Finite Group Databasebeta

← #37571607

Subgroups — #37571607

Order 24448.

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

normal subgroups only

This group has 256 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#27133317C211yesmaximal, characteristic, complemented
2#27133009C211yesmaximal, characteristic, complemented
3#27132855C211yesmaximal, characteristic, complemented
4#27132410C211yesmaximal, characteristic, complemented
5D4⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic, complemented
7#459×C191C211yesmaximal, characteristic, complemented
8C2×#15690286—21no—
9C2×#15690272—21no—
10#20259322C2211yescharacteristic
11#20259306—22no—
12#20259302—21no—
13#20259296—21no—
14C4×D764C2211yescharacteristic
15C2×C4⋊Dic191—21no—
16C4⋊(C191⋊C8)C2211yescharacteristic
17#20259243—22no—
18#20259243—21no—
19C4×C191⋊C8C2211yescharacteristic
20C2×D8×C191—21no—
21#127×C191C2211yescharacteristic
22C4×D4×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24#102×C191—21no—
25#15690286—42no—
26#15690286—42no—
27C2×Dic382—21no—
28C2×D764D411yescharacteristic
29C2×C22⋊D191—22no—
30C22×Dic191—22no—
31C2×C4×D191—21no—
32#15690272—42no—
33C2×C191⋊C8—21no—
34C2×C191⋊C8—21no—
35C4⋊Dic191—22no—
36C4⋊Dic191—21no—
37C4⋊Dic191D411yescharacteristic
38C4⋊Dic191D412yes—
39(C2×D191)⋊C4—22no—
40(C2×D191)⋊C4—21no—
41C2×D4×C191—21no—
42C2×D4×C191—21no—
43C2×D4×C191—21no—
44C2×D4×C191D411yescharacteristic
45C22×C4×C191—21no—
46D8×C191—42no—
47C2×C8×C191—21no—
48C4⋊C4×C191D411yescharacteristic
49C22⋊C4×C191—21no—
50C42×C191C2311yescharacteristic
51C22×D191—21no—
52Dic382—41no—
53Dic382—41no—
54D764—21no—
55D764—21no—
56C22⋊D191—44no—
57C2×Dic191—21no—
58C2×Dic191—22no—
59C2×Dic191—44no—
60C4×D191—41no—
61C4×D191—41no—
62C191⋊C8—41no—
63C191⋊C8—41no—
64C23×C191—21no—
65C23×C191—41no—
66D4×C191—21no—
67D4×C191—42no—
68D4×C191—41no—
69D4×C191—41no—
70D4×C191—41no—
71D4×C191—41no—
72D4×C191D812yescomplemented
73C2×C4×C191—21no—
74C2×C4×C191—41no—
75C2×C4×C191—41no—
76C2×C4×C191C2×D411yescharacteristic
77C2×C4×C191C2×D411yescharacteristic
78C2×C4×C191C2×D411yescharacteristic
79C8×C191—41no—
80C22×C191—22no—
81C22×C191—41no—
82C22×C191—41no—
83C22×C191—81no—
84C22×C191—81no—
85C22×C191—81no—
86C22×C191#12011yescharacteristic
87C2×D191—41no—
88C2×D191—41no—
89C2×D191—41no—
90C4×C191—21no—
91C4×C191—41no—
92C4×C191C2×D811yescharacteristic
93C4×C191#13511yescharacteristic
94C4×C191AGL(1,ℤ/8)11yescharacteristic
95C4×C191AGL(1,ℤ/8)11yescharacteristic
96Dic191—41no—
97Dic191—42no—
98C2×C191—42no—
99C2×C191—161no—
100C2×C191#44711yescharacteristic
101C2×C191#44911yescharacteristic
102C2×C191#45411yescharacteristic
103D191—81no—
104C191#162511yescharacteristic, complemented, Sylow
105#1625—1911nomaximal, Sylow
106#547—1911no—
107#486—1911no—
108#459—1911no—
109#459D19111yescharacteristic, complemented
110#332—1911no—
111D4⋊C8—1911no—
112D4⋊C8—1911no—
113C2×#41—3821no—
114C2×D8—21no—
115C2×D8—3821no—
116C4⋊Q8—1911no—
117#127C2×D19111yescharacteristic
118#123—3822no—
119C22⋊Q8—3821no—
120#121—3821no—
121C4×D4—1911no—
122C4×D4C2×D19111yescharacteristic
123C2×C4⋊C4—3821no—
124C4⋊C8—1911no—
125C4⋊C8C2×D19111yescharacteristic
126#102—21no—
127#102—3821no—
128#102—3822no—
129C4×C8—1911no—
130#41—7642no—
131#41—7642no—
132C2×Q8—3821no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—3822no—
137C2×D4—1911no—
138C2×D4C22⋊D19111yescharacteristic
139C22×C4—21no—
140C22×C4—3822no—
141C22×C4—3821no—
142D8—42no—
143D8—7642no—
144C2×C8—21no—
145C2×C8—3821no—
146C2×C8—3821no—
147C4⋊C4—3822no—
148C4⋊C4—3821no—
149C4⋊C4—1911no—
150C4⋊C4—1912no—
151C4⋊C4C22⋊D19111yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3822no—
154C22⋊C4—3821no—
155C42C22×D19111yescharacteristic
156C23—21no—
157C23—41no—
158C23—3821no—
159Q8—7641no—
160Q8—7641no—
161D4—21no—
162D4—42no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—3821no—
168D4—3821no—
169D4—7644no—
170D4#1569027212yescomplemented
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—3821no—
175C2×C4—3822no—
176C2×C4—7644no—
177C2×C4—7641no—
178C2×C4—7641no—
179C2×C4D4×D19111yescharacteristic
180C2×C4D4×D19111yescharacteristic
181C2×C4C2×C22⋊D19111yescharacteristic
182C8—41no—
183C8—7641no—
184C8—7641no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—7641no—
192C22—7641no—
193C22—7641no—
194C22#2025929011yescharacteristic
195C4—21no—
196C4—41no—
197C4—7641no—
198C4—7642no—
199C4C2×#1569027211yescharacteristic
200C4#2025934111yescharacteristic
201C4#2025934611yescharacteristic
202C4#2025934711yescharacteristic
203C2—42no—
204C2—161no—
205C2—15281no—
206C2#2713279611yescharacteristic
207C2#2713280511yescharacteristic
208C2#2713283411yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#459×C191C2
socleC22×C191#120
hypercenter#459D191
hypocenterC191#1625
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #27133317C2semidirect
2. #27133009C2semidirect
3. #27132855C2semidirect
4. #27132410C2semidirect
5. D4⋊(C191⋊C8)C2semidirect
6. #27132390C2semidirect
7. #459×C191C2semidirect
72. D4×C191D8semidirect
104. C191#1625semidirect
109. #459D191semidirect
170. D4#15690272semidirect

Maximal subgroups

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

subgroupindex
#271333172
#271330092
#271328552
#271324102
D4⋊(C191⋊C8)2
#271323902
#459×C1912
#1625191