Finite Group Databasebeta

← C3×#10936 #10629543

Subgroups — C3×#10936 #10629543

Order 768.

1258 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 1258 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×#381C213yesmaximal
2C3×#2273C211yesmaximal, characteristic
3C2×C3×#342C213yesmaximal
4#10936C311yesmaximal, characteristic, direct product factor, Sylow
5C24×C3×C4C2211yescharacteristic
6C22×C3×C22⋊C4C2213yes—
7C22×C3×C22⋊C4C2213yes—
8C3×#381—26no—
9C3×#381C4112yescomplemented
10C3×C4×C22⋊C4—26no—
11C2×C3×#95—26no—
12C3×#342—26no—
13C2×#381C2×C313yes—
14#2273C2×C311yescharacteristic
15C2×#342C2×C313yes—
16C25×C3C2311yescharacteristic
17C23×C3×C4—23no—
18C23×C3×C4—23no—
19C23×C3×C4—26no—
20C23×C3×C4—23no—
21C23×C3×C4C2×C416yes—
22C23×C3×C4D416yes—
23C2×C3×C22⋊C4—26no—
24C2×C3×C22⋊C4—26no—
25C2×C3×C22⋊C4—26no—
26C2×C3×C22⋊C4—26no—
27C2×C3×C22⋊C4—23no—
28C2×C3×C22⋊C4C2×C416yes—
29C2×C3×C22⋊C4C2×C416yes—
30C2×C3×C22⋊C4D416yes—
31C2×C3×C42—24no—
32C3×#95—24no—
33C3×#95—224no—
34C24×C4C22×C311yescharacteristic
35C22×C22⋊C4C22×C313yes—
36C22×C22⋊C4C22×C313yes—
37#381—26no—
38#381C3×C4112yescomplemented
39C4×C22⋊C4—26no—
40C2×#95—26no—
41#342—26no—
42C24×C3—44no—
43C24×C3C22×C413yes—
44C24×C3C2×D413yes—
45C24×C3C2×D413yes—
46C24×C3#4116yes—
47C22×C3×C4—26no—
48C22×C3×C4—26no—
49C22×C3×C4—212no—
50C22×C3×C4—212no—
51C22×C3×C4—26no—
52C22×C3×C4—26no—
53C22×C3×C4—26no—
54C22×C3×C4—41no—
55C22×C3×C4—46no—
56C22×C3×C4—43no—
57C22×C3×C4—43no—
58C22×C3×C4—42no—
59C22×C3×C4—41no—
60C22×C3×C4—412no—
61C22×C3×C4—412no—
62C22×C3×C4—46no—
63C22×C3×C4C4214yescomplemented
64C22×C3×C4C22⋊C4112yes—
65C3×C22⋊C4—212no—
66C3×C22⋊C4—212no—
67C3×C22⋊C4—212no—
68C3×C22⋊C4—43no—
69C3×C22⋊C4—43no—
70C3×C42—84no—
71C25C23×C311yescharacteristic
72C23×C4—23no—
73C23×C4—23no—
74C23×C4—26no—
75C23×C4—23no—
76C23×C4C2×C3×C416yes—
77C23×C4C3×D416yes—
78C2×C22⋊C4—26no—
79C2×C22⋊C4—26no—
80C2×C22⋊C4—26no—
81C2×C22⋊C4—26no—
82C2×C22⋊C4—23no—
83C2×C22⋊C4C2×C3×C416yes—
84C2×C22⋊C4C2×C3×C416yes—
85C2×C22⋊C4C3×D416yes—
86C2×C42—24no—
87#95—24no—
88#95—224no—
89C23×C3—212no—
90C23×C3—46no—
91C23×C3—46no—
92C23×C3—412no—
93C23×C3C2×C4211yescharacteristic
94C23×C3C2×C22⋊C413yes—
95C23×C3#11713yes—
96C23×C3C4×D416yes—
97C23×C3C4×D416yes—
98C23×C3#12011yescharacteristic
99C23×C3#12116yes—
100C23×C3#12313yes—
101C23×C3#12413yes—
102C23×C3#12612yes—
103C23×C3#12711yescharacteristic
104C2×C3×C4—212no—
105C2×C3×C4—212no—
106C2×C3×C4—46no—
107C2×C3×C4—412no—
108C2×C3×C4—46no—
109C2×C3×C4—412no—
110C2×C3×C4—412no—
111C2×C3×C4—412no—
112C2×C3×C4—46no—
113C2×C3×C4—86no—
114C2×C3×C4—812no—
115C24—44no—
116C24C22×C3×C413yes—
117C24C2×C3×D413yes—
118C24C2×C3×D413yes—
119C24C3×#4116yes—
120C22×C4—26no—
121C22×C4—26no—
122C22×C4—212no—
123C22×C4—212no—
124C22×C4—26no—
125C22×C4—26no—
126C22×C4—26no—
127C22×C4—41no—
128C22×C4—46no—
129C22×C4—43no—
130C22×C4—43no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—412no—
134C22×C4—412no—
135C22×C4—46no—
136C22×C4C3×C4214yescomplemented
137C22×C4C3×C22⋊C4112yes—
138C22⋊C4—212no—
139C22⋊C4—212no—
140C22⋊C4—212no—
141C22⋊C4—43no—
142C22⋊C4—43no—
143C42—84no—
144C22×C3—26no—
145C22×C3—212no—
146C22×C3—26no—
147C22×C3—212no—
148C22×C3—41no—
149C22×C3—43no—
150C22×C3—46no—
151C22×C3—43no—
152C22×C3—42no—
153C22×C3—41no—
154C22×C3C4×C22⋊C413yes—
155C22×C3#38613yes—
156C22×C3#38816yes—
157C22×C3#39013yes—
158C22×C3#45814yes—
159C3×C4—412no—
160C3×C4—44no—
161C3×C4—86no—
162C23—212no—
163C23—46no—
164C23—46no—
165C23—412no—
166C23C2×C3×C4211yescharacteristic
167C23C2×C3×C22⋊C413yes—
168C23C3×#11713yes—
169C23C3×C4×D416yes—
170C23C3×C4×D416yes—
171C23C3×#12011yescharacteristic
172C23C3×#12116yes—
173C23C3×#12313yes—
174C23C3×#12413yes—
175C23C3×#12612yes—
176C23C3×#12711yescharacteristic
177C2×C4—212no—
178C2×C4—212no—
179C2×C4—46no—
180C2×C4—412no—
181C2×C4—46no—
182C2×C4—412no—
183C2×C4—412no—
184C2×C4—412no—
185C2×C4—46no—
186C2×C4—86no—
187C2×C4—812no—
188C2×C3—23no—
189C2×C3—23no—
190C2×C3—26no—
191C2×C3#143711yescharacteristic
192C2×C3#189016yes—
193C22—26no—
194C22—212no—
195C22—26no—
196C22—212no—
197C22—41no—
198C22—43no—
199C22—46no—
200C22—43no—
201C22—42no—
202C22—41no—
203C22C3×C4×C22⋊C413yes—
204C22C3×#38613yes—
205C22C3×#38816yes—
206C22C3×#39013yes—
207C22C3×#45814yes—
208C4—412no—
209C4—44no—
210C4—86no—
211C3#1093611yescharacteristic, direct product factor, Sylow
212C2—23no—
213C2—23no—
214C2—26no—
215C2C3×#143711yescharacteristic
216C2C3×#189016yes—
subgroupG/H
center, socleC23×C3#120
commutatorC23C2×C3×C42
Frattini, cosocleC25C23×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
4. #10936C3direct (N × H)
9. C3×#381C4semidirect
38. #381C3×C4semidirect
63. C22×C3×C4C42semidirect
136. C22×C4C3×C42semidirect
211. C3#10936direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3812
C3×#22732
C2×C3×#3422
#109363