Finite Group Databasebeta

← C3×#23565 #10642172

Subgroups — C3×#23565 #10642172

Order 768.

1334 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 366 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×#543C211yesmaximal, characteristic
2C3×#2315C213yesmaximal
3C3×#2062C218yesmaximal, complemented
4C2×C3×#342C213yesmaximal
5#23565C311yesmaximal, characteristic, direct product factor, Sylow
6C3×#543—26no—
7C3×#543C2214yes—
8C2×C3×C22⋊Q8—26no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×C3×C22⋊C4C2213yes—
11C22×C3×C22⋊C4C2213yes—
12C3×#385—26no—
13C3×#385C22112yes—
14C3×#380—26no—
15C3×#342C22112yes—
16C2×#543C2×C311yescharacteristic
17#2315C2×C313yes—
18#2062C2×C318yescomplemented
19C2×#342C2×C313yes—
20C25×C3C2311yescharacteristic
21C22×C3×Q8—22no—
22C23×C3×C4—23no—
23C23×C3×C4D416yes—
24C23×C3×C4C2312yes—
25C3×C22⋊Q8—224no—
26C3×C22⋊Q8—224no—
27C2×C3×C4⋊C4—26no—
28C2×C3×C4⋊C4—212no—
29C2×C3×C22⋊C4—23no—
30C2×C3×C22⋊C4—26no—
31C2×C3×C22⋊C4—26no—
32C2×C3×C22⋊C4—26no—
33C2×C3×C22⋊C4—23no—
34C2×C3×C22⋊C4—23no—
35C2×C3×C22⋊C4D416yes—
36C2×C3×C22⋊C4Q8112yescomplemented
37C2×C3×C22⋊C4C2316yes—
38C2×C3×C22⋊C4C2316yes—
39C3×#95—212no—
40#543—26no—
41#543C22×C314yes—
42C2×C22⋊Q8—26no—
43C22×C22⋊C4C22×C311yescharacteristic
44C22×C22⋊C4C22×C313yes—
45C22×C22⋊C4C22×C313yes—
46#385—26no—
47#385C22×C3112yes—
48#380—26no—
49#342C22×C3112yes—
50C24×C3—82no—
51C24×C3C2×D413yes—
52C24×C3C2×D413yes—
53C24×C3C2×Q816yes—
54C24×C3#4112yes—
55C24×C3C2411yescharacteristic
56C2×C3×Q8—28no—
57C2×C3×Q8—84no—
58C22×C3×C4—26no—
59C22×C3×C4—26no—
60C22×C3×C4—26no—
61C22×C3×C4—212no—
62C22×C3×C4—26no—
63C22×C3×C4—82no—
64C22×C3×C4—86no—
65C22×C3×C4—86no—
66C22×C3×C4C2×D4112yes—
67C3×C4⋊C4—224no—
68C3×C4⋊C4—812no—
69C3×C22⋊C4—212no—
70C3×C22⋊C4—212no—
71C3×C22⋊C4—212no—
72C3×C22⋊C4—26no—
73C3×C22⋊C4—43no—
74C3×C22⋊C4—41no—
75C3×C22⋊C4C2×Q8112yescomplemented
76C25C23×C311yescharacteristic
77C22×Q8—22no—
78C23×C4—23no—
79C23×C4C3×D416yes—
80C23×C4C23×C312yes—
81C22⋊Q8—224no—
82C22⋊Q8—224no—
83C2×C4⋊C4—26no—
84C2×C4⋊C4—212no—
85C2×C22⋊C4—26no—
86C2×C22⋊C4—26no—
87C2×C22⋊C4—23no—
88C2×C22⋊C4—26no—
89C2×C22⋊C4—23no—
90C2×C22⋊C4—23no—
91C2×C22⋊C4C3×D416yes—
92C2×C22⋊C4C3×Q8112yescomplemented
93C2×C22⋊C4C23×C316yes—
94C2×C22⋊C4C23×C316yes—
95#95—212no—
96C23×C3—46no—
97C23×C3—46no—
98C23×C3—42no—
99C23×C3—86no—
100C23×C3—82no—
101C23×C3#12013yes—
102C23×C3#12011yescharacteristic
103C23×C3C22⋊Q816yes—
104C23×C3C22⋊Q816yes—
105C23×C3C4⋊Q816yes—
106C23×C3C4⋊Q816yes—
107C23×C3C22×D413yes—
108C23×C3C22×Q813yes—
109C23×C3C2×#4111yescharacteristic
110C3×Q8—88no—
111C2×C3×C4—212no—
112C2×C3×C4—212no—
113C2×C3×C4—412no—
114C2×C3×C4—412no—
115C2×C3×C4—412no—
116C2×C3×C4—412no—
117C2×C3×C4—42no—
118C2×C3×C4—86no—
119C2×C3×C4—812no—
120C24—82no—
121C24C2×C3×D413yes—
122C24C2×C3×D413yes—
123C24C2×C3×Q816yes—
124C24C3×#4112yes—
125C24C24×C311yescharacteristic
126C2×Q8—28no—
127C2×Q8—84no—
128C22×C4—26no—
129C22×C4—26no—
130C22×C4—26no—
131C22×C4—212no—
132C22×C4—26no—
133C22×C4—82no—
134C22×C4—86no—
135C22×C4—86no—
136C22×C4C2×C3×D4112yes—
137C4⋊C4—224no—
138C4⋊C4—812no—
139C22⋊C4—212no—
140C22⋊C4—212no—
141C22⋊C4—212no—
142C22⋊C4—26no—
143C22⋊C4—43no—
144C22⋊C4—41no—
145C22⋊C4C2×C3×Q8112yescomplemented
146C22×C3—22no—
147C22×C3—26no—
148C22×C3—26no—
149C22×C3—46no—
150C22×C3—46no—
151C22×C3—46no—
152C22×C3—46no—
153C22×C3—43no—
154C22×C3—41no—
155C22×C3#39516yes—
156C22×C3#39512yes—
157C22×C3C2×#12011yescharacteristic
158C22×C3C2×C22⋊Q813yes—
159C22×C3C2×C4⋊Q813yes—
160C3×C4—412no—
161C3×C4—412no—
162C3×C4—82no—
163C23—46no—
164C23—46no—
165C23—42no—
166C23—86no—
167C23—82no—
168C23C3×#12013yes—
169C23C3×#12011yescharacteristic
170C23C3×C22⋊Q816yes—
171C23C3×C22⋊Q816yes—
172C23C3×C4⋊Q816yes—
173C23C3×C4⋊Q816yes—
174C23C22×C3×D413yes—
175C23C22×C3×Q813yes—
176C23C2×C3×#4111yescharacteristic
177Q8—88no—
178C2×C4—212no—
179C2×C4—212no—
180C2×C4—412no—
181C2×C4—412no—
182C2×C4—412no—
183C2×C4—412no—
184C2×C4—42no—
185C2×C4—86no—
186C2×C4—812no—
187C2×C3—23no—
188C2×C3—23no—
189C2×C3—22no—
190C2×C3—26no—
191C2×C3C2×#39511yescharacteristic
192C2×C3#302612yes—
193C22—22no—
194C22—26no—
195C22—26no—
196C22—46no—
197C22—46no—
198C22—46no—
199C22—46no—
200C22—43no—
201C22—41no—
202C22C3×#39516yes—
203C22C3×#39512yes—
204C22C2×C3×#12011yescharacteristic
205C22C2×C3×C22⋊Q813yes—
206C22C2×C3×C4⋊Q813yes—
207C4—412no—
208C4—412no—
209C4—82no—
210C3#2356511yescharacteristic, direct product factor, Sylow
211C2—23no—
212C2—22no—
213C2—26no—
214C2—23no—
215C2C2×C3×#39511yescharacteristic
216C2C3×#302612yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2062C2semidirect
5. #23565C3direct (N × H)
18. #2062C2×C3semidirect
36. C2×C3×C22⋊C4Q8semidirect
75. C3×C22⋊C4C2×Q8semidirect
92. C2×C22⋊C4C3×Q8semidirect
145. C22⋊C4C2×C3×Q8semidirect
210. C3#23565direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5432
C3×#23152
C3×#20622
C2×C3×#3422
#235653