Finite Group Databasebeta

← #37571805

Subgroups — #37571805

Order 24448.

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

normal subgroups only

This group has 240 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#27133316C211yesmaximal, characteristic
2#27133059C211yesmaximal, characteristic, complemented
3#27132944C211yesmaximal, characteristic, complemented
4#27132869C211yesmaximal, characteristic, complemented
5#27132390C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic, complemented
7#332×C191C211yesmaximal, characteristic, complemented
8C2×#15690288—21no—
9C2×#15690276—21no—
10#20259322C2211yescharacteristic
11#20259307—22no—
12#20259298—21no—
13#20259297—21no—
14C4×D764C2211yescharacteristic
15C4×D764C2211yescharacteristic
16C4⋊C4×D191—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
25#15690288—42no—
26#15690288—42no—
27C2×Dic382—21no—
28C2×D764—22no—
29C2×D764D411yescharacteristic
30C2×D764D411yescharacteristic
31C2×C4×D191—22no—
32C2×C4×D191—21no—
33C2×C4×D191—21no—
34#15690276—42no—
35C2×C191⋊C8—21no—
36C4⋊Dic191—21no—
37C4⋊Dic191—21no—
38C4⋊Dic191D411yescharacteristic
39C4⋊Dic191D411yescharacteristic
40Dic191⋊C4—22no—
41(C2×D191)⋊C4—22no—
42(C2×D191)⋊C4—21no—
43(C2×D191)⋊C4—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—
51C22×D191—21no—
52Dic382—41no—
53Dic382—41no—
54D764—21no—
55D764—21no—
56D764—21no—
57D764—44no—
58D764SD1612yescomplemented
59C2×Dic191—21no—
60C2×Dic191—21no—
61C2×Dic191—41no—
62C4×D191—44no—
63C4×D191—41no—
64C4×D191—41no—
65C4×D191—41no—
66C4×D191—41no—
67C191⋊C8—22no—
68Q8×C191—41no—
69Q8×C191—41no—
70C2×C4×C191—22no—
71C2×C4×C191C2×D411yescharacteristic
72C2×C4×C191C2×D411yescharacteristic
73C2×C4×C191C2×D411yescharacteristic
74C8×C191—41no—
75C8×C191—41no—
76C22×C191#12011yescharacteristic
77C2×D191—22no—
78C2×D191—41no—
79C2×D191—41no—
80C2×D191—41no—
81C2×D191—41no—
82C2×D191—41no—
83C4×C191—21no—
84C4×C191—42no—
85C4×C191C2×SD1611yescharacteristic
86C4×C191#13511yescharacteristic
87C4×C191AGL(1,ℤ/8)11yescharacteristic
88C4×C191AGL(1,ℤ/8)11yescharacteristic
89Dic191—41no—
90Dic191—41no—
91Dic191—81no—
92C2×C191#44911yescharacteristic
93C2×C191#45011yescharacteristic
94C2×C191#45611yescharacteristic
95D191—42no—
96D191—81no—
97C191#165411yescharacteristic, complemented, Sylow
98#1654—1911nomaximal, Sylow
99#547—1911no—
100#498—1911no—
101#474—1911no—
102#461—1911no—
103#332D19111yescharacteristic, complemented
104D4⋊C8—1911no—
105D4⋊C8—1911no—
106C2×#41—3821no—
107C2×SD16—3821no—
108C4⋊Q8—1911no—
109C4⋊Q8C2×D19111yescharacteristic
110#123—3822no—
111C22⋊Q8—3821no—
112#121—3821no—
113C4×D4—1911no—
114C4×D4—1911no—
115C2×C4⋊C4—3821no—
116#107—3821no—
117#106—3822no—
118C4⋊C8C2×D19111yescharacteristic
119C4⋊C8C2×D19111yescharacteristic
120#103—3821no—
121#102—3821no—
122C4×C8—1911no—
123#41—7642no—
124#41—7642no—
125C2×Q8—21no—
126C2×Q8—3821no—
127C2×D4—3822no—
128C2×D4—1911no—
129C2×D4—1911no—
130C22×C4—3822no—
131C22×C4—3821no—
132C22×C4—3821no—
133SD16—7642no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—3821no—
137C4⋊C4—21no—
138C4⋊C4—3822no—
139C4⋊C4—3821no—
140C4⋊C4—3821no—
141C4⋊C4—1911no—
142C4⋊C4—1911no—
143C4⋊C4D76412yes—
144C22⋊C4—3822no—
145C22⋊C4—3821no—
146C22⋊C4—3821no—
147C42C22×D19111yescharacteristic
148C23—3821no—
149C23—3821no—
150Q8—41no—
151Q8—41no—
152Q8—7641no—
153Q8—7641no—
154D4—3821no—
155D4—3821no—
156D4—3821no—
157D4—1912no—
158D4—7644no—
159C2×C4—22no—
160C2×C4—3821no—
161C2×C4—3821no—
162C2×C4—7644no—
163C2×C4—7641no—
164C2×C4—7641no—
165C2×C4—7641no—
166C2×C4—7641no—
167C2×C4—7641no—
168C2×C4D4×D19111yescharacteristic
169C2×C4D4×D19111yescharacteristic
170C2×C4C2×D76411yescharacteristic
171C8—41no—
172C8—41no—
173C8—3822no—
174C22—3822no—
175C22—7641no—
176C22—7641no—
177C22—7641no—
178C22—7641no—
179C22—7641no—
180C22#2025929211yescharacteristic
181C4—21no—
182C4—42no—
183C4—15281no—
184C4—7641no—
185C4—7641no—
186C4SD16×D19111yescharacteristic
187C4#2025934511yescharacteristic
188C4#2025934611yescharacteristic
189C4#2025934911yescharacteristic
190C2—15281no—
191C2—7642no—
192C2#2713281111yescharacteristic
193C2#2713281511yescharacteristic
194C2#2713284811yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#332×C191C2
socleC22×C191#120
prime radical#27133316C2
hypercenter#332D191
hypocenterC191#1654
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
3. #27132944C2semidirect
4. #27132869C2semidirect
5. #27132390C2semidirect
6. #27132390C2semidirect
7. #332×C191C2semidirect
58. D764SD16semidirect
97. C191#1654semidirect
103. #332D191semidirect

Maximal subgroups

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

subgroupindex
#271333162
#271330592
#271329442
#271328692
#271323902
#332×C1912
#1654191