Finite Group Databasebeta

← #37544947

Subgroups — #37544947

Order 23168.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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#26041808C211yesmaximal, characteristic
3#26041688C211yesmaximal, characteristic, complemented
4#26041685C211yesmaximal, characteristic, complemented
5#26041638C211yesmaximal, characteristic, complemented
6#26041613C212yesmaximal, complemented
7#26041596C212yesmaximal
8#26041562C212yesmaximal, complemented
9#26041557C212yesmaximal, complemented
10#26041334C211yesmaximal, characteristic, complemented
11#489×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181—21no—
13#19153816C2211yescharacteristic
14#19153815—21no—
15#19153815C2212yes—
16#19153814—21no—
17#19153809—21no—
18#19153808C2211yescharacteristic
19#19153798—21no—
20#19153792C2211yescharacteristic
21#19153787C2211yescharacteristic
22Q8×Dic181C2211yescharacteristic
23D4×Dic181C2211yescharacteristic
24#19153776—21no—
25#19153772—21no—
26C4⋊C4×D181—21no—
27C4⋊C4×D181C2212yes—
28C42×D181C2211yescharacteristic
29#19153750C2212yes—
30#19153749C2212yes—
31#19153748C2212yes—
32#19153747C2212yes—
33#19153745C2212yes—
34#19153740C2212yes—
35#19153737C2212yes—
36#19153728C2212yes—
37#19153727C2211yescharacteristic
38C4⋊Q8×C181C2211yescharacteristic
39#124×C181C2211yescharacteristic
40#103×C181C2212yes—
41#102×C181C2212yes—
42#97×C181C2211yescharacteristic
43C2×Dic362—22no—
44C2×Dic362—21no—
45Q8×D181—44no—
46C2×C22⋊D181—21no—
47C22×Dic181—21no—
48C2×C4×D181—22no—
49C2×C4×D181D411yescharacteristic
50C2×C4×D181D411yescharacteristic
51C2×C4×D181C2311yescharacteristic
52C2×#13089678C2312yes—
53C4⋊Dic181—21no—
54C4⋊Dic181—21no—
55C4⋊Dic181C2312yes—
56C4⋊Dic181C2311yescharacteristic
57C4⋊Dic181C2311yescharacteristic
58#15668428—22no—
59#15668428—22no—
60#15668428—21no—
61#15668428—21no—
62#15668428—21no—
63C22⋊Dic181—21no—
64C22⋊Dic181—21no—
65#15668426—21no—
66C4×Dic181—21no—
67C4×Dic181D411yescharacteristic
68C4×Dic181D411yescharacteristic
69C4×Dic181C2311yescharacteristic
70C2×Q8×C181—21no—
71C2×Q8×C181C2311yescharacteristic
72C2×D4×C181C2311yescharacteristic
73C2×C8×C181C2312yes—
74C4⋊C4×C181—21no—
75C4⋊C4×C181C2312yes—
76C22⋊C4×C181—21no—
77C22⋊C4×C181—21no—
78C42×C181C2311yescharacteristic
79C22×D181C2×D411yescharacteristic
80Dic362—42no—
81Dic362—44no—
82C22⋊D181—42no—
83C2×Dic181—21no—
84C2×Dic181—21no—
85C2×Dic181—22no—
86C2×Dic181—42no—
87C2×Dic181C2×D411yescharacteristic
88C2×Dic181C2×D411yescharacteristic
89C2×Dic181C2×D411yescharacteristic
90C4×D181—22no—
91C4×D181—22no—
92C4×D181—22no—
93C4×D181—44no—
94#13089678—22no—
95C23×C181—21no—
96Q8×C181—22no—
97Q8×C181—42no—
98D4×C181—22no—
99C2×C4×C181—21no—
100C2×C4×C181—22no—
101C2×C4×C181C2×D411yescharacteristic
102C2×C4×C181C2×D411yescharacteristic
103C2×C4×C181C2411yescharacteristic
104C8×C181—22no—
105C22×C181—41no—
106C22×C181—41no—
107C22×C181—41no—
108C22×C181C22×D411yescharacteristic
109C2×D181—21no—
110C2×D181—21no—
111C2×D181#13712yes—
112C4×C181—21no—
113C4×C181—21no—
114C4×C181—41no—
115C4×C181—42no—
116C4×C1812−1+412yes—
117Dic181—21no—
118Dic181—21no—
119Dic181—42no—
120Dic181—41no—
121Dic181—41no—
122Dic181AGL(1,ℤ/8)12yes—
123C2×C181—81no—
124C2×C181#53611yescharacteristic
125C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C181C2×#13711yescharacteristic
127D181—22no—
128C181#323911yescharacteristic, complemented, Sylow
129#3239—1811nomaximal, Sylow
130C2×C4⋊Q8—1811no—
131#529—1811no—
132#501—1811no—
133#501—1811no—
134#489—1811no—
135#489D18111yescharacteristic, complemented
136#484—1812no—
137#481—1812no—
138#475—1812no—
139#474—1812no—
140#432—1811no—
141C22×Q8—3621no—
142C4⋊Q8—3621no—
143C4⋊Q8—3621no—
144C4⋊Q8—1812no—
145C4⋊Q8—1811no—
146C4⋊Q8C2×D18111yescharacteristic
147#126—3621no—
148#125—1811no—
149#124C2×D18111yescharacteristic
150#123—3621no—
151C22⋊Q8—1811no—
152#121—1811no—
153C4×Q8—1811no—
154C4×D4—3621no—
155C4×D4—1811no—
156#117—3621no—
157C2×C4⋊C4—3621no—
158C2×C4⋊C4—1812no—
159C2×C42—1811no—
160#107—1812no—
161#107—1812no—
162#106—1812no—
163#106—1812no—
164C4⋊C8—1812no—
165#103—1812no—
166#103C2×D18112yes—
167#102—1812no—
168#102C2×D18112yes—
169C22⋊C8—1812no—
170#97—1811no—
171#97C2×D18111yescharacteristic
172C2×Q8—21no—
173C2×Q8—3622no—
174C2×Q8—3621no—
175C2×Q8—7244no—
176C2×Q8C22×D18111yescharacteristic
177C2×D4—3621no—
178C2×D4C22×D18111yescharacteristic
179C22×C4—3622no—
180C22×C4—3621no—
181C22×C4—1811no—
182C22×C4—1811no—
183C22×C4—1811no—
184C2×C8—1812no—
185C2×C8C22×D18112yes—
186C4⋊C4—21no—
187C4⋊C4—3622no—
188C4⋊C4—3622no—
189C4⋊C4—3621no—
190C4⋊C4—3621no—
191C4⋊C4—3621no—
192C4⋊C4—3621no—
193C4⋊C4—3621no—
194C4⋊C4—1812no—
195C4⋊C4—1811no—
196C4⋊C4—1811no—
197C4⋊C4C22×D18112yes—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—3621no—
201C22⋊C4—3621no—
202C22⋊C4—3621no—
203C42—3621no—
204C42—1811no—
205C42—1811no—
206C42—1811no—
207C42C22×D18111yescharacteristic
208C23—21no—
209C23—1811no—
210Q8—22no—
211Q8—42no—
212Q8—7242no—
213Q8—7244no—
214D4—22no—
215D4—7242no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—3622no—
219C2×C4—3622no—
220C2×C4—3622no—
221C2×C4—3621no—
222C2×C4—3621no—
223C2×C4—3622no—
224C2×C4—1811no—
225C2×C4—1811no—
226C2×C4—1811no—
227C2×C4—7244no—
228C2×C4—7242no—
229C2×C4D4×D18111yescharacteristic
230C2×C4D4×D18111yescharacteristic
231C2×C4C23×D18111yescharacteristic
232C8—22no—
233C8—3622no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—3621no—
238C22—3621no—
239C22—1812no—
240C22C2×D4×D18111yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—3621no—
246C4—3621no—
247C4—1812no—
248C4—7242no—
249C4—7241no—
250C4—7241no—
251C4#1915386312yes—
252C2—81no—
253C2—3622no—
254C2#2604184911yescharacteristic
255C2AGL(1,ℤ/8)×D18111yescharacteristic
256C2#137×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#489×C181C2
socleC22×C181C22×D4
prime radical#19153787C22
hypercenter#489D181
hypocenterC181#3239
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D181C2semidirect
3. #26041688C2semidirect
4. #26041685C2semidirect
5. #26041638C2semidirect
6. #26041613C2semidirect
8. #26041562C2semidirect
9. #26041557C2semidirect
10. #26041334C2semidirect
11. #489×C181C2semidirect
128. C181#3239semidirect
135. #489D181semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1812
#260418082
#260416882
#260416852
#260416382
#260416132
#260415962
#260415622
#260415572
#260413342
#489×C1812
#3239181