Finite Group Databasebeta

← C3×#10935 #10629542

Subgroups — C3×#10935 #10629542

Order 768.

2594 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 2594 conjugacy classes of subgroups — too many to draw. Only its 302 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
1C2×C3×#379C213yesmaximal
2C3×#2273C211yesmaximal, characteristic, complemented
3C2×C3×#342C213yesmaximal, complemented
4#10935C311yesmaximal, characteristic, direct product factor, Sylow
5C26×C3C2211yescharacteristic
6C22×C3×C22⋊C4—26no—
7C22×C3×C22⋊C4C2213yes—
8C22×C3×C22⋊C4C2213yes—
9C3×#379—26no—
10C3×#379C4112yescomplemented
11C3×C4×C22⋊C4—26no—
12C3×#342—26no—
13C2×#379C2×C313yes—
14#2273C2×C311yescharacteristic, complemented
15C2×#342C2×C313yescomplemented
16C25×C3—23no—
17C25×C3—23no—
18C25×C3—26no—
19C25×C3—48no—
20C25×C3C2×C416yes—
21C25×C3C2311yescharacteristic
22C23×C3×C4—23no—
23C23×C3×C4D416yes—
24C2×C3×C22⋊C4—26no—
25C2×C3×C22⋊C4—26no—
26C2×C3×C22⋊C4—26no—
27C2×C3×C22⋊C4—26no—
28C2×C3×C22⋊C4—23no—
29C2×C3×C22⋊C4—224no—
30C2×C3×C22⋊C4—424no—
31C2×C3×C22⋊C4C2×C416yes—
32C2×C3×C22⋊C4C2×C416yes—
33C2×C3×C22⋊C4D416yes—
34C2×C3×C42—24no—
35C3×#95—24no—
36C26C22×C311yescharacteristic
37C22×C22⋊C4—26no—
38C22×C22⋊C4C22×C313yes—
39C22×C22⋊C4C22×C313yes—
40#379—26no—
41#379C3×C4112yescomplemented
42C4×C22⋊C4—26no—
43#342—26no—
44C24×C3—224no—
45C24×C3—26no—
46C24×C3—212no—
47C24×C3—26no—
48C24×C3—212no—
49C24×C3—424no—
50C24×C3—44no—
51C24×C3—412no—
52C24×C3—424no—
53C24×C3—448no—
54C24×C3—41no—
55C24×C3—46no—
56C24×C3—43no—
57C24×C3—43no—
58C24×C3—42no—
59C24×C3—41no—
60C24×C3C4214yescomplemented
61C24×C3C22×C413yes—
62C24×C3C2×D413yes—
63C24×C3C2×D413yes—
64C24×C3#4116yes—
65C22×C3×C4—26no—
66C22×C3×C4—26no—
67C22×C3×C4—26no—
68C22×C3×C4—412no—
69C22×C3×C4—412no—
70C22×C3×C4—46no—
71C22×C3×C4C22⋊C4112yescomplemented
72C3×C22⋊C4—212no—
73C3×C22⋊C4—212no—
74C3×C22⋊C4—212no—
75C3×C22⋊C4—448no—
76C3×C22⋊C4—43no—
77C3×C22⋊C4—43no—
78C3×C42—84no—
79C25—23no—
80C25—23no—
81C25—26no—
82C25—48no—
83C25C2×C3×C416yes—
84C25C23×C311yescharacteristic
85C23×C4—23no—
86C23×C4C3×D416yes—
87C2×C22⋊C4—26no—
88C2×C22⋊C4—26no—
89C2×C22⋊C4—26no—
90C2×C22⋊C4—26no—
91C2×C22⋊C4—23no—
92C2×C22⋊C4—224no—
93C2×C22⋊C4—424no—
94C2×C22⋊C4C2×C3×C416yes—
95C2×C22⋊C4C2×C3×C416yes—
96C2×C22⋊C4C3×D416yes—
97C2×C42—24no—
98#95—24no—
99C23×C3—212no—
100C23×C3—224no—
101C23×C3—248no—
102C23×C3—212no—
103C23×C3—48no—
104C23×C3—46no—
105C23×C3—46no—
106C23×C3—412no—
107C23×C3—412no—
108C23×C3—448no—
109C23×C3—424no—
110C23×C3—424no—
111C23×C3—424no—
112C23×C3—448no—
113C23×C3—424no—
114C23×C3—416no—
115C23×C3—48no—
116C23×C3—46no—
117C23×C3—412no—
118C23×C3—46no—
119C23×C3—48no—
120C23×C3C2×C4211yescharacteristic
121C23×C3C2×C22⋊C413yes—
122C23×C3#11713yes—
123C23×C3C4×D416yes—
124C23×C3C4×D416yes—
125C23×C3#12011yescharacteristic
126C23×C3#12116yes—
127C23×C3#12313yes—
128C23×C3#12413yes—
129C23×C3#12612yes—
130C23×C3#12711yescharacteristic
131C2×C3×C4—212no—
132C2×C3×C4—412no—
133C2×C3×C4—412no—
134C2×C3×C4—412no—
135C2×C3×C4—46no—
136C2×C3×C4—86no—
137C2×C3×C4—812no—
138C24—224no—
139C24—26no—
140C24—212no—
141C24—26no—
142C24—212no—
143C24—424no—
144C24—44no—
145C24—412no—
146C24—424no—
147C24—448no—
148C24—41no—
149C24—46no—
150C24—43no—
151C24—43no—
152C24—42no—
153C24—41no—
154C24C3×C4214yescomplemented
155C24C22×C3×C413yes—
156C24C2×C3×D413yes—
157C24C2×C3×D413yes—
158C24C3×#4116yes—
159C22×C4—26no—
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—412no—
163C22×C4—412no—
164C22×C4—46no—
165C22×C4C3×C22⋊C4112yescomplemented
166C22⋊C4—212no—
167C22⋊C4—212no—
168C22⋊C4—212no—
169C22⋊C4—448no—
170C22⋊C4—43no—
171C22⋊C4—43no—
172C42—84no—
173C22×C3—26no—
174C22×C3—212no—
175C22×C3—26no—
176C22×C3—212no—
177C22×C3—224no—
178C22×C3—41no—
179C22×C3—43no—
180C22×C3—46no—
181C22×C3—43no—
182C22×C3—42no—
183C22×C3—41no—
184C22×C3—424no—
185C22×C3—412no—
186C22×C3—448no—
187C22×C3—424no—
188C22×C3—44no—
189C22×C3C4×C22⋊C413yes—
190C22×C3#38613yes—
191C22×C3#38816yes—
192C22×C3#39013yes—
193C22×C3B(4,𝔽2)14yes—
194C3×C4—412no—
195C3×C4—86no—
196C23—212no—
197C23—224no—
198C23—248no—
199C23—212no—
200C23—48no—
201C23—46no—
202C23—46no—
203C23—412no—
204C23—412no—
205C23—448no—
206C23—424no—
207C23—424no—
208C23—424no—
209C23—448no—
210C23—424no—
211C23—416no—
212C23—48no—
213C23—46no—
214C23—412no—
215C23—46no—
216C23—48no—
217C23C2×C3×C4211yescharacteristic
218C23C2×C3×C22⋊C413yes—
219C23C3×#11713yes—
220C23C3×C4×D416yes—
221C23C3×C4×D416yes—
222C23C3×#12011yescharacteristic
223C23C3×#12116yes—
224C23C3×#12313yes—
225C23C3×#12413yes—
226C23C3×#12612yes—
227C23C3×#12711yescharacteristic
228C2×C4—212no—
229C2×C4—412no—
230C2×C4—412no—
231C2×C4—412no—
232C2×C4—46no—
233C2×C4—86no—
234C2×C4—812no—
235C2×C3—23no—
236C2×C3—23no—
237C2×C3—26no—
238C2×C3—48no—
239C2×C3#143711yescharacteristic
240C2×C3#188916yes—
241C22—26no—
242C22—212no—
243C22—26no—
244C22—212no—
245C22—224no—
246C22—41no—
247C22—43no—
248C22—46no—
249C22—43no—
250C22—42no—
251C22—41no—
252C22—424no—
253C22—412no—
254C22—448no—
255C22—424no—
256C22—44no—
257C22C3×C4×C22⋊C413yes—
258C22C3×#38613yes—
259C22C3×#38816yes—
260C22C3×#39013yes—
261C22C3×B(4,𝔽2)14yes—
262C4—412no—
263C4—86no—
264C3#1093511yescharacteristic, direct product factor, Sylow
265C2—23no—
266C2—23no—
267C2—26no—
268C2—48no—
269C2C3×#143711yescharacteristic
270C2C3×#188916yes—
subgroupG/H
center, socleC23×C3#120
commutatorC23C2×C3×C42
Frattini, cosocleC25C23×C3
prime radicalC26×C3C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2273C2semidirect
3. C2×C3×#342C2semidirect
4. #10935C3direct (N × H)
10. C3×#379C4semidirect
14. #2273C2×C3semidirect
15. C2×#342C2×C3semidirect
41. #379C3×C4semidirect
60. C24×C3C42semidirect
71. C22×C3×C4C22⋊C4semidirect
154. C24C3×C42semidirect
165. C22×C4C3×C22⋊C4semidirect
264. C3#10935direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3792
C3×#22732
C2×C3×#3422
#109353