Finite Group Databasebeta

← #37544866

Subgroups — #37544866

Order 23168.

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

normal subgroups only

This group has 360 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#26041834C211yesmaximal, characteristic
2C4⋊Q8×D181C211yesmaximal, characteristic
3#26041675C211yesmaximal, characteristic, complemented
4#26041631C211yesmaximal, characteristic, complemented
5#26041630C211yesmaximal, characteristic, complemented
6#26041611C214yesmaximal
7#26041563C214yesmaximal, complemented
8#26041330C211yesmaximal, characteristic
9#498×C181C211yesmaximal, characteristic, complemented
10C2×Q8×D181—21no—
11#19153816C2211yescharacteristic
12#19153816C2211yescharacteristic
13#19153815—21no—
14#19153815C2212yes—
15#19153814—21no—
16#19153807—21no—
17#19153796C2212yes—
18#19153792—21no—
19Q8×Dic181—21no—
20#19153781C2212yes—
21#19153774—21no—
22C4⋊C4×D181—21no—
23C4⋊C4×D181C2212yes—
24C42×D181C2211yescharacteristic
25#19153747C2214yes—
26#19153745C2212yes—
27#19153741C2214yes—
28#19153739C2214yes—
29#19153728C2212yes—
30C4×#13089678C2211yescharacteristic
31C4⋊Q8×C181C2211yescharacteristic
32C4⋊Q8×C181C2211yescharacteristic
33#106×C181C2214yes—
34C4×C8×C181C2211yescharacteristic
35C2×Dic362—22no—
36C2×Dic362—21no—
37C2×Dic362C2312yes—
38Q8×D181—44no—
39C2×C4×D181—22no—
40C2×C4×D181D411yescharacteristic
41C2×C4×D181D411yescharacteristic
42C2×C4×D181C2311yescharacteristic
43C2×#13089678C2312yes—
44C4⋊Dic181—22no—
45C4⋊Dic181—21no—
46C4⋊Dic181—21no—
47C4⋊Dic181—21no—
48C4⋊Dic181C2312yes—
49#15668428—22no—
50#15668428—22no—
51#15668428—22no—
52#15668426—22no—
53C4×Dic181—22no—
54C4×Dic181D411yescharacteristic
55C4×Dic181D411yescharacteristic
56C4×Dic181C2311yescharacteristic
57C2×Q8×C181—21no—
58C2×Q8×C181—21no—
59C2×C8×C181C2312yes—
60C4⋊C4×C181—21no—
61C4⋊C4×C181—21no—
62C4⋊C4×C181C2312yes—
63C4⋊C4×C181C2312yes—
64C42×C181C2311yescharacteristic
65C22×D181C2×D411yescharacteristic
66Dic362—24no—
67Dic362—42no—
68Dic362—44no—
69C2×Dic181—22no—
70C2×Dic181—22no—
71C2×Dic181C2×D411yescharacteristic
72C2×Dic181C2×D411yescharacteristic
73C2×Dic181C2×D411yescharacteristic
74C4×D181—22no—
75C4×D181—22no—
76C4×D181—44no—
77C4×D181SD1614yes—
78#13089678—22no—
79Q8×C181—42no—
80Q8×C181—42no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181C2×D411yescharacteristic
84C2×C4×C181C2×D411yescharacteristic
85C2×C4×C181C2411yescharacteristic
86C8×C181—22no—
87C22×C181C22×D411yescharacteristic
88C2×D181—21no—
89C2×D181—21no—
90C2×D181C2×SD1612yes—
91C4×C181—42no—
92C4×C181—42no—
93C4×C181C2×SD1612yes—
94C4×C181#13712yes—
95C4×C1812−1+412yes—
96Dic181—21no—
97Dic181—21no—
98Dic181—42no—
99Dic181—42no—
100Dic181C2×SD1612yes—
101C2×C181#53611yescharacteristic
102C2×C181C22×SD1611yescharacteristic
103C2×C181C2×#13711yescharacteristic
104D181—22no—
105C181#322911yescharacteristic, complemented, Sylow
106#3229—1811nomaximal, Sylow
107#533—1811no—
108C2×C4⋊Q8—1811no—
109#498—1811no—
110#498D18111yescharacteristic, complemented
111#487—1811no—
112#487—1811no—
113#483—1814no—
114#475—1814no—
115#431—1811no—
116C22×Q8—3621no—
117C4⋊Q8—3621no—
118C4⋊Q8—3621no—
119C4⋊Q8—1812no—
120C4⋊Q8—1811no—
121C4⋊Q8—1811no—
122C4⋊Q8C2×D18111yescharacteristic
123C4⋊Q8C2×D18111yescharacteristic
124#125—3621no—
125C22⋊Q8—3621no—
126C22⋊Q8—1812no—
127C4×Q8—3621no—
128C4×Q8—1812no—
129#117—3621no—
130C2×C4⋊C4—3621no—
131C2×C4⋊C4—1812no—
132C2×C42—1811no—
133#106—1814no—
134#106C2×D18114yes—
135C4⋊C8—1812no—
136#103—1814no—
137#103—1814no—
138C22⋊C8—1812no—
139C4×C8—1811no—
140C4×C8C2×D18111yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—3622no—
144C2×Q8—3621no—
145C2×Q8—1812no—
146C2×Q8—7244no—
147C22×C4—3622no—
148C22×C4—1811no—
149C22×C4—1811no—
150C22×C4—1811no—
151C2×C8—1812no—
152C2×C8C22×D18112yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—3622no—
156C4⋊C4—3622no—
157C4⋊C4—3622no—
158C4⋊C4—3622no—
159C4⋊C4—3621no—
160C4⋊C4—3621no—
161C4⋊C4—3621no—
162C4⋊C4—1812no—
163C4⋊C4C22×D18112yes—
164C4⋊C4C22×D18112yes—
165C22⋊C4—3622no—
166C42—3622no—
167C42—1811no—
168C42—1811no—
169C42—1811no—
170C42C22×D18111yescharacteristic
171C23—1811no—
172Q8—42no—
173Q8—42no—
174Q8—3624no—
175Q8—7244no—
176Q8—7242no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—3622no—
180C2×C4—3622no—
181C2×C4—3622no—
182C2×C4—3622no—
183C2×C4—1811no—
184C2×C4—1811no—
185C2×C4—1811no—
186C2×C4—1814no—
187C2×C4—7244no—
188C2×C4D4×D18111yescharacteristic
189C2×C4D4×D18111yescharacteristic
190C2×C4C23×D18111yescharacteristic
191C8—22no—
192C8—3622no—
193C22—3621no—
194C22—3621no—
195C22—1812no—
196C22C2×D4×D18111yescharacteristic
197C4—42no—
198C4—42no—
199C4—3621no—
200C4—3621no—
201C4—1812no—
202C4—7242no—
203C4—7242no—
204C4SD16×D18112yes—
205C4#1915384512yes—
206C4#1915386412yes—
207C2—3622no—
208C2#2604185411yescharacteristic
209C2C2×SD16×D18111yescharacteristic
210C2C2×#1915384511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#498×C181C2
socleC22×C181C22×D4
prime radicalC22×D181C2×D4
hypercenter#498D181
hypocenterC181#3229
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
3. #26041675C2semidirect
4. #26041631C2semidirect
5. #26041630C2semidirect
7. #26041563C2semidirect
9. #498×C181C2semidirect
105. C181#3229semidirect
110. #498D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
C4⋊Q8×D1812
#260416752
#260416312
#260416302
#260416112
#260415632
#260413302
#498×C1812
#3229181