Finite Group Databasebeta

← #37571704

Subgroups — #37571704

Order 24448.

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

normal subgroups only

This group has 212 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#27133398C211yesmaximal, characteristic
2#27133011C211yesmaximal, characteristic
3#27132945C211yesmaximal, characteristic, complemented
4#27132406C211yesmaximal, characteristic
5Q8⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
6#27132393C211yesmaximal, characteristic, complemented
7#462×C191C211yesmaximal, characteristic, complemented
8#20259319C2211yescharacteristic
9#20259311—21no—
10#20259310—21no—
11#20259307—22no—
12C4×Dic382C2211yescharacteristic
13C4×D764—21no—
14#20259284—21no—
15#20259253—21no—
16C4⋊(C191⋊C8)C2211yescharacteristic
17#20259245—21no—
18#20259242—22no—
19C4×C191⋊C8C2211yescharacteristic
20C2×Q16×C191—21no—
21C4⋊Q8×C191C2211yescharacteristic
22C4×Q8×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24#103×C191—21no—
25C2×Dic382D411yescharacteristic
26C2×D764—21no—
27C2×D764—21no—
28C2×D764D412yes—
29C2×C4×D191—22no—
30C2×C191⋊C8—21no—
31C2×C191⋊C8—21no—
32C4⋊Dic191D411yescharacteristic
33Dic191⋊C4—21no—
34(C2×D191)⋊C4—22no—
35(C2×D191)⋊C4—22no—
36(C2×D191)⋊C4—22no—
37C4×Dic191—21no—
38C2×Q8×C191—21no—
39C2×Q8×C191D411yescharacteristic
40Q16×C191—42no—
41C2×C8×C191—21no—
42C4⋊C4×C191—21no—
43C4⋊C4×C191—21no—
44C4⋊C4×C191—21no—
45C4⋊C4×C191D411yescharacteristic
46C42×C191—21no—
47C42×C191C2311yescharacteristic
48C22×D191—22no—
49Dic382—21no—
50Dic382—21no—
51D764—42no—
52D764—41no—
53D764—41no—
54D764—41no—
55D764—41no—
56C2×Dic191—21no—
57C2×Dic191—21no—
58C4×D191—82no—
59C191⋊C8—41no—
60C191⋊C8—41no—
61Q8×C191—21no—
62Q8×C191—21no—
63Q8×C191—41no—
64Q8×C191—41no—
65C2×C4×C191—21no—
66C2×C4×C191—21no—
67C2×C4×C191—41no—
68C2×C4×C191C2×D411yescharacteristic
69C2×C4×C191C2×D411yescharacteristic
70C2×C4×C191C2×D411yescharacteristic
71C8×C191—41no—
72C22×C191#12011yescharacteristic
73C2×D191—42no—
74C2×D191—42no—
75C2×D191—42no—
76C4×C191—21no—
77C4×C191—41no—
78C4×C191—41no—
79C4×C191—81no—
80C4×C191#13511yescharacteristic
81C4×C191#13511yescharacteristic
82C4×C191AGL(1,ℤ/8)11yescharacteristic
83C4×C191#13711yescharacteristic
84Dic191—41no—
85Dic191—41no—
86C2×C191#44911yescharacteristic
87C2×C191#45211yescharacteristic
88C2×C191#45511yescharacteristic
89D191—82no—
90C191#164011yescharacteristic, complemented, Sylow
91#1640—1911nomaximal, Sylow
92#559—1911no—
93#486—1911no—
94#475—1911no—
95#462D19111yescharacteristic, complemented
96#331—1911no—
97Q8⋊C8—1911no—
98Q8⋊C8—1911no—
99C2×Q16—21no—
100C4⋊Q8C2×D19111yescharacteristic
101#127—1911no—
102#124—3821no—
103#124—3821no—
104#123—3822no—
105C4×Q8—1911no—
106C4×Q8C2×D19111yescharacteristic
107C4×D4—3821no—
108C4×D4—3821no—
109#106—3821no—
110C4⋊C8—1911no—
111C4⋊C8C2×D19111yescharacteristic
112#103—21no—
113#103—3821no—
114#102—3822no—
115C4×C8—1911no—
116C2×Q8—21no—
117C2×Q8—1911no—
118C2×Q8C22⋊D19111yescharacteristic
119C2×D4—3821no—
120C2×D4—3821no—
121C2×D4—1912no—
122C22×C4—3822no—
123Q16—42no—
124C2×C8—21no—
125C2×C8—3821no—
126C2×C8—3821no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—3821no—
131C4⋊C4—1911no—
132C4⋊C4C22⋊D19111yescharacteristic
133C22⋊C4—3822no—
134C22⋊C4—3822no—
135C22⋊C4—3822no—
136C42—21no—
137C42—3821no—
138C42C22×D19111yescharacteristic
139C23—3822no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—3821no—
145Q8—3821no—
146D4—7642no—
147D4—7641no—
148D4—7641no—
149D4—7641no—
150D4—7641no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—3821no—
155C2×C4—3821no—
156C2×C4—15282no—
157C2×C4D4×D19111yescharacteristic
158C2×C4D4×D19111yescharacteristic
159C2×C4C2×C22⋊D19111yescharacteristic
160C8—41no—
161C8—7641no—
162C8—7641no—
163C22—7642no—
164C22—7642no—
165C22—7642no—
166C22#2025929011yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—7641no—
172C4—7641no—
173C4#2025934211yescharacteristic
174C4#2025934511yescharacteristic
175C4#2025934811yescharacteristic
176C4#2025935411yescharacteristic
177C2—15282no—
178C2#2713280611yescharacteristic
179C2#2713282411yescharacteristic
180C2#2713283911yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#462×C191C2
socleC22×C191#120
prime radical#20259319C22
hypercenter#462D191
hypocenterC191#1640
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #27132945C2semidirect
5. Q8⋊(C191⋊C8)C2semidirect
6. #27132393C2semidirect
7. #462×C191C2semidirect
90. C191#1640semidirect
95. #462D191semidirect

Maximal subgroups

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

subgroupindex
#271333982
#271330112
#271329452
#271324062
Q8⋊(C191⋊C8)2
#271323932
#462×C1912
#1640191