Finite Group Databasebeta

← #37583469

Subgroups — #37583469

Order 24448.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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#27133218C211yesmaximal, characteristic
2C4⋊Q8×D191C211yesmaximal, characteristic
3#27133043C211yesmaximal, characteristic, complemented
4#27133042C211yesmaximal, characteristic, complemented
5#27133015C211yesmaximal, characteristic, complemented
6#27132872C214yesmaximal, complemented
7#27132820C214yesmaximal
8#27132714C211yesmaximal, characteristic
9#487×C191C211yesmaximal, characteristic, complemented
10C2×Q8×D191C2212yes—
11C2×Dic764C2214yes—
12C2×#15690277C2214yes—
13#20259322C2211yescharacteristic
14#20259322C2211yescharacteristic
15#20259321—21no—
16#20259321—21no—
17#20259320C2212yes—
18#20259313—21no—
19#20259302C2212yes—
20#20259298—21no—
21Q8×Dic191—21no—
22#20259287C2212yes—
23#20259280—21no—
24C4⋊C4×D191—21no—
25C4⋊C4×D191—21no—
26C42×D191C2211yescharacteristic
27Dic191⋊C8C2212yes—
28#20259245C2214yes—
29(C2×D191)⋊C8C2212yes—
30C4×C191⋊C8C2211yescharacteristic
31C4⋊Q8×C191C2211yescharacteristic
32C4⋊Q8×C191C2211yescharacteristic
33#103×C191C2214yes—
34C4×C8×C191C2211yescharacteristic
35C2×Dic382—22no—
36C2×Dic382C2312yes—
37C2×Dic382C2312yes—
38Q8×D191—44no—
39C2×C4×D191—22no—
40C2×C4×D191D411yescharacteristic
41C2×C4×D191D411yescharacteristic
42C2×C4×D191C2311yescharacteristic
43Dic764—44no—
44#15690277—44no—
45C2×C191⋊C8C2312yes—
46C4⋊Dic191—22no—
47C4⋊Dic191—21no—
48C4⋊Dic191—21no—
49C4⋊Dic191—21no—
50C4⋊Dic191—21no—
51Dic191⋊C4—22no—
52Dic191⋊C4—22no—
53Dic191⋊C4—22no—
54(C2×D191)⋊C4—22no—
55C4×Dic191—22no—
56C4×Dic191D411yescharacteristic
57C4×Dic191D411yescharacteristic
58C4×Dic191C2311yescharacteristic
59C2×Q8×C191—21no—
60C2×Q8×C191C2312yes—
61C2×C8×C191C2312yes—
62C4⋊C4×C191—21no—
63C4⋊C4×C191—21no—
64C4⋊C4×C191—21no—
65C4⋊C4×C191C2312yes—
66C42×C191C2311yescharacteristic
67C22×D191C2×D411yescharacteristic
68Dic382—24no—
69Dic382—24no—
70Dic382—44no—
71C2×Dic191—22no—
72C2×Dic191—22no—
73C2×Dic191C2×D411yescharacteristic
74C2×Dic191C2×D411yescharacteristic
75C2×Dic191C2×D411yescharacteristic
76C4×D191—22no—
77C4×D191—22no—
78C4×D191—44no—
79C4×D191Q1614yescomplemented
80C191⋊C8—22no—
81Q8×C191—24no—
82Q8×C191—42no—
83C2×C4×C191—22no—
84C2×C4×C191—22no—
85C2×C4×C191C2×D411yescharacteristic
86C2×C4×C191C2×D411yescharacteristic
87C2×C4×C191C2411yescharacteristic
88C8×C191—22no—
89C22×C191C22×D411yescharacteristic
90C2×D191—21no—
91C2×D191—21no—
92C2×D191C2×Q1612yes—
93C4×C191—42no—
94C4×C191—42no—
95C4×C191C2×Q1612yes—
96C4×C191#13712yes—
97C4×C191H(5,𝔽2)12yes—
98Dic191—21no—
99Dic191—21no—
100Dic191—42no—
101Dic191—42no—
102Dic191C2×Q1612yes—
103C2×C191#53511yescharacteristic
104C2×C191C22×Q1611yescharacteristic
105C2×C191C2×#13711yescharacteristic
106D191—22no—
107C191#320911yescharacteristic, complemented, Sylow
108#3209—1911nomaximal, Sylow
109#533—1911no—
110C2×C4⋊Q8—1911no—
111#494—1911no—
112#494—1911no—
113#487—1911no—
114#487D19111yescharacteristic, complemented
115#462—1914no—
116#451—1914no—
117#431—1911no—
118C22×Q8—1912no—
119C2×Q16—1914no—
120C2×Q16—1914no—
121C4⋊Q8—3821no—
122C4⋊Q8—3821no—
123C4⋊Q8—1912no—
124C4⋊Q8—1911no—
125C4⋊Q8—1911no—
126C4⋊Q8C2×D19111yescharacteristic
127C4⋊Q8C2×D19111yescharacteristic
128#125—3821no—
129C22⋊Q8—3821no—
130C22⋊Q8—1912no—
131C4×Q8—3821no—
132C4×Q8—1912no—
133#117—3821no—
134C2×C4⋊C4—3821no—
135C2×C4⋊C4—3821no—
136C2×C42—1911no—
137C4⋊C8—1912no—
138#103—1914no—
139#103C2×D19114yes—
140C22⋊C8—1912no—
141C4×C8—1911no—
142C4×C8C2×D19111yescharacteristic
143C2×Q8—21no—
144C2×Q8—3822no—
145C2×Q8—1912no—
146C2×Q8—1912no—
147C2×Q8—7644no—
148C2×Q8C22×D19112yes—
149C22×C4—3822no—
150C22×C4—1911no—
151C22×C4—1911no—
152C22×C4—1911no—
153Q16—7644no—
154Q16—7644no—
155C2×C8—1912no—
156C2×C8C22×D19112yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—3822no—
161C4⋊C4—3822no—
162C4⋊C4—3822no—
163C4⋊C4—3822no—
164C4⋊C4—3821no—
165C4⋊C4—3821no—
166C4⋊C4—3821no—
167C4⋊C4—3821no—
168C4⋊C4C22×D19112yes—
169C22⋊C4—3822no—
170C42—3822no—
171C42—1911no—
172C42—1911no—
173C42—1911no—
174C42C22×D19111yescharacteristic
175C23—1911no—
176Q8—24no—
177Q8—42no—
178Q8—3824no—
179Q8—3824no—
180Q8—7644no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—3822no—
184C2×C4—3822no—
185C2×C4—3822no—
186C2×C4—3822no—
187C2×C4—1911no—
188C2×C4—1911no—
189C2×C4—1911no—
190C2×C4—1914no—
191C2×C4—7644no—
192C2×C4D4×D19111yescharacteristic
193C2×C4D4×D19111yescharacteristic
194C2×C4C23×D19111yescharacteristic
195C8—22no—
196C8—3822no—
197C22—3821no—
198C22—3821no—
199C22—1912no—
200C22C2×D4×D19111yescharacteristic
201C4—42no—
202C4—42no—
203C4—3821no—
204C4—3821no—
205C4—1912no—
206C4—7642no—
207C4—7642no—
208C4Q16×D19112yes—
209C4#2025935112yes—
210C4#2025936812yes—
211C2—3822no—
212C2#2713323111yescharacteristic
213C2C2×Q16×D19111yescharacteristic
214C2C2×#2025935111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#487×C191C2
socleC22×C191C22×D4
prime radicalC22×D191C2×D4
hypercenter#487D191
hypocenterC191#3209
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #27133043C2semidirect
4. #27133042C2semidirect
5. #27133015C2semidirect
6. #27132872C2semidirect
9. #487×C191C2semidirect
79. C4×D191Q16semidirect
107. C191#3209semidirect
114. #487D191semidirect

Maximal subgroups

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

subgroupindex
#271332182
C4⋊Q8×D1912
#271330432
#271330422
#271330152
#271328722
#271328202
#271327142
#487×C1912
#3209191