Finite Group Databasebeta

← #37544950

Subgroups — #37544950

Order 23168.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C4⋊Q8×D181C211yesmaximal, characteristic, complemented
2#26041811C211yesmaximal, characteristic
3#26041687C212yesmaximal, complemented
4#26041637C212yesmaximal, complemented
5#26041616C212yesmaximal, complemented
6#26041596C212yesmaximal
7#26041565C211yesmaximal, characteristic, complemented
8#26041561C211yesmaximal, characteristic, complemented
9#26041559C211yesmaximal, characteristic, complemented
10#26041335C211yesmaximal, characteristic, complemented
11#474×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181—21no—
13#19153816C2211yescharacteristic
14#19153815—21no—
15#19153815C2212yes—
16#19153814—21no—
17#19153809—21no—
18#19153807C2211yescharacteristic
19#19153803C2211yescharacteristic
20#19153798—21no—
21#19153796C2211yescharacteristic
22#19153787C2211yescharacteristic
23C4×Dic362C2211yescharacteristic
24C4×C22⋊D181—21no—
25#19153775—21no—
26C4⋊C4×D181—21no—
27C4⋊C4×D181C2212yes—
28C42×D181C2211yescharacteristic
29#19153749C2212yes—
30#19153748C2212yes—
31#19153747C2212yes—
32C4⋊#13089678C2211yescharacteristic
33#19153741C2212yes—
34#19153739C2212yes—
35#19153737C2212yes—
36#19153728C2212yes—
37#19153726C2212yes—
38C4⋊Q8×C181C2211yescharacteristic
39C4×D4×C181C2211yescharacteristic
40#107×C181C2212yes—
41C4⋊C8×C181C2211yescharacteristic
42#102×C181C2212yes—
43C2×Dic362—22no—
44C2×Dic362—21no—
45C2×Dic362C2311yescharacteristic
46Q8×D181—44no—
47C2×C22⋊D181—21no—
48C2×C4×D181—22no—
49C2×C4×D181D411yescharacteristic
50C2×C4×D181D411yescharacteristic
51C2×C4×D181C2311yescharacteristic
52C2×#13089678C2312yes—
53C4⋊Dic181—21no—
54C4⋊Dic181—21no—
55C4⋊Dic181—21no—
56C4⋊Dic181C2312yes—
57C4⋊Dic181C2311yescharacteristic
58#15668428—22no—
59#15668428—22no—
60#15668428—21no—
61#15668428—21no—
62C22⋊Dic181—21no—
63C22⋊Dic181—21no—
64C22⋊Dic181—21no—
65#15668426—21no—
66C4×Dic181—21no—
67C4×Dic181D411yescharacteristic
68C4×Dic181D411yescharacteristic
69C4×Dic181C2311yescharacteristic
70C2×Q8×C181—21no—
71C2×D4×C181C2311yescharacteristic
72C22×C4×C181—21no—
73C2×C8×C181C2312yes—
74C4⋊C4×C181—21no—
75C4⋊C4×C181C2312yes—
76C4⋊C4×C181C2311yescharacteristic
77C22⋊C4×C181—21no—
78C42×C181C2311yescharacteristic
79C22×D181C2×D411yescharacteristic
80Dic362—21no—
81Dic362—21no—
82Dic362—44no—
83Dic362—42no—
84C22⋊D181—42no—
85C2×Dic181—21no—
86C2×Dic181—21no—
87C2×Dic181—22no—
88C2×Dic181C2×D411yescharacteristic
89C2×Dic181C2×D411yescharacteristic
90C2×Dic181C2×D411yescharacteristic
91C4×D181—22no—
92C4×D181—22no—
93C4×D181—21no—
94C4×D181—21no—
95C4×D181—44no—
96#13089678—22no—
97C23×C181—21no—
98Q8×C181—42no—
99D4×C181—21no—
100D4×C181—21no—
101C2×C4×C181—21no—
102C2×C4×C181—22no—
103C2×C4×C181—42no—
104C2×C4×C181C2×D411yescharacteristic
105C2×C4×C181C2×D411yescharacteristic
106C2×C4×C181C2411yescharacteristic
107C8×C181—22no—
108C22×C181—41no—
109C22×C181—41no—
110C22×C181—41no—
111C22×C181C22×D411yescharacteristic
112C2×D181—21no—
113C2×D181—21no—
114C2×D181#13712yes—
115C4×C181—21no—
116C4×C181—41no—
117C4×C181—42no—
118C4×C181AGL(1,ℤ/8)12yes—
119C4×C1812−1+411yescharacteristic
120C4×C1812−1+411yescharacteristic
121Dic181—21no—
122Dic181—21no—
123Dic181—21no—
124Dic181—41no—
125Dic181—42no—
126Dic181—41no—
127C2×C181—81no—
128C2×C181#53611yescharacteristic
129C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
130C2×C181C2×#13711yescharacteristic
131D181—22no—
132C181#323911yescharacteristic, complemented, Sylow
133#3239—1811nomaximal, Sylow
134C2×C4⋊Q8—1811no—
135#529—1811no—
136#501—1812no—
137#489—1812no—
138#484—1812no—
139#481—1812no—
140#475—1811no—
141#475—1811no—
142#474—1811no—
143#474D18111yescharacteristic, complemented
144#432—1811no—
145C22×Q8—3621no—
146C4⋊Q8—3621no—
147C4⋊Q8—3621no—
148C4⋊Q8—1812no—
149C4⋊Q8—1811no—
150C4⋊Q8C2×D18111yescharacteristic
151#126—3621no—
152#125—1811no—
153#124—1811no—
154#123—3621no—
155C22⋊Q8—1811no—
156#121—1811no—
157C4×Q8—1811no—
158C4×D4—3621no—
159C4×D4C2×D18111yescharacteristic
160#117—3621no—
161C2×C4⋊C4—3621no—
162C2×C4⋊C4—1812no—
163C2×C42—1811no—
164#107—1812no—
165#107C2×D18112yes—
166#106—1812no—
167#106—1812no—
168C4⋊C8—1811no—
169C4⋊C8C2×D18111yescharacteristic
170#103—1812no—
171#103—1812no—
172#102—1812no—
173#102C2×D18112yes—
174C22⋊C8—1812no—
175#97—1812no—
176C2×Q8—21no—
177C2×Q8—3622no—
178C2×Q8—3621no—
179C2×Q8—1811no—
180C2×Q8—7244no—
181C2×D4—3621no—
182C2×D4C22×D18111yescharacteristic
183C22×C4—21no—
184C22×C4—3622no—
185C22×C4—1811no—
186C22×C4—1811no—
187C22×C4—1811no—
188C2×C8—1812no—
189C2×C8C22×D18112yes—
190C4⋊C4—21no—
191C4⋊C4—3622no—
192C4⋊C4—3622no—
193C4⋊C4—3621no—
194C4⋊C4—3621no—
195C4⋊C4—3621no—
196C4⋊C4—3621no—
197C4⋊C4—3621no—
198C4⋊C4—1812no—
199C4⋊C4—1811no—
200C4⋊C4C22×D18112yes—
201C4⋊C4C22×D18111yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—3621no—
204C22⋊C4—3621no—
205C22⋊C4—3621no—
206C22⋊C4—3621no—
207C42—3621no—
208C42—1811no—
209C42—1811no—
210C42—1811no—
211C42C22×D18111yescharacteristic
212C23—21no—
213C23—1811no—
214Q8—42no—
215Q8—3621no—
216Q8—3621no—
217Q8—7244no—
218Q8—7242no—
219D4—21no—
220D4—21no—
221D4—7242no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—3622no—
226C2×C4—3622no—
227C2×C4—3621no—
228C2×C4—3621no—
229C2×C4—3621no—
230C2×C4—3621no—
231C2×C4—3622no—
232C2×C4—1811no—
233C2×C4—1811no—
234C2×C4—1811no—
235C2×C4—7244no—
236C2×C4D4×D18111yescharacteristic
237C2×C4D4×D18111yescharacteristic
238C2×C4C23×D18111yescharacteristic
239C8—22no—
240C8—3622no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—3621no—
245C22—3621no—
246C22—1812no—
247C22C2×D4×D18111yescharacteristic
248C4—21no—
249C4—41no—
250C4—42no—
251C4—3621no—
252C4—3621no—
253C4—3621no—
254C4—7242no—
255C4—7241no—
256C4—7241no—
257C4#1915384012yes—
258C4#1915386311yescharacteristic
259C4#1915386411yescharacteristic
260C2—81no—
261C2—3622no—
262C2#2604185211yescharacteristic
263C2C2×#1915384011yescharacteristic
264C2#137×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#474×C181C2
socleC22×C181C22×D4
prime radical#19153787C22
hypercenter#474D181
hypocenterC181#3239
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D181C2semidirect
3. #26041687C2semidirect
4. #26041637C2semidirect
5. #26041616C2semidirect
7. #26041565C2semidirect
8. #26041561C2semidirect
9. #26041559C2semidirect
10. #26041335C2semidirect
11. #474×C181C2semidirect
132. C181#3239semidirect
143. #474D181semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1812
#260418112
#260416872
#260416372
#260416162
#260415962
#260415652
#260415612
#260415592
#260413352
#474×C1812
#3239181