Finite Group Databasebeta

← C2×C3×#2062 #10642170

Subgroups — C2×C3×#2062 #10642170

Order 768.

1614 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 1614 conjugacy classes of subgroups — too many to draw. Only its 382 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
2C2×C3×#385C213yesmaximal
3C3×#2062C218yesmaximal, direct product factor
4C2×C3×#342C213yesmaximal
5C2×#2062C311yesmaximal, characteristic, direct product factor, Sylow
6C24×C3×C4C2211yescharacteristic
7C3×#543—26no—
8C3×#543C2214yes—
9C2×C3×C22⋊Q8—23no—
10C2×C3×C22⋊Q8—23no—
11C22×C3×C4⋊C4—23no—
12C22×C3×C22⋊C4C2213yes—
13C22×C3×C22⋊C4C2213yes—
14C3×#385—26no—
15C3×#385C22112yes—
16C2×C3×#95—23no—
17C3×#342C22112yes—
18C2×#543C2×C311yescharacteristic
19C2×#385C2×C313yes—
20#2062C2×C318yesdirect product factor
21C2×#342C2×C313yes—
22C25×C3C2311yescharacteristic
23C22×C3×Q8—21no—
24C22×C3×Q8—21no—
25C23×C3×C4—23no—
26C23×C3×C4—23no—
27C23×C3×C4—23no—
28C23×C3×C4—44no—
29C23×C3×C4D413yes—
30C23×C3×C4D413yes—
31C23×C3×C4C2312yes—
32C3×C22⋊Q8—212no—
33C3×C22⋊Q8—212no—
34C3×C22⋊Q8—212no—
35C3×C22⋊Q8—212no—
36C2×C3×C4⋊C4—23no—
37C2×C3×C4⋊C4—23no—
38C2×C3×C4⋊C4—212no—
39C2×C3×C4⋊C4—412no—
40C2×C3×C22⋊C4—23no—
41C2×C3×C22⋊C4—26no—
42C2×C3×C22⋊C4—26no—
43C2×C3×C22⋊C4—26no—
44C2×C3×C22⋊C4D416yes—
45C2×C3×C22⋊C4Q8112yescomplemented
46C2×C3×C22⋊C4C2316yes—
47C2×C3×C22⋊C4C2316yes—
48C3×#95—212no—
49C24×C4C22×C311yescharacteristic
50#543—26no—
51#543C22×C314yes—
52C2×C22⋊Q8—23no—
53C2×C22⋊Q8—23no—
54C22×C4⋊C4—23no—
55C22×C22⋊C4C22×C313yes—
56C22×C22⋊C4C22×C313yes—
57#385—26no—
58#385C22×C3112yes—
59C2×#95—23no—
60#342C22×C3112yes—
61C24×C3—42no—
62C24×C3—42no—
63C24×C3C2×D413yes—
64C24×C3C2×D413yes—
65C24×C3C2×Q816yes—
66C24×C3#4112yes—
67C24×C3C2411yescharacteristic
68C2×C3×Q8—24no—
69C2×C3×Q8—24no—
70C2×C3×Q8—82no—
71C2×C3×Q8—82no—
72C22×C3×C4—212no—
73C22×C3×C4—23no—
74C22×C3×C4—23no—
75C22×C3×C4—212no—
76C22×C3×C4—26no—
77C22×C3×C4—26no—
78C22×C3×C4—23no—
79C22×C3×C4—23no—
80C22×C3×C4—44no—
81C22×C3×C4—412no—
82C22×C3×C4—46no—
83C22×C3×C4—46no—
84C22×C3×C4—46no—
85C22×C3×C4—43no—
86C22×C3×C4—41no—
87C22×C3×C4—46no—
88C22×C3×C4—46no—
89C22×C3×C4C2×D416yes—
90C22×C3×C4C2×D416yes—
91C3×C4⋊C4—212no—
92C3×C4⋊C4—212no—
93C3×C4⋊C4—424no—
94C3×C22⋊C4—212no—
95C3×C22⋊C4—212no—
96C3×C22⋊C4—212no—
97C3×C22⋊C4—26no—
98C3×C22⋊C4C2×Q8112yescomplemented
99C25C23×C311yescharacteristic
100C22×Q8—21no—
101C22×Q8—21no—
102C23×C4—23no—
103C23×C4—23no—
104C23×C4—23no—
105C23×C4—44no—
106C23×C4C3×D413yes—
107C23×C4C3×D413yes—
108C23×C4C23×C312yes—
109C22⋊Q8—212no—
110C22⋊Q8—212no—
111C22⋊Q8—212no—
112C22⋊Q8—212no—
113C2×C4⋊C4—23no—
114C2×C4⋊C4—23no—
115C2×C4⋊C4—212no—
116C2×C4⋊C4—412no—
117C2×C22⋊C4—26no—
118C2×C22⋊C4—26no—
119C2×C22⋊C4—23no—
120C2×C22⋊C4—26no—
121C2×C22⋊C4C3×D416yes—
122C2×C22⋊C4C3×Q8112yescomplemented
123C2×C22⋊C4C23×C316yes—
124C2×C22⋊C4C23×C316yes—
125#95—212no—
126C23×C3—26no—
127C23×C3—26no—
128C23×C3—43no—
129C23×C3—41no—
130C23×C3—43no—
131C23×C3—412no—
132C23×C3—44no—
133C23×C3—41no—
134C23×C3#12013yes—
135C23×C3#12011yescharacteristic
136C23×C3C22⋊Q816yes—
137C23×C3C22⋊Q816yes—
138C23×C3C4⋊Q816yes—
139C23×C3C4⋊Q816yes—
140C23×C3C22×D413yes—
141C23×C3C22×Q813yes—
142C23×C3C2×#4111yescharacteristic
143C3×Q8—84no—
144C3×Q8—84no—
145C2×C3×C4—212no—
146C2×C3×C4—26no—
147C2×C3×C4—26no—
148C2×C3×C4—212no—
149C2×C3×C4—46no—
150C2×C3×C4—424no—
151C2×C3×C4—46no—
152C2×C3×C4—46no—
153C2×C3×C4—412no—
154C2×C3×C4—42no—
155C2×C3×C4—44no—
156C2×C3×C4—46no—
157C2×C3×C4—46no—
158C2×C3×C4—46no—
159C2×C3×C4—46no—
160C2×C3×C4—83no—
161C2×C3×C4—83no—
162C24—42no—
163C24—42no—
164C24C2×C3×D413yes—
165C24C2×C3×D413yes—
166C24C2×C3×Q816yes—
167C24C3×#4112yes—
168C24C24×C311yescharacteristic
169C2×Q8—24no—
170C2×Q8—24no—
171C2×Q8—82no—
172C2×Q8—82no—
173C22×C4—212no—
174C22×C4—23no—
175C22×C4—23no—
176C22×C4—212no—
177C22×C4—26no—
178C22×C4—26no—
179C22×C4—23no—
180C22×C4—23no—
181C22×C4—44no—
182C22×C4—412no—
183C22×C4—46no—
184C22×C4—46no—
185C22×C4—46no—
186C22×C4—43no—
187C22×C4—41no—
188C22×C4—46no—
189C22×C4—46no—
190C22×C4C2×C3×D416yes—
191C22×C4C2×C3×D416yes—
192C4⋊C4—212no—
193C4⋊C4—212no—
194C4⋊C4—424no—
195C22⋊C4—212no—
196C22⋊C4—212no—
197C22⋊C4—212no—
198C22⋊C4—26no—
199C22⋊C4C2×C3×Q8112yescomplemented
200C22×C3—212no—
201C22×C3—212no—
202C22×C3—23no—
203C22×C3—23no—
204C22×C3—23no—
205C22×C3—23no—
206C22×C3—46no—
207C22×C3—46no—
208C22×C3—43no—
209C22×C3—41no—
210C22×C3#39516yes—
211C22×C3#39512yes—
212C22×C3B(4,𝔽2)12yes—
213C22×C3#45812yes—
214C22×C3C2×#12011yescharacteristic
215C22×C3C2×C22⋊Q813yes—
216C22×C3C2×C4⋊Q813yes—
217C3×C4—412no—
218C3×C4—44no—
219C3×C4—46no—
220C3×C4—46no—
221C23—26no—
222C23—26no—
223C23—43no—
224C23—41no—
225C23—43no—
226C23—412no—
227C23—44no—
228C23—41no—
229C23C3×#12013yes—
230C23C3×#12011yescharacteristic
231C23C3×C22⋊Q816yes—
232C23C3×C22⋊Q816yes—
233C23C3×C4⋊Q816yes—
234C23C3×C4⋊Q816yes—
235C23C22×C3×D413yes—
236C23C22×C3×Q813yes—
237C23C2×C3×#4111yescharacteristic
238Q8—84no—
239Q8—84no—
240C2×C4—212no—
241C2×C4—26no—
242C2×C4—26no—
243C2×C4—212no—
244C2×C4—46no—
245C2×C4—46no—
246C2×C4—412no—
247C2×C4—424no—
248C2×C4—46no—
249C2×C4—44no—
250C2×C4—46no—
251C2×C4—46no—
252C2×C4—42no—
253C2×C4—46no—
254C2×C4—46no—
255C2×C4—83no—
256C2×C4—83no—
257C2×C3—23no—
258C2×C3—23no—
259C2×C3—26no—
260C2×C3#206214yesdirect product factor
261C2×C3C2×#39511yescharacteristic
262C2×C3C2×B(4,𝔽2)11yescharacteristic
263C2×C3C2×#45811yescharacteristic
264C22—212no—
265C22—212no—
266C22—23no—
267C22—23no—
268C22—23no—
269C22—23no—
270C22—46no—
271C22—46no—
272C22—43no—
273C22—41no—
274C22C3×#39516yes—
275C22C3×#39512yes—
276C22C3×B(4,𝔽2)12yes—
277C22C3×#45812yes—
278C22C2×C3×#12011yescharacteristic
279C22C2×C3×C22⋊Q813yes—
280C22C2×C3×C4⋊Q813yes—
281C4—412no—
282C4—44no—
283C4—46no—
284C4—46no—
285C3C2×#206211yescharacteristic, direct product factor, Sylow
286C2—23no—
287C2—26no—
288C2—23no—
289C2C3×#206214yesdirect product factor
290C2C2×C3×#39511yescharacteristic
291C2C2×C3×B(4,𝔽2)11yescharacteristic
292C2C2×C3×#45811yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2062C2direct (N × H)
5. C2×#2062C3direct (N × H)
20. #2062C2×C3direct (N × H)
45. C2×C3×C22⋊C4Q8semidirect
98. C3×C22⋊C4C2×Q8semidirect
122. C2×C22⋊C4C3×Q8semidirect
199. C22⋊C4C2×C3×Q8semidirect
260. C2×C3#2062direct (N × H)
285. C3C2×#2062direct (N × H)
289. C2C3×#2062direct (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
C2×C3×#3852
C3×#20622
C2×C3×#3422
C2×#20623