Finite Group Databasebeta

← #37549036

Subgroups — #37549036

Order 23168.

234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C2×C4⋊#13089685C211yesmaximal, characteristic, complemented
2C4×#15668458C211yesmaximal, characteristic, complemented
3#26042129C212yesmaximal, complemented
4D4⋊#13089685C212yesmaximal, complemented
5#26041810C211yesmaximal, characteristic
6C2×#15668458—21no—
7C22×#13089685—21no—
8C4⋊#13089685—21no—
9C4⋊#13089685C2212yes—
10#19153871—21no—
11C4×#13089685C2212yes—
12#19153809—21no—
13#19153807C411yescharacteristic
14#19153803C411yescharacteristic
15#19153797—21no—
16#19153795C411yescharacteristic
17#19153781C2211yescharacteristic
18D4×Dic181—21no—
19D4×Dic181C2211yescharacteristic
20#19153772—21no—
21C2×C4×Dic181C2211yescharacteristic
22#121×C181C411yescharacteristic
23#15668458—24no—
24C2×#13089685—22no—
25C2×#13089685—22no—
26C2×#13089685—42no—
27C2×Dic362C2×C411yescharacteristic
28C22×Dic181—21no—
29C22×Dic181D411yescharacteristic
30C22×Dic181D411yescharacteristic
31C4⋊Dic181—21no—
32C4⋊Dic181C2×C411yescharacteristic
33#15668428—21no—
34#15668428—21no—
35#15668428—21no—
36C22⋊Dic181—21no—
37C22⋊Dic181—21no—
38C22⋊Dic181—21no—
39C22⋊Dic181—21no—
40C4×Dic181—21no—
41C4×Dic181C2×C411yescharacteristic
42C4×Dic181D411yescharacteristic
43C4×Dic181D411yescharacteristic
44C4×Dic181C2311yescharacteristic
45C2×D4×C181—21no—
46C2×D4×C181C2×C411yescharacteristic
47C22×C4×C181C2×C411yescharacteristic
48C4⋊C4×C181C2×C411yescharacteristic
49C22⋊C4×C181—21no—
50#13089685—24no—
51#13089685—42no—
52Dic362—22no—
53C2×Dic181—22no—
54C2×Dic181—21no—
55C2×Dic181—21no—
56C2×Dic181—21no—
57C2×Dic181—21no—
58C2×Dic181—41no—
59C2×Dic181—41no—
60C2×Dic181C22⋊C411yescharacteristic
61C2×Dic181C22⋊C411yescharacteristic
62C2×Dic181C2×D411yescharacteristic
63C2×Dic181C2×D411yescharacteristic
64C23×C181—21no—
65C23×C181C22⋊C411yescharacteristic
66D4×C181—22no—
67D4×C181—42no—
68C2×C4×C181—22no—
69C2×C4×C181—21no—
70C2×C4×C181C22⋊C411yescharacteristic
71C2×C4×C181C22×C411yescharacteristic
72C22×C181—21no—
73C22×C181—21no—
74C22×C181—41no—
75C22×C181—41no—
76C22×C181—41no—
77C22×C181#10412yes—
78C22×C181C2×C22⋊C411yescharacteristic
79C4×C181—21no—
80C4×C181—41no—
81C4×C181#13112yes—
82Dic181—21no—
83Dic181—21no—
84Dic181—21no—
85Dic181—41no—
86Dic181—41no—
87Dic181AGL(1,ℤ/8)11yescharacteristic
88Dic181#13711yescharacteristic
89C2×C181—22no—
90C2×C181—81no—
91C2×C181#40811yescharacteristic
92C2×C181#41711yescharacteristic
93C2×C181C2×#10411yescharacteristic
94C181#148311yescharacteristic, complemented, Sylow
95#1483—1811nomaximal, Sylow
96#529—1811no—
97C2×C4⋊C8—1811no—
98C4×M16—1811no—
99Q8⋊C8—1812no—
100D4⋊C8—1812no—
101C2×M16—3621no—
102C22×C8—3621no—
103#126—3621no—
104#125—1811no—
105#124—1811no—
106#123—3621no—
107C22⋊Q8—1811no—
108#121#1062438211yescharacteristic
109C4×Q8—1811no—
110C4×D4—3621no—
111C4×D4—1811no—
112#117—3621no—
113C2×C42—1811no—
114C4⋊C8—3621no—
115C4⋊C8—1812no—
116#97—3621no—
117C4×C8—1812no—
118C2×Q8—1811no—
119C2×D4—21no—
120C2×D4C2×#1062438211yescharacteristic
121C22×C4—3621no—
122C22×C4—1811no—
123C22×C4—1811no—
124C22×C4C2×#1062438211yescharacteristic
125M16—3624no—
126C2×C8—3622no—
127C2×C8—3622no—
128C2×C8—7242no—
129C4⋊C4—3621no—
130C4⋊C4—3621no—
131C4⋊C4—3621no—
132C4⋊C4—3621no—
133C4⋊C4—1811no—
134C4⋊C4C2×#1062438211yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—3621no—
137C22⋊C4—3621no—
138C22⋊C4—3621no—
139C22⋊C4—3621no—
140C42—3621no—
141C42—1811no—
142C42—1811no—
143C42—1811no—
144C42—1811no—
145C23—21no—
146C23C22⋊#1062438211yescharacteristic
147Q8—3622no—
148D4—22no—
149D4—42no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—3622no—
153C2×C4—3621no—
154C2×C4—3621no—
155C2×C4—3621no—
156C2×C4—3621no—
157C2×C4—1811no—
158C2×C4—1811no—
159C2×C4—1811no—
160C2×C4—1811no—
161C2×C4—7241no—
162C2×C4—7241no—
163C2×C4C22⋊#1062438211yescharacteristic
164C2×C4C22×#1062438211yescharacteristic
165C8—3624no—
166C8—7242no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1915388012yes—
173C22C2×C22⋊#1062438211yescharacteristic
174C4—21no—
175C4—41no—
176C4—3621no—
177C4—3621no—
178C4—3621no—
179C4—1811no—
180C4—1811no—
181C4—7241no—
182C4—7241no—
183C4#1915390312yes—
184C2—22no—
185C2—81no—
186C2#2604225211yescharacteristic
187C2#2604227011yescharacteristic
188C2C2×#1915388011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting, prime radical#121×C181C4
socleC22×C181C2×C22⋊C4
cosocleC4×Dic181C23
hypercenter#121#10624382
hypocenterC181#1483
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#13089685C2semidirect
2. C4×#15668458C2semidirect
3. #26042129C2semidirect
4. D4⋊#13089685C2semidirect
94. C181#1483semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#130896852
C4×#156684582
#260421292
D4⋊#130896852
#260418102
#1483181