Finite Group Databasebeta

← #37571791

Subgroups — #37571791

Order 24448.

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

normal subgroups only

This group has 200 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#27133387C211yesmaximal, characteristic
2#27133059C211yesmaximal, characteristic, complemented
3#27132944C211yesmaximal, characteristic
4#27132874C211yesmaximal, characteristic, complemented
5#27132393C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic
7#332×C191C211yesmaximal, characteristic, complemented
8C2×Dic764—21no—
9#20259322C2211yescharacteristic
10#20259317—22no—
11#20259311—21no—
12#20259302—21no—
13Q8×Dic191—21no—
14C4×Dic382C2211yescharacteristic
15C4×D764C2211yescharacteristic
16#20259280—21no—
17#20259256—21no—
18#20259253—22no—
19#20259247—21no—
20#20259244—21no—
21C4×C191⋊C8C2211yescharacteristic
22C4⋊Q8×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4⋊C8×C191C2211yescharacteristic
25C2×Dic382—21no—
26C2×Dic382D411yescharacteristic
27C2×D764D411yescharacteristic
28C2×C4×D191—21no—
29Dic764—42no—
30C2×C191⋊C8—21no—
31C4⋊Dic191—22no—
32C4⋊Dic191—21no—
33C4⋊Dic191—21no—
34C4⋊Dic191D411yescharacteristic
35C4⋊Dic191D411yescharacteristic
36Dic191⋊C4—21no—
37(C2×D191)⋊C4—22no—
38(C2×D191)⋊C4—22no—
39(C2×D191)⋊C4—21no—
40C4×Dic191—22no—
41C4×Dic191—21no—
42C2×Q8×C191—21no—
43C2×C8×C191—21no—
44C2×C8×C191—21no—
45C4⋊C4×C191—21no—
46C4⋊C4×C191D412yes—
47C42×C191C2311yescharacteristic
48C22×D191—21no—
49Dic382—21no—
50Dic382—21no—
51Dic382—41no—
52Dic382—41no—
53D764—21no—
54D764—21no—
55C2×Dic191—21no—
56C2×Dic191—21no—
57C2×Dic191—21no—
58C2×Dic191—41no—
59C4×D191—41no—
60C4×D191—41no—
61C191⋊C8—22no—
62Q8×C191—41no—
63Q8×C191—41no—
64C2×C4×C191—22no—
65C2×C4×C191C2×D411yescharacteristic
66C2×C4×C191C2×D411yescharacteristic
67C2×C4×C191C2×D411yescharacteristic
68C8×C191—41no—
69C8×C191—41no—
70C22×C191#12011yescharacteristic
71C2×D191—41no—
72C2×D191—41no—
73C2×D191—41no—
74C4×C191—21no—
75C4×C191—42no—
76C4×C191#13511yescharacteristic
77C4×C191#13511yescharacteristic
78C4×C191AGL(1,ℤ/8)11yescharacteristic
79C4×C191#13711yescharacteristic
80Dic191—41no—
81Dic191—41no—
82Dic191—41no—
83Dic191—81no—
84C2×C191#44911yescharacteristic
85C2×C191#45211yescharacteristic
86C2×C191#45611yescharacteristic
87D191—81no—
88C191#165211yescharacteristic, complemented, Sylow
89#1652—1911nomaximal, Sylow
90#556—1911no—
91#498—1911no—
92#474—1911no—
93#462—1911no—
94#332D19111yescharacteristic, complemented
95Q8⋊C8—1911no—
96D4⋊C8—1911no—
97C2×Q16—3821no—
98C4⋊Q8—1911no—
99C4⋊Q8C2×D19111yescharacteristic
100#126—3822no—
101#124—3821no—
102C22⋊Q8—3821no—
103C4×Q8—3821no—
104C4×Q8—1911no—
105C4×D4—1911no—
106#117—3821no—
107#107—3821no—
108#106—3822no—
109C4⋊C8C2×D19111yescharacteristic
110C4⋊C8C2×D19111yescharacteristic
111#103—3821no—
112#102—3821no—
113C4×C8—1911no—
114C2×Q8—21no—
115C2×Q8—3821no—
116C2×Q8—1911no—
117C2×D4—1911no—
118C22×C4—3821no—
119Q16—7642no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3821no—
123C4⋊C4—21no—
124C4⋊C4—3822no—
125C4⋊C4—3821no—
126C4⋊C4—3821no—
127C4⋊C4—3821no—
128C4⋊C4—1911no—
129C4⋊C4—1911no—
130C4⋊C4D76412yes—
131C22⋊C4—3822no—
132C22⋊C4—3822no—
133C22⋊C4—3821no—
134C42—3822no—
135C42—3821no—
136C42C22×D19111yescharacteristic
137C23—3821no—
138Q8—41no—
139Q8—41no—
140Q8—3821no—
141Q8—3821no—
142Q8—7641no—
143Q8—7641no—
144D4—3821no—
145D4—3821no—
146C2×C4—22no—
147C2×C4—3821no—
148C2×C4—3821no—
149C2×C4—3821no—
150C2×C4—7641no—
151C2×C4—7641no—
152C2×C4—7641no—
153C2×C4D4×D19111yescharacteristic
154C2×C4D4×D19111yescharacteristic
155C2×C4C2×D76411yescharacteristic
156C8—41no—
157C8—41no—
158C8—3822no—
159C22—7641no—
160C22—7641no—
161C22—7641no—
162C22#2025929211yescharacteristic
163C4—21no—
164C4—42no—
165C4—15281no—
166C4—7641no—
167C4—7641no—
168C4—7641no—
169C4#2025934111yescharacteristic
170C4#2025934511yescharacteristic
171C4#2025934911yescharacteristic
172C4#2025935111yescharacteristic
173C2—15281no—
174C2#2713281111yescharacteristic
175C2#2713282611yescharacteristic
176C2#2713284811yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#332×C191C2
socleC22×C191#120
prime radicalC2×D764D4
hypercenter#332D191
hypocenterC191#1652
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27133059C2semidirect
4. #27132874C2semidirect
5. #27132393C2semidirect
7. #332×C191C2semidirect
88. C191#1652semidirect
94. #332D191semidirect

Maximal subgroups

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

subgroupindex
#271333872
#271330592
#271329442
#271328742
#271323932
#271323902
#332×C1912
#1652191