Finite Group Databasebeta

← #37546631

Subgroups — #37546631

Order 23168.

340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic362C211yesmaximal, characteristic, complemented
2#26041689C212yesmaximal, complemented
3#26041682C211yesmaximal, characteristic, complemented
4#26041583C212yesmaximal, complemented
5#26041571C211yesmaximal, characteristic, complemented
6#26041569C211yesmaximal, characteristic
7#26041559C212yesmaximal
8#26041410C211yesmaximal, characteristic, complemented
9#26041379C212yesmaximal
10C4×#15668437C211yesmaximal, characteristic
11#554×C181C211yesmaximal, characteristic
12C2×#15668437C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153816C2212yes—
15#19153815—21no—
16#19153815—22no—
17Q8×Dic181—21no—
18C4×Dic362C2211yescharacteristic
19C4×Dic362C2212yes—
20#19153749C2212yes—
21#19153749C2212yes—
22#19153749C2212yes—
23#19153747C2211yescharacteristic
24#19153747C2212yes—
25C4⋊#13089678C2212yes—
26C4⋊#13089678C2211yescharacteristic
27#19153739C2211yescharacteristic
28#19153739C2212yes—
29#19153737C2211yescharacteristic
30#19153737C2212yes—
31#19153727C2212yes—
32C4×#13089678C2211yescharacteristic
33C4⋊Q8×C181C2211yescharacteristic
34#125×C181C2212yes—
35C22⋊Q8×C181—21no—
36C22⋊Q8×C181—22no—
37C4×Q8×C181C2211yescharacteristic
38C4×D4×C181C2211yescharacteristic
39C4×D4×C181C2212yes—
40C2×C4⋊C4×C181—21no—
41C2×Dic362—21no—
42C2×Dic362—22no—
43C2×Dic362C2311yescharacteristic
44#15668437Q814yescomplemented
45C2×#13089678C2311yescharacteristic
46C2×#13089678C2312yes—
47C2×#13089678C2311yescharacteristic
48C4⋊Dic181—22no—
49C4⋊Dic181—21no—
50C4⋊Dic181C2311yescharacteristic
51C4⋊Dic181C2312yes—
52#15668428—21no—
53#15668428—22no—
54C4×Dic181—22no—
55C4×Dic181—21no—
56C2×Q8×C181—21no—
57C2×Q8×C181D411yescharacteristic
58C2×D4×C181C2311yescharacteristic
59C22×C4×C181—21no—
60C22×C4×C181—22no—
61C4⋊C4×C181—22no—
62C4⋊C4×C181—21no—
63C4⋊C4×C181—22no—
64C4⋊C4×C181D411yescharacteristic
65C4⋊C4×C181D412yes—
66C4⋊C4×C181C2311yescharacteristic
67C4⋊C4×C181C2312yes—
68C22⋊C4×C181—21no—
69C22⋊C4×C181—22no—
70C42×C181C2312yes—
71C42×C181C2311yescharacteristic
72Dic362—21no—
73Dic362—82no—
74Dic362—81no—
75Dic362C2×Q812yes—
76C2×Dic181—21no—
77C2×Dic181—22no—
78C2×Dic181—21no—
79#13089678—21no—
80#13089678—22no—
81#13089678C2×Q812yes—
82C23×C181—21no—
83Q8×C181—22no—
84Q8×C181—81no—
85D4×C181—21no—
86D4×C181C2×Q812yes—
87C2×C4×C181—21no—
88C2×C4×C181—22no—
89C2×C4×C181—42no—
90C2×C4×C181—82no—
91C2×C4×C181C2×D412yes—
92C2×C4×C181C2×D411yescharacteristic
93C2×C4×C181C2×D412yes—
94C2×C4×C181C2×D411yescharacteristic
95C2×C4×C181C2411yescharacteristic
96C22×C181—22no—
97C22×C181—41no—
98C22×C181—41no—
99C22×C181C22×D411yescharacteristic
100C4×C181—22no—
101C4×C181—22no—
102C4×C181—21no—
103C4×C181—41no—
104C4×C181—42no—
105C4×C181#13712yes—
106C4×C181C22×Q811yescharacteristic
107C4×C1812−1+411yescharacteristic
108Dic181—22no—
109Dic181—41no—
110Dic181—42no—
111C2×C181—42no—
112C2×C181D4×Q811yescharacteristic
113C2×C181C2×#13711yescharacteristic
114C2×C181#57611yescharacteristic
115C181#338511yescharacteristic, complemented, Sylow
116#3385—1811nomaximal, Sylow
117Q82—1811no—
118#554D18111yescharacteristic
119#501—1812no—
120#500—1811no—
121#479—1812no—
122#477—1811no—
123#476—1811no—
124#474—1812no—
125#446—1811no—
126#440—1812no—
127C4×SD16—1811no—
128C2×SD16—1811no—
129C4⋊Q8—3621no—
130C4⋊Q8—3622no—
131C4⋊Q8—1812no—
132C4⋊Q8—1811no—
133C4⋊Q8C2×D18111yescharacteristic
134#125C2×D18112yes—
135C22⋊Q8—21no—
136C22⋊Q8—22no—
137C4×Q8—3621no—
138C4×Q8—1811no—
139C4×Q8—1812no—
140C4×Q8C2×D18111yescharacteristic
141C4×D4C2×D18111yescharacteristic
142C4×D4C2×D18112yes—
143C2×C4⋊C4—21no—
144#107—1812no—
145#107—1812no—
146#107—1812no—
147#106—1811no—
148#106—1812no—
149C4⋊C8—1812no—
150C4⋊C8—1811no—
151#103—1811no—
152#103—1812no—
153#102—1811no—
154#102—1812no—
155#97—1812no—
156C4×C8—1811no—
157C2×Q8—21no—
158C2×Q8—3621no—
159C2×Q8—3622no—
160C2×Q8—1811no—
161C2×Q8C22⋊D18111yescharacteristic
162C2×D4C22×D18111yescharacteristic
163C22×C4—21no—
164C22×C4—22no—
165SD16—1814no—
166C2×C8—1811no—
167C2×C8—1812no—
168C2×C8—1811no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—3622no—
173C4⋊C4—3621no—
174C4⋊C4—3621no—
175C4⋊C4—3622no—
176C4⋊C4—1811no—
177C4⋊C4—1812no—
178C4⋊C4C22⋊D18111yescharacteristic
179C4⋊C4C22⋊D18112yes—
180C4⋊C4C22×D18111yescharacteristic
181C4⋊C4C22×D18112yes—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C42—3622no—
185C42—3621no—
186C42C22×D18111yescharacteristic
187C42C22×D18112yes—
188C23—21no—
189Q8—22no—
190Q8—81no—
191Q8—3621no—
192Q8—14482no—
193Q8—14481no—
194Q8—1812no—
195D4—21no—
196D4Q8×D18112yes—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—3621no—
202C2×C4—3622no—
203C2×C4—3621no—
204C2×C4C2×C22⋊D18112yes—
205C2×C4C2×C22⋊D18111yescharacteristic
206C2×C4C2×C22⋊D18112yes—
207C2×C4C2×C22⋊D18111yescharacteristic
208C2×C4C23×D18111yescharacteristic
209C8—3621no—
210C8—3622no—
211C8—1812no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22C22×C22⋊D18111yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—42no—
221C4—3622no—
222C4—7241no—
223C4—7242no—
224C4#1915384612yes—
225C4C2×Q8×D18111yescharacteristic
226C4#1915386411yescharacteristic
227C2—42no—
228C2Q8×C22⋊D18111yescharacteristic
229C2C2×#1915384611yescharacteristic
230C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#554×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181C23
hypercenter#554D181
hypocenterC181#3385
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. Q8×Dic362C2semidirect
2. #26041689C2semidirect
3. #26041682C2semidirect
4. #26041583C2semidirect
5. #26041571C2semidirect
8. #26041410C2semidirect
44. #15668437Q8semidirect
115. C181#3385semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3622
#260416892
#260416822
#260415832
#260415712
#260415692
#260415592
#260414102
#260413792
C4×#156684372
#554×C1812
#3385181