Finite Group Databasebeta

← #3459⋊C9 #11919829

Subgroups — #3459⋊C9 #11919829

Order 1152.

466 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 44 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#543⋊C9C211yesmaximal, characteristic, complemented
2C3×#3459C311yesmaximal, characteristic
3C2×#41⋊C9—42nomaximal
4C2×C4×C22⋊C9—41nomaximal
5C4×Q8⋊C9—42nomaximal
6C3×#543C2×C311yescharacteristic
7C3×#539—32no—
8C3×#538—32no—
9C3×#534—31no—
10C3×#529—32no—
11C3×#525—32no—
12C2×C3×C4×D4—31no—
13C22×C22⋊C9—41no—
14#41⋊C9—44no—
15C2×Q8⋊C9—42no—
16C2×Q8⋊C9—42no—
17C4×C22⋊C9—42no—
18#3459C911yescharacteristic, complemented, Sylow
19C2×C3×#41A412yes—
20C22×C3×D4—31no—
21C23×C3×C4—31no—
22C23×C3×C4A411yescharacteristic
23C3×#126—34no—
24C3×#125—32no—
25C3×#124—32no—
26C3×#123—34no—
27C3×#123—34no—
28C3×C22⋊Q8—32no—
29C3×C22⋊Q8—32no—
30C3×C22⋊Q8—32no—
31C3×#121—32no—
32C3×#121—32no—
33C3×#121—32no—
34C3×#120—34no—
35C3×C4×Q8A412yes—
36C3×C4×D4—34no—
37C3×C4×D4—34no—
38C3×C4×D4—32no—
39C3×#117—32no—
40C2×C3×C4⋊C4—31no—
41C2×C3×C22⋊C4—31no—
42C2×C3×C22⋊C4—31no—
43C2×C3×C42—31no—
44C2×C22⋊C9—41no—
45C2×C22⋊C9—41no—
46C2×C22⋊C9—41no—
47C2×C4×C9—161no—
48Q8⋊C9—42no—
49Q8⋊C9—42no—
50#543C2×C911yescharacteristic
51#539—32no—
52#538—32no—
53#534—31no—
54#529—32no—
55#525—32no—
56C2×C4×D4—31no—
57C24×C3—31no—
58C24×C3C2×A411yescharacteristic
59C3×#41—44no—
60C2×C3×Q8C2×A412yes—
61C2×C3×Q8C2×A412yes—
62C2×C3×D4—34no—
63C2×C3×D4—32no—
64C2×C3×D4—32no—
65C2×C3×D4—62no—
66C2×C3×D4—61no—
67C2×C3×D4—61no—
68C22×C3×C4—31no—
69C22×C3×C4—31no—
70C22×C3×C4—31no—
71C22×C3×C4—32no—
72C22×C3×C4—31no—
73C22×C3×C4—32no—
74C22×C3×C4—31no—
75C22×C3×C4—42no—
76C22×C3×C4—122no—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—34no—
79C3×C4⋊C4—32no—
80C3×C4⋊C4—32no—
81C3×C4⋊C4—32no—
82C3×C22⋊C4—34no—
83C3×C22⋊C4—34no—
84C3×C22⋊C4—34no—
85C3×C22⋊C4—34no—
86C3×C42—32no—
87C3×C42—32no—
88C22×C9—161no—
89C22⋊C9—41no—
90C4×C9—162no—
91C2×#41C22⋊C912yescomplemented
92C22×D4—31no—
93C23×C4—31no—
94C23×C4C22⋊C911yescharacteristic
95#126—34no—
96#125—32no—
97#124—32no—
98#123—34no—
99#123—34no—
100C22⋊Q8—32no—
101C22⋊Q8—32no—
102C22⋊Q8—32no—
103#121—32no—
104#121—32no—
105#121—32no—
106#120—34no—
107C4×Q8C22⋊C912yescomplemented
108C4×D4—34no—
109C4×D4—34no—
110C4×D4—32no—
111#117—32no—
112C2×C4⋊C4—31no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4—31no—
115C2×C42—31no—
116C23×C3—31no—
117C23×C3—31no—
118C23×C3—32no—
119C23×C3—41no—
120C23×C3—41no—
121C23×C3—61no—
122C23×C3—61no—
123C23×C3—121no—
124C23×C3—121no—
125C23×C3#41⋊C311yescharacteristic
126C3×Q8—42no—
127C3×Q8—42no—
128C3×D4—64no—
129C3×D4—64no—
130C3×D4—122no—
131C3×D4—122no—
132C2×C3×C4—32no—
133C2×C3×C4—32no—
134C2×C3×C4—31no—
135C2×C3×C4—32no—
136C2×C3×C4—61no—
137C2×C3×C4—61no—
138C2×C3×C4—62no—
139C2×C3×C4—64no—
140C2×C3×C4—121no—
141C2×C3×C4—122no—
142C2×C3×C4—121no—
143C2×C3×C4—121no—
144C2×C3×C4C22⋊A411yescharacteristic
145C2×C9—161no—
146C2×C9—161no—
147C2×C9—161no—
148C24—31no—
149C24C2×C22⋊C911yescharacteristic
150#41—44no—
151C2×Q8C2×C22⋊C912yes—
152C2×Q8C2×C22⋊C912yes—
153C2×D4—34no—
154C2×D4—32no—
155C2×D4—32no—
156C2×D4—61no—
157C2×D4—62no—
158C2×D4—61no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—32no—
163C22×C4—31no—
164C22×C4—32no—
165C22×C4—31no—
166C22×C4—42no—
167C22×C4—122no—
168C4⋊C4—32no—
169C4⋊C4—34no—
170C4⋊C4—32no—
171C4⋊C4—32no—
172C4⋊C4—32no—
173C22⋊C4—34no—
174C22⋊C4—34no—
175C22⋊C4—34no—
176C22⋊C4—34no—
177C42—32no—
178C42—32no—
179C22×C3—31no—
180C22×C3—31no—
181C22×C3—41no—
182C22×C3—61no—
183C22×C3—61no—
184C22×C3—62no—
185C22×C3—62no—
186C22×C3—62no—
187C22×C3—121no—
188C22×C3—122no—
189C22×C3—122no—
190C22×C3C2×C22⋊A411yescharacteristic
191C3×C4—62no—
192C3×C4—61no—
193C3×C4—62no—
194C3×C4—62no—
195C3×C4#99712yes—
196C9—161noSylow
197C23—31no—
198C23—31no—
199C23—32no—
200C23—41no—
201C23—41no—
202C23—61no—
203C23—61no—
204C23—121no—
205C23—121no—
206C23#41⋊C911yescharacteristic, complemented
207Q8—42no—
208Q8—42no—
209D4—64no—
210D4—64no—
211D4—122no—
212D4—122no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—31no—
216C2×C4—32no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—62no—
220C2×C4—64no—
221C2×C4—122no—
222C2×C4—121no—
223C2×C4—121no—
224C2×C4—121no—
225C2×C4C22⋊(C22⋊C9)11yescharacteristic
226C2×C3—61no—
227C2×C3—61no—
228C2×C3—122no—
229C2×C3#594811yescharacteristic
230C2×C3C2×#99711yescharacteristic
231C2×C3#595011yescharacteristic
232C22—31no—
233C22—31no—
234C22—41no—
235C22—61no—
236C22—61no—
237C22—62no—
238C22—62no—
239C22—62no—
240C22—121no—
241C22—122no—
242C22—122no—
243C22C2×C22⋊(C22⋊C9)11yescharacteristic
244C4—62no—
245C4—61no—
246C4—62no—
247C4—62no—
248C4#6370312yes—
249C3#3459⋊C311yescharacteristic
250C2—61no—
251C2—61no—
252C2—122no—
253C2#1059312311yescharacteristic
254C2C2×#6370311yescharacteristic
255C2#1059312511yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C22⋊A4
commutator, hypocenter#543C2×C9
Frattini, socleC22×C3C2×C22⋊A4
Fitting, prime radicalC3×#3459C3
cosocleC3×#543C2×C3
LCSG ▷ #543
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. #543⋊C9C2semidirect
18. #3459C9semidirect
91. C2×#41C22⋊C9semidirect
107. C4×Q8C22⋊C9semidirect
206. C23#41⋊C9semidirect

Maximal subgroups

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

subgroupindex
#543⋊C92
C3×#34593
C2×#41⋊C94
C2×C4×C22⋊C94
C4×Q8⋊C94