Finite Group Databasebeta

← #37571770

Subgroups — #37571770

Order 24448.

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

normal subgroups only

This group has 234 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#27133328C211yesmaximal, characteristic
2#27133066C211yesmaximal, characteristic, complemented
3#27132944C211yesmaximal, characteristic
4#27132874C211yesmaximal, characteristic, complemented
5#27132393C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic
7#332×C191C211yesmaximal, characteristic, complemented
8C2×Q8×D191—21no—
9C2×Dic764—21no—
10#20259322C2211yescharacteristic
11#20259321—21no—
12#20259300—22no—
13#20259298—21no—
14C4×Dic382C2211yescharacteristic
15C4×D764C2211yescharacteristic
16#20259284—21no—
17#20259256—21no—
18#20259255—22no—
19#20259247—21no—
20#20259244—21no—
21C4×C191⋊C8C2211yescharacteristic
22C4⋊Q8×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4⋊C8×C191C2211yescharacteristic
25C2×Dic382—22no—
26C2×Dic382—21no—
27C2×Dic382D411yescharacteristic
28Q8×D191—42no—
29Q8×D191—42no—
30C2×D764D411yescharacteristic
31C2×C4×D191—22no—
32C2×C4×D191—21no—
33Dic764—42no—
34C2×C191⋊C8—21no—
35C4⋊Dic191—21no—
36C4⋊Dic191—21no—
37C4⋊Dic191D411yescharacteristic
38C4⋊Dic191D411yescharacteristic
39Dic191⋊C4—22no—
40Dic191⋊C4—21no—
41(C2×D191)⋊C4—22no—
42(C2×D191)⋊C4—21no—
43C4×Dic191—21no—
44C2×Q8×C191—21no—
45C2×C8×C191—21no—
46C2×C8×C191—21no—
47C4⋊C4×C191—21no—
48C4⋊C4×C191D412yes—
49C42×C191C2311yescharacteristic
50C22×D191—21no—
51Dic382—21no—
52Dic382—44no—
53Dic382—41no—
54Dic382—41no—
55Dic382D812yes—
56D764—21no—
57D764Q1612yescomplemented
58C2×Dic191—21no—
59C2×Dic191—21no—
60C2×Dic191—21no—
61C2×Dic191—41no—
62C4×D191—44no—
63C4×D191—41no—
64C4×D191—41no—
65C191⋊C8—22no—
66Q8×C191—41no—
67Q8×C191—41no—
68C2×C4×C191—22no—
69C2×C4×C191C2×D411yescharacteristic
70C2×C4×C191C2×D411yescharacteristic
71C2×C4×C191C2×D411yescharacteristic
72C8×C191—41no—
73C8×C191—41no—
74C22×C191#12011yescharacteristic
75C2×D191—22no—
76C2×D191—41no—
77C2×D191—41no—
78C4×C191—21no—
79C4×C191—42no—
80C4×C191C2×D811yescharacteristic
81C4×C191C2×Q1611yescharacteristic
82C4×C191AGL(1,ℤ/8)11yescharacteristic
83C4×C191#13711yescharacteristic
84Dic191—22no—
85Dic191—41no—
86Dic191—41no—
87Dic191—81no—
88C2×C191#44711yescharacteristic
89C2×C191#45111yescharacteristic
90C2×C191#45611yescharacteristic
91D191—42no—
92C191#164911yescharacteristic, complemented, Sylow
93#1649—1911nomaximal, Sylow
94D4×Q8—1911no—
95#500—1911no—
96#474—1911no—
97#462—1911no—
98#332D19111yescharacteristic, complemented
99Q8⋊C8—1911no—
100D4⋊C8—1911no—
101C22×Q8—3821no—
102C2×Q16—3821no—
103C4⋊Q8—3821no—
104C4⋊Q8—1911no—
105C4⋊Q8C2×D19111yescharacteristic
106C22⋊Q8—3821no—
107C22⋊Q8—3822no—
108C4×Q8—1911no—
109C4×D4—3821no—
110C4×D4—1911no—
111#107—3821no—
112#107—3822no—
113C4⋊C8C2×D19111yescharacteristic
114C4⋊C8C2×D19111yescharacteristic
115#103—3821no—
116#102—3821no—
117C4×C8—1911no—
118C2×Q8—21no—
119C2×Q8—3822no—
120C2×Q8—3821no—
121C2×Q8—1911no—
122C2×Q8—7642no—
123C2×Q8—7642no—
124C2×D4—1911no—
125C22×C4—3822no—
126C22×C4—3821no—
127Q16—7642no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3821no—
131C4⋊C4—21no—
132C4⋊C4—3822no—
133C4⋊C4—3821no—
134C4⋊C4—3821no—
135C4⋊C4—3821no—
136C4⋊C4—1911no—
137C4⋊C4—1911no—
138C4⋊C4D76412yes—
139C22⋊C4—3822no—
140C22⋊C4—3821no—
141C42—3821no—
142C42C22×D19111yescharacteristic
143C23—3821no—
144Q8—41no—
145Q8—41no—
146Q8—3821no—
147Q8—1912no—
148Q8—7644no—
149Q8—7641no—
150Q8—7641no—
151D4—3821no—
152D4—1912no—
153C2×C4—22no—
154C2×C4—3821no—
155C2×C4—3821no—
156C2×C4—3821no—
157C2×C4—7644no—
158C2×C4—7641no—
159C2×C4—7641no—
160C2×C4—7641no—
161C2×C4D4×D19111yescharacteristic
162C2×C4D4×D19111yescharacteristic
163C2×C4C2×D76411yescharacteristic
164C8—41no—
165C8—41no—
166C8—3822no—
167C22—3822no—
168C22—7641no—
169C22—7641no—
170C22#2025929211yescharacteristic
171C4—21no—
172C4—42no—
173C4—3822no—
174C4—15281no—
175C4—7641no—
176C4—7641no—
177C4D8×D19111yescharacteristic
178C4Q16×D19111yescharacteristic
179C4#2025934911yescharacteristic
180C4#2025935111yescharacteristic
181C2—7642no—
182C2#2713279811yescharacteristic
183C2#2713282011yescharacteristic
184C2#2713284811yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#332×C191C2
socleC22×C191#120
prime radicalC2×D764D4
hypercenter#332D191
hypocenterC191#1649
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. #27133066C2semidirect
4. #27132874C2semidirect
5. #27132393C2semidirect
7. #332×C191C2semidirect
57. D764Q16semidirect
92. C191#1649semidirect
98. #332D191semidirect

Maximal subgroups

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

subgroupindex
#271333282
#271330662
#271329442
#271328742
#271323932
#271323902
#332×C1912
#1649191