Finite Group Databasebeta

← #37532949

Subgroups — #37532949

Order 23168.

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

normal subgroups only

This group has 212 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#26041629C211yesmaximal, characteristic, complemented
2#26041559C211yesmaximal, characteristic, complemented
3#26041484C211yesmaximal, characteristic, complemented
4#26041028C211yesmaximal, characteristic, complemented
5D4⋊#13089678C211yesmaximal, characteristic, complemented
6D4⋊#13089678C211yesmaximal, characteristic, complemented
7#555×C181C211yesmaximal, characteristic
8C2×#15668437—21no—
9#19153816C2211yescharacteristic
10#19153749—21no—
11C4⋊#13089678C2211yescharacteristic
12C4⋊#13089678C2211yescharacteristic
13#19153740—22no—
14#19153739—21no—
15#19153737—21no—
16C4×#13089678C2211yescharacteristic
17C4⋊Q8×C181C2211yescharacteristic
18#124×C181—22no—
19#121×C181—21no—
20#121×C181—21no—
21C4×D4×C181C2211yescharacteristic
22C4×D4×C181C2211yescharacteristic
23#117×C181—21no—
24#117×C181—21no—
25C2×Dic362—21no—
26#15668437—42no—
27C2×#13089678—21no—
28C2×#13089678—21no—
29C2×#13089678—21no—
30C4⋊Dic181—21no—
31C4⋊Dic181—21no—
32C2×Q8×C181D412yes—
33C2×D4×C181—22no—
34C2×D4×C181D411yescharacteristic
35C2×D4×C181D411yescharacteristic
36C22×C4×C181—21no—
37C22×C4×C181—21no—
38C4⋊C4×C181—21no—
39C4⋊C4×C181—21no—
40C4⋊C4×C181D411yescharacteristic
41C4⋊C4×C181D411yescharacteristic
42C22⋊C4×C181—22no—
43C22⋊C4×C181—22no—
44C22⋊C4×C181—21no—
45C22⋊C4×C181—21no—
46C42×C181—22no—
47C42×C181C2311yescharacteristic
48Dic362—41no—
49Dic362—41no—
50C2×Dic181—41no—
51#13089678—41no—
52#13089678—41no—
53#13089678—41no—
54C23×C181—21no—
55C23×C181—21no—
56Q8×C181—42no—
57D4×C181—21no—
58D4×C181—21no—
59D4×C181—21no—
60D4×C181—21no—
61D4×C181—82no—
62C2×C4×C181—21no—
63C2×C4×C181—21no—
64C2×C4×C181—22no—
65C2×C4×C181—41no—
66C2×C4×C181—41no—
67C2×C4×C181—41no—
68C2×C4×C181—41no—
69C2×C4×C181C2×D411yescharacteristic
70C2×C4×C181C2×D411yescharacteristic
71C2×C4×C181C2×D411yescharacteristic
72C22×C181—41no—
73C22×C181—41no—
74C22×C181—41no—
75C22×C181—41no—
76C22×C181—41no—
77C22×C181—41no—
78C22×C181#12011yescharacteristic
79C4×C181—21no—
80C4×C181—41no—
81C4×C181—41no—
82C4×C181—42no—
83C4×C181#13511yescharacteristic
84C4×C181#13511yescharacteristic
85C4×C181AGL(1,ℤ/8)11yescharacteristic
86C4×C181#13711yescharacteristic
87Dic181—81no—
88C2×C181—81no—
89C2×C181—81no—
90C2×C181#44911yescharacteristic
91C2×C181#45211yescharacteristic
92C2×C181#45511yescharacteristic
93C181#164611yescharacteristic, complemented, Sylow
94#1646—1811nomaximal, Sylow
95#555D18111yescharacteristic
96#487—1811no—
97#474—1811no—
98#461—1811no—
99#333—1811no—
100D4⋊C8—1811no—
101D4⋊C8—1811no—
102C2×SD16—3621no—
103C4⋊Q8—1811no—
104C4⋊Q8C2×D18111yescharacteristic
105#124—22no—
106#121—21no—
107#121—21no—
108C4×D4C2×D18111yescharacteristic
109C4×D4C2×D18111yescharacteristic
110#117—21no—
111#117—21no—
112#107—3621no—
113C4⋊C8—1811no—
114C4⋊C8—1811no—
115#103—3622no—
116#103—3621no—
117#102—3621no—
118C4×C8—1811no—
119C2×Q8—3621no—
120C2×Q8C22⋊D18112yes—
121C2×D4—22no—
122C2×D4C22⋊D18111yescharacteristic
123C2×D4C22⋊D18111yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—7242no—
127C2×C8—3621no—
128C2×C8—3621no—
129C2×C8—3621no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3621no—
133C4⋊C4—3621no—
134C4⋊C4C22⋊D18111yescharacteristic
135C4⋊C4C22⋊D18111yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D18111yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—7241no—
146Q8—7241no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—7241no—
160C2×C4C2×C22⋊D18111yescharacteristic
161C2×C4C2×C22⋊D18111yescharacteristic
162C2×C4C2×C22⋊D18111yescharacteristic
163C8—7241no—
164C8—7241no—
165C8—7241no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1915378511yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—14481no—
178C4#1915383611yescharacteristic
179C4#1915383611yescharacteristic
180C4#1915384111yescharacteristic
181C4#1915384611yescharacteristic
182C2—81no—
183C2—81no—
184C2#2604142411yescharacteristic
185C2#2604144111yescharacteristic
186C2#2604145711yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting, prime radical#555×C181C2
socleC22×C181#120
hypercenter#555D181
hypocenterC181#1646
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #26041629C2semidirect
2. #26041559C2semidirect
3. #26041484C2semidirect
4. #26041028C2semidirect
5. D4⋊#13089678C2semidirect
6. D4⋊#13089678C2semidirect
93. C181#1646semidirect

Maximal subgroups

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

subgroupindex
#260416292
#260415592
#260414842
#260410282
D4⋊#130896782
#555×C1812
#1646181