Finite Group Databasebeta

← #37571721

Subgroups — #37571721

Order 24448.

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

normal subgroups only

This group has 250 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#27133333C211yesmaximal, characteristic
2#27133009C211yesmaximal, characteristic, complemented
3#27132949C211yesmaximal, characteristic, complemented
4#27132869C211yesmaximal, characteristic, complemented
5#27132393C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic, complemented
7#331×C191C211yesmaximal, characteristic, complemented
8C2×#15690286—21no—
9C2×#15690276—21no—
10#20259322C2211yescharacteristic
11#20259318—21no—
12#20259294—22no—
13#20259293—21no—
14C4×Dic382C2211yescharacteristic
15C4×D764C2211yescharacteristic
16D4×Dic191—21no—
17#20259254—21no—
18#20259247—21no—
19#20259247—21no—
20#20259243—22no—
21C4×C191⋊C8C2211yescharacteristic
22#127×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4⋊C8×C191C2211yescharacteristic
25#15690286—42no—
26#15690286—42no—
27C2×Dic382—21no—
28C2×Dic382D411yescharacteristic
29C2×D764D411yescharacteristic
30C2×C22⋊D191—22no—
31C2×C22⋊D191—22no—
32C22×Dic191—22no—
33C2×C4×D191—21no—
34#15690276—42no—
35C2×C191⋊C8—21no—
36C4⋊Dic191—21no—
37C4⋊Dic191—21no—
38C4⋊Dic191D411yescharacteristic
39C4⋊Dic191D411yescharacteristic
40Dic191⋊C4—21no—
41C22⋊Dic191—22no—
42(C2×D191)⋊C4—21no—
43C4×Dic191—21no—
44C2×D4×C191—21no—
45C2×D4×C191—21no—
46C2×D4×C191D412yes—
47C2×C8×C191—21no—
48C2×C8×C191—21no—
49C42×C191C2311yescharacteristic
50C22×D191—21no—
51Dic382—21no—
52Dic382—41no—
53Dic382—41no—
54Dic382SD1612yescomplemented
55D764—21no—
56D764—21no—
57C22⋊D191—44no—
58C22⋊D191—82no—
59C2×Dic191—21no—
60C2×Dic191—21no—
61C2×Dic191—21no—
62C2×Dic191—44no—
63C2×Dic191—41no—
64C4×D191—41no—
65C4×D191—41no—
66C191⋊C8—41no—
67C23×C191—22no—
68D4×C191—42no—
69D4×C191—41no—
70D4×C191—41no—
71D4×C191—41no—
72D4×C191—41no—
73C2×C4×C191C2×D411yescharacteristic
74C2×C4×C191C2×D411yescharacteristic
75C2×C4×C191C2×D411yescharacteristic
76C8×C191—41no—
77C8×C191—41no—
78C22×C191—42no—
79C22×C191—42no—
80C22×C191—42no—
81C22×C191#12011yescharacteristic
82C2×D191—41no—
83C2×D191—41no—
84C2×D191—41no—
85C4×C191—21no—
86C4×C191C2×SD1611yescharacteristic
87C4×C191#13511yescharacteristic
88C4×C191AGL(1,ℤ/8)11yescharacteristic
89C4×C191AGL(1,ℤ/8)11yescharacteristic
90Dic191—22no—
91Dic191—41no—
92Dic191—41no—
93Dic191—81no—
94C2×C191—82no—
95C2×C191#44911yescharacteristic
96C2×C191#45011yescharacteristic
97C2×C191#45511yescharacteristic
98D191—81no—
99C191#164211yescharacteristic, complemented, Sylow
100#1642—1911nomaximal, Sylow
101#550—1911no—
102#486—1911no—
103#475—1911no—
104#461—1911no—
105#331D19111yescharacteristic, complemented
106Q8⋊C8—1911no—
107D4⋊C8—1911no—
108C2×#41—3821no—
109C2×SD16—3821no—
110C4⋊Q8—1911no—
111#127—3821no—
112#127C2×D19111yescharacteristic
113#121—3822no—
114#121—3821no—
115C4×Q8—1911no—
116C4×D4—3821no—
117C4×D4—1911no—
118#106—3821no—
119C4⋊C8C2×D19111yescharacteristic
120C4⋊C8C2×D19111yescharacteristic
121#103—3821no—
122#103—3821no—
123#102—3822no—
124C4×C8—1911no—
125#41—7642no—
126#41—7642no—
127C2×Q8—3821no—
128C2×Q8—1911no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3822no—
132C2×D4—3822no—
133C2×D4—1911no—
134C2×D4D76412yes—
135C22×C4—3822no—
136C22×C4—3821no—
137SD16—7642no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3821no—
141C4⋊C4—3821no—
142C4⋊C4—3821no—
143C4⋊C4—3821no—
144C4⋊C4—1911no—
145C4⋊C4—1911no—
146C22⋊C4—3822no—
147C22⋊C4—3821no—
148C42—3821no—
149C42C22×D19111yescharacteristic
150C23—22no—
151C23—3821no—
152Q8—3821no—
153Q8—1912no—
154Q8—7641no—
155Q8—7641no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—3821no—
162D4—3821no—
163D4—15282no—
164D4—7644no—
165C2×C4—3821no—
166C2×C4—3821no—
167C2×C4—3821no—
168C2×C4—7644no—
169C2×C4—7641no—
170C2×C4—7641no—
171C2×C4—7641no—
172C2×C4D4×D19111yescharacteristic
173C2×C4D4×D19111yescharacteristic
174C2×C4C2×D76411yescharacteristic
175C8—41no—
176C8—41no—
177C8—7641no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—7641no—
182C22—7641no—
183C22—7641no—
184C22#2025929211yescharacteristic
185C4—21no—
186C4—3822no—
187C4—15281no—
188C4—7641no—
189C4—7641no—
190C4SD16×D19111yescharacteristic
191C4#2025934311yescharacteristic
192C4#2025934611yescharacteristic
193C4#2025934611yescharacteristic
194C2—82no—
195C2—15281no—
196C2#2713280911yescharacteristic
197C2#2713281511yescharacteristic
198C2#2713284311yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#331×C191C2
socleC22×C191#120
prime radical#27133333C2
hypercenter#331D191
hypocenterC191#1642
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27133009C2semidirect
3. #27132949C2semidirect
4. #27132869C2semidirect
5. #27132393C2semidirect
6. #27132390C2semidirect
7. #331×C191C2semidirect
54. Dic382SD16semidirect
99. C191#1642semidirect
105. #331D191semidirect

Maximal subgroups

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

subgroupindex
#271333332
#271330092
#271329492
#271328692
#271323932
#271323902
#331×C1912
#1642191