36C22⋊Dic191—22no—
37C22⋊Dic191—22no—
38C4×Dic191—22no—
39C4×Dic191—21no—
40C2×D4×C191D411yescharacteristic
41C22×C4×C191—21no—
42C2×C8×C191—21no—
43C2×C8×C191—21no—
44C4⋊C4×C191D411yescharacteristic
45C22⋊C4×C191—21no—
46C42×C191C2311yescharacteristic
47Dic382—21no—
48Dic382—21no—
49Dic382—41no—
50Dic382—41no—
51Dic382—42no—
52C2×Dic191—21no—
53C2×Dic191—21no—
54C2×Dic191—22no—
55C2×Dic191—41no—
56C191⋊C8—41no—
57C23×C191—21no—
58D4×C191—21no—
59D4×C191—21no—
60C2×C4×C191—21no—
61C2×C4×C191—41no—
62C2×C4×C191—41no—
63C2×C4×C191C2×D411yescharacteristic
64C2×C4×C191C2×D411yescharacteristic
65C2×C4×C191C2×D411yescharacteristic
66C8×C191—41no—
67C8×C191—41no—
68C22×C191—41no—
69C22×C191—41no—
70C22×C191—41no—
71C22×C191#12011yescharacteristic
72C4×C191—21no—
73C4×C191—41no—
74C4×C191#13511yescharacteristic
75C4×C191#13511yescharacteristic
76C4×C191#13711yescharacteristic
77C4×C191#13711yescharacteristic
78Dic191—41no—
79Dic191—42no—
80Dic191—41no—
81Dic191—81no—
82C2×C191—81no—
83C2×C191#45211yescharacteristic
84C2×C191#45211yescharacteristic
85C2×C191#45411yescharacteristic
86C191#163811yescharacteristic, complemented, Sylow
87#1638—1911nomaximal, Sylow
88#556—1911no—
89#487—1911no—
90#477—1911no—
91#476—1911no—
92#332—1911no—
93Q8⋊C8—1911no—
94D4⋊C8D19111yescharacteristic
95C4⋊Q8—1911no—
96C4⋊Q8—1911no—
97#126—3822no—
98#124—3821no—
99C22⋊Q8—3821no—
100C4×Q8—3821no—
101C4×Q8—1911no—
102C4×D4C2×D19111yescharacteristic
103#117—3821no—
104#107—3821no—
105#106—3821no—
106C4⋊C8—1911no—
107C4⋊C8C2×D19111yescharacteristic
108#103—3821no—
109#103—3822no—
110#102—3821no—
111C4×C8C2×D19111yescharacteristic
112C2×Q8—3821no—
113C2×Q8—3821no—
114C2×Q8—1911no—
115C2×D4D76411yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3821no—
120C4⋊C4—3822no—
121C4⋊C4—3821no—
122C4⋊C4—3821no—
123C4⋊C4—3821no—
124C4⋊C4—3821no—
125C4⋊C4—1912no—
126C4⋊C4—1911no—
127C4⋊C4D76411yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—3822no—
130C22⋊C4—3822no—
131C42—3822no—
132C42—3821no—
133C42C22×D19111yescharacteristic
134C23—21no—
135Q8—3821no—
136Q8—3821no—
137Q8—7642no—
138Q8—7641no—
139Q8—7641no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—3821no—
146C2×C4—3821no—
147C2×C4—3822no—
148C2×C4—7641no—
149C2×C4D4×D19111yescharacteristic
150C2×C4D4×D19111yescharacteristic
151C2×C4C2×D76411yescharacteristic
152C8—41no—
153C8—41no—
154C8—7641no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#2025929211yescharacteristic
159C4—21no—
160C4—41no—
161C4—15281no—
162C4—7641no—
163C4—7642no—
164C4—7641no—
165C4#2025934111yescharacteristic
166C4#2025934411yescharacteristic
167C4#2025935111yescharacteristic
168C4#2025935511yescharacteristic
169C2—81no—
170C2#2713282611yescharacteristic
171C2#2713282711yescharacteristic
172C2#2713283711yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
FittingD4⋊C8×C191C2
socleC22×C191#120
prime radicalC2×D4×C191D4
hypercenterD4⋊C8D191
hypocenterC191#1638
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27133014C2semidirect
3. #27132959C2semidirect
5. #27132408C2semidirect
6. #27132395C2semidirect
86. C191#1638semidirect

Maximal subgroups

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

subgroupindex
#271333832
#271330142
#271329592
#271329532
#271324082
#271323952
D4⋊C8×C1912
#1638191