Finite Group Databasebeta

← #37549064

Subgroups — #37549064

Order 23168.

252 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 68 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#26042301C211yesmaximal, characteristic
2C4×#15668459C211yesmaximal, characteristic
3#26042129C212yesmaximal, complemented
4Q8⋊#13089685C212yesmaximal, complemented
5C4×Q8×D181C211yesmaximal, characteristic
6C2×#15668459—21no—
7C4⋊#13089685C2212yes—
8#19153873—21no—
9#19153871—21no—
10#19153871—21no—
11C4×#13089685C2212yes—
12C2×Q8×D181C411yescharacteristic
13Q8×Dic181C2211yescharacteristic
14C4×Dic362—21no—
15C4×Dic362C411yescharacteristic
16#19153781—21no—
17#19153781C2211yescharacteristic
18C4⋊C4×D181—21no—
19C4⋊C4×D181C411yescharacteristic
20C42×D181—21no—
21C42×D181C2211yescharacteristic
22C4×Q8×C181C411yescharacteristic
23#15668459—24no—
24C2×#13089685—22no—
25C2×#13089685—22no—
26C2×Dic362—21no—
27C2×Dic362C2×C411yescharacteristic
28Q8×D181—22no—
29Q8×D181—22no—
30C2×C4×D181—21no—
31C2×C4×D181—21no—
32C2×C4×D181C2×C411yescharacteristic
33C2×C4×D181D411yescharacteristic
34C2×C4×D181D411yescharacteristic
35C4⋊Dic181—21no—
36C4⋊Dic181C2×C411yescharacteristic
37#15668428—21no—
38#15668428—21no—
39#15668428—21no—
40C4×Dic181—21no—
41C4×Dic181—21no—
42C4×Dic181—21no—
43C4×Dic181D411yescharacteristic
44C4×Dic181D411yescharacteristic
45C4×Dic181C2311yescharacteristic
46C2×Q8×C181C2×C411yescharacteristic
47C4⋊C4×C181—21no—
48C4⋊C4×C181C2×C411yescharacteristic
49C42×C181—21no—
50C42×C181C2×C411yescharacteristic
51#13089685—24no—
52#13089685—42no—
53C22×D181C22⋊C411yescharacteristic
54Dic362—24no—
55Dic362—21no—
56Dic362M1612yescomplemented
57C2×Dic181—21no—
58C2×Dic181—21no—
59C2×Dic181C22⋊C411yescharacteristic
60C2×Dic181C2×D411yescharacteristic
61C2×Dic181C2×D411yescharacteristic
62C4×D181—24no—
63C4×D181—22no—
64C4×D181—21no—
65C4×D181—21no—
66C4×D181—22no—
67C4×D181—42no—
68Q8×C181—21no—
69Q8×C181M1612yescomplemented
70C2×C4×C181—21no—
71C2×C4×C181—21no—
72C2×C4×C181C22⋊C411yescharacteristic
73C2×C4×C181C22⋊C411yescharacteristic
74C2×C4×C181C22×C411yescharacteristic
75C22×C181C2×C22⋊C411yescharacteristic
76C2×D181—21no—
77C2×D181—21no—
78C2×D181#10412yes—
79C4×C181—22no—
80C4×C181—21no—
81C4×C181—21no—
82C4×C181—41no—
83C4×C181C2×M1611yescharacteristic
84C4×C181C2×M1611yescharacteristic
85Dic181—22no—
86Dic181—21no—
87Dic181—21no—
88Dic181—41no—
89Dic181#13712yes—
90C2×C181#40711yescharacteristic
91C2×C181#41911yescharacteristic
92C2×C181C2×#10411yescharacteristic
93D181—22no—
94C181#149111yescharacteristic, complemented, Sylow
95#1491—1811nomaximal, Sylow
96C2×C4×Q8—1811no—
97#432—1811no—
98C4×M16—1811no—
99Q8⋊C8—1812no—
100Q8⋊C8—1812no—
101C22×Q8—1811no—
102C2×M16—3621no—
103C4×Q8—3621no—
104C4×Q8—3621no—
105C4×Q8—1811no—
106C4×Q8—1811no—
107C4×Q8—1811no—
108C4×Q8#1062438211yescharacteristic
109C2×C4⋊C4—3621no—
110C2×C4⋊C4—1811no—
111C2×C42—3621no—
112C2×C42—1811no—
113C4⋊C8—1812no—
114C22⋊C8—3621no—
115#97—3621no—
116#97—3621no—
117C4×C8—1812no—
118C2×Q8—3621no—
119C2×Q8—3622no—
120C2×Q8—3622no—
121C2×Q8—1811no—
122C2×Q8C2×#1062438211yescharacteristic
123C22×C4—3621no—
124C22×C4—3621no—
125C22×C4—1811no—
126C22×C4—1811no—
127C22×C4—1811no—
128M16—3624no—
129C2×C8—3622no—
130C2×C8—3622no—
131C4⋊C4—21no—
132C4⋊C4—3621no—
133C4⋊C4—3621no—
134C4⋊C4—3621no—
135C4⋊C4—3621no—
136C4⋊C4—1811no—
137C4⋊C4C2×#1062438211yescharacteristic
138C42—21no—
139C42—3621no—
140C42—3621no—
141C42—3621no—
142C42—1811no—
143C42—1811no—
144C42—1811no—
145C42C2×#1062438211yescharacteristic
146C23—1811no—
147Q8—21no—
148Q8—3624no—
149Q8—3621no—
150Q8—1812no—
151Q8#1566845912yescomplemented
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—3624no—
155C2×C4—3622no—
156C2×C4—3622no—
157C2×C4—3621no—
158C2×C4—3621no—
159C2×C4—3621no—
160C2×C4—3621no—
161C2×C4—1811no—
162C2×C4—1811no—
163C2×C4—1811no—
164C2×C4—7242no—
165C2×C4C22⋊#1062438211yescharacteristic
166C2×C4C22⋊#1062438211yescharacteristic
167C2×C4C22×#1062438211yescharacteristic
168C8—3624no—
169C8—7242no—
170C22—3621no—
171C22—3621no—
172C22—1812no—
173C22C2×C22⋊#1062438211yescharacteristic
174C4—22no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—3622no—
179C4—3621no—
180C4—3621no—
181C4—1812no—
182C4—7241no—
183C4D181⋊M1611yescharacteristic
184C4C2×#1566845911yescharacteristic
185C2—3622no—
186C2#2604225011yescharacteristic
187C2#2604227211yescharacteristic
188C2D181⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC4×Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC22×D181C22⋊C4
cosocleC4×Dic181C23
hypercenterC4×Q8#10624382
hypocenterC181#1491
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #26042129C2semidirect
4. Q8⋊#13089685C2semidirect
56. Dic362M16semidirect
69. Q8×C181M16semidirect
94. C181#1491semidirect
151. Q8#15668459semidirect

Maximal subgroups

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

subgroupindex
#260423012
C4×#156684592
#260421292
Q8⋊#130896852
C4×Q8×D1812
#1491181