Finite Group Databasebeta

← #37546374

Subgroups — #37546374

Order 23168.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#26042003C211yesmaximal, characteristic, complemented
2#26041986C211yesmaximal, characteristic, complemented
3#26041635C212yesmaximal, complemented
4#26041629C211yesmaximal, characteristic, complemented
5#26041584C212yesmaximal, complemented
6#26041570C211yesmaximal, characteristic, complemented
7#26041496C212yesmaximal
8#26041403C211yesmaximal, characteristic, complemented
9SD16×Dic181C212yesmaximal, complemented
10C4×#15668434C211yesmaximal, characteristic
11#461×C181C211yesmaximal, characteristic, complemented
12C2×#15668434C2211yescharacteristic
13#19153816C2211yescharacteristic
14#19153811—22no—
15#19153807C2212yes—
16#19153805C2212yes—
17#19153796—21no—
18#19153795—21no—
19#19153793—22no—
20Q8×Dic181C2212yes—
21C4×Dic362C2211yescharacteristic
22C4×D724C2211yescharacteristic
23D4×Dic181C2212yes—
24#19153774—21no—
25C2×C4⋊Dic181—21no—
26#19153749C2211yescharacteristic
27#19153748C2212yes—
28#19153745C2212yes—
29#19153740C2212yes—
30#19153740C2212yes—
31#19153738C2212yes—
32#19153737C2211yescharacteristic
33#19153736C2211yescharacteristic
34C4×#13089678C2211yescharacteristic
35C8×Dic181C2212yes—
36C2×SD16×C181C2212yes—
37C4⋊Q8×C181C2211yescharacteristic
38C4×D4×C181C2211yescharacteristic
39C4⋊C8×C181C2211yescharacteristic
40#103×C181C2212yes—
41C2×Dic362—21no—
42C2×Dic362D411yescharacteristic
43C2×D724C2311yescharacteristic
44C22×Dic181—22no—
45C2×C4×D181—21no—
46#15668434—22no—
47C2×#13089678C2311yescharacteristic
48C2×#13089678C2311yescharacteristic
49C4⋊Dic181—22no—
50C4⋊Dic181—22no—
51C4⋊Dic181—21no—
52C4⋊Dic181D411yescharacteristic
53C4⋊Dic181C2311yescharacteristic
54C4⋊Dic181C2312yes—
55#15668428—22no—
56#15668428D412yes—
57C22⋊Dic181—22no—
58#15668426—22no—
59#15668426—22no—
60#15668426—21no—
61C4×Dic181—22no—
62C4×Dic181C2312yes—
63C2×Q8×C181C2312yes—
64C2×D4×C181C2311yescharacteristic
65C22×C4×C181—21no—
66SD16×C181—24no—
67C2×C8×C181C2312yes—
68C4⋊C4×C181—21no—
69C4⋊C4×C181—21no—
70C4⋊C4×C181C2311yescharacteristic
71C22⋊C4×C181—21no—
72C42×C181C2311yescharacteristic
73C22×D181—21no—
74Dic362—22no—
75Dic362—81no—
76D724—21no—
77D724—21no—
78C2×Dic181—21no—
79C2×Dic181—22no—
80C2×Dic181—82no—
81C2×Dic181C2×D412yes—
82C2×Dic181C2×D412yes—
83C4×D181—42no—
84#13089678—21no—
85#13089678—21no—
86C23×C181—21no—
87Q8×C181—22no—
88Q8×C181—22no—
89D4×C181—21no—
90D4×C181#4112yes—
91C2×C4×C181—21no—
92C2×C4×C181—22no—
93C2×C4×C181—42no—
94C2×C4×C181C2×D411yescharacteristic
95C2×C4×C181C2×D411yescharacteristic
96C2×C4×C181C2411yescharacteristic
97C8×C181—22no—
98C22×C181—22no—
99C22×C181—41no—
100C22×C181—41no—
101C22×C181C22×D411yescharacteristic
102C2×D181—41no—
103C2×D181—41no—
104C2×D181—41no—
105C4×C181—21no—
106C4×C181—42no—
107C4×C181—41no—
108C4×C181#13512yes—
109C4×C181C2×#4111yescharacteristic
110C4×C1812−1+411yescharacteristic
111Dic181—22no—
112Dic181—22no—
113Dic181—41no—
114Dic181—42no—
115C2×C181—42no—
116C2×C181#54811yescharacteristic
117C2×C181C2×#13511yescharacteristic
118C2×C181#57711yescharacteristic
119D181—81no—
120C181#336011yescharacteristic, complemented, Sylow
121#3360—1811nomaximal, Sylow
122#556—1811no—
123#554—1811no—
124#488—1812no—
125#487—1811no—
126#479—1812no—
127#476—1811no—
128#463—1812no—
129#461D18111yescharacteristic, complemented
130C8×Q8—1811no—
131C4×SD16—1812no—
132C4×D8—1811no—
133C2×SD16C2×D18112yes—
134C2×D8—1811no—
135C4⋊Q8—1811no—
136C4⋊Q8C2×D18111yescharacteristic
137#126—3622no—
138#125—1812no—
139#124—1812no—
140C22⋊Q8—3622no—
141C22⋊Q8—3621no—
142C22⋊Q8—3621no—
143C4×Q8—1812no—
144C4×Q8—1811no—
145C4×D4—1812no—
146C4×D4—1811no—
147C4×D4C2×D18111yescharacteristic
148#117—3621no—
149C2×C4⋊C4—3621no—
150#107—1811no—
151#106—1812no—
152C4⋊C8—1812no—
153C4⋊C8C2×D18111yescharacteristic
154#103—1812no—
155#103—1812no—
156#103C2×D18112yes—
157#102—1811no—
158#102—1812no—
159#102—1811no—
160C4×C8—1812no—
161C4×C8—1811no—
162C2×Q8—3621no—
163C2×Q8—1811no—
164C2×Q8C22×D18112yes—
165C2×D4—1811no—
166C2×D4C22×D18111yescharacteristic
167C22×C4—21no—
168C22×C4—3622no—
169C22×C4—3621no—
170SD16—24no—
171D8—3622no—
172C2×C8—1811no—
173C2×C8—1811no—
174C2×C8C22×D18112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—3622no—
178C4⋊C4—3622no—
179C4⋊C4—3622no—
180C4⋊C4—3621no—
181C4⋊C4—1812no—
182C4⋊C4—1811no—
183C4⋊C4—1811no—
184C4⋊C4—1812no—
185C4⋊C4C22×D18111yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—3622no—
188C22⋊C4—3622no—
189C22⋊C4—3622no—
190C22⋊C4—3621no—
191C42—3622no—
192C42—1812no—
193C42C22×D18111yescharacteristic
194C23—21no—
195C23—3621no—
196Q8—22no—
197Q8—22no—
198Q8—3622no—
199Q8—14481no—
200D4—21no—
201D4—3621no—
202D4—3621no—
203D4#1566844812yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—3621no—
208C2×C4—3622no—
209C2×C4—14482no—
210C2×C4—1812no—
211C2×C4—1812no—
212C2×C4—7242no—
213C2×C4D4×D18111yescharacteristic
214C2×C4D4×D18111yescharacteristic
215C2×C4C23×D18111yescharacteristic
216C8—22no—
217C8—3621no—
218C8—3621no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—7241no—
223C22—7241no—
224C22—7241no—
225C22C2×D4×D18111yescharacteristic
226C4—21no—
227C4—42no—
228C4—41no—
229C4—3622no—
230C4—3622no—
231C4—7241no—
232C4—7242no—
233C4#1915383712yes—
234C4C2×#1566844811yescharacteristic
235C4#1915386311yescharacteristic
236C2—42no—
237C2—14481no—
238C2#2604193511yescharacteristic
239C2C2×#1915383711yescharacteristic
240C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#461×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668434C22
hypercenter#461D181
hypocenterC181#3360
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #26042003C2semidirect
2. #26041986C2semidirect
3. #26041635C2semidirect
4. #26041629C2semidirect
5. #26041584C2semidirect
6. #26041570C2semidirect
8. #26041403C2semidirect
9. SD16×Dic181C2semidirect
11. #461×C181C2semidirect
120. C181#3360semidirect
129. #461D181semidirect

Maximal subgroups

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

subgroupindex
#260420032
#260419862
#260416352
#260416292
#260415842
#260415702
#260414962
#260414032
SD16×Dic1812
C4×#156684342
#461×C1812
#3360181