Finite Group Databasebeta

← C22×C3×#343 #10638963

Subgroups — C22×C3×#343 #10638963

Order 768.

2494 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C212yesmaximal
2C23×C3×C4⋊C4C211yesmaximal, characteristic
3C2×C3×#343C2112yesmaximal, direct product factor
4C22×#343C311yesmaximal, characteristic, direct product factor, Sylow
5C24×C3×C4C412yes—
6C24×C3×C4C2211yescharacteristic
7C22×C3×M16—28no—
8C22×C3×M16—26no—
9C22×C3×M16C4116yescomplemented
10C22×C3×M16C22112yes—
11C23×C3×C8—22no—
12C22×C3×C4⋊C4—26no—
13C22×C3×C4⋊C4—22no—
14C22×C3×C4⋊C4C416yes—
15C22×C3×C4⋊C4C2216yes—
16C3×#343C22116yesdirect product factor
17C23×M16C2×C312yes—
18C23×C4⋊C4C2×C311yescharacteristic
19C2×#343C2×C3112yesdirect product factor
20C25×C3C2×C411yescharacteristic
21C23×C3×C4—26no—
22C23×C3×C4—22no—
23C23×C3×C4—48no—
24C23×C3×C4C2×C4112yes—
25C23×C3×C4C2×C413yes—
26C23×C3×C4C2×C418yes—
27C23×C3×C4D414yes—
28C23×C3×C4D416yes—
29C23×C3×C4D412yes—
30C23×C3×C4Q814yes—
31C23×C3×C4C2313yes—
32C2×C3×M16—248no—
33C2×C3×M16—224no—
34C2×C3×M16—212no—
35C2×C3×M16C2×C4148yescomplemented
36C2×C3×M16C2318yes—
37C22×C3×C8—216no—
38C22×C3×C8—212no—
39C2×C3×C4⋊C4—212no—
40C2×C3×C4⋊C4—212no—
41C2×C3×C4⋊C4—212no—
42C2×C3×C4⋊C4—212no—
43C2×C3×C4⋊C4C2×C4112yes—
44C2×C3×C4⋊C4C2314yes—
45C24×C4C3×C412yes—
46C24×C4C22×C311yescharacteristic
47C22×M16—28no—
48C22×M16—26no—
49C22×M16C3×C4116yescomplemented
50C22×M16C22×C3112yes—
51C23×C8—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—26no—
54C22×C4⋊C4C3×C416yes—
55C22×C4⋊C4C22×C316yes—
56#343C22×C3116yesdirect product factor
57C24×C3—24no—
58C24×C3—24no—
59C24×C3C4214yes—
60C24×C3C22⋊C413yes—
61C24×C3C22⋊C411yescharacteristic
62C24×C3C4⋊C414yes—
63C24×C3C22×C413yes—
64C22×C3×C4—212no—
65C22×C3×C4—212no—
66C22×C3×C4—424no—
67C22×C3×C4—48no—
68C22×C3×C4—424no—
69C22×C3×C4C42112yes—
70C22×C3×C4C22⋊C416yes—
71C22×C3×C4C22⋊C416yes—
72C22×C3×C4C22⋊C418yes—
73C22×C3×C4C22⋊C4124yes—
74C22×C3×C4C4⋊C4112yes—
75C22×C3×C4C4⋊C4124yes—
76C22×C3×C4C4⋊C418yes—
77C22×C3×C4C22×C418yes—
78C22×C3×C4C22×C413yes—
79C22×C3×C4C22×C4112yes—
80C22×C3×C4C2×D416yes—
81C22×C3×C4C2×D416yes—
82C22×C3×C4C2×D416yes—
83C22×C3×C4C2×Q816yes—
84C22×C3×C4C2411yescharacteristic
85C3×M16—248no—
86C3×M16—28no—
87C3×M16C22×C4116yescomplemented
88C2×C3×C8—248no—
89C2×C3×C8—28no—
90C3×C4⋊C4—224no—
91C3×C4⋊C4—28no—
92C25C2×C3×C411yescharacteristic
93C23×C4—26no—
94C23×C4—22no—
95C23×C4—48no—
96C23×C4C2×C3×C4112yes—
97C23×C4C2×C3×C413yes—
98C23×C4C2×C3×C418yes—
99C23×C4C3×D414yes—
100C23×C4C3×D412yes—
101C23×C4C3×D416yes—
102C23×C4C3×Q814yes—
103C23×C4C23×C313yes—
104C2×M16—248no—
105C2×M16—224no—
106C2×M16—212no—
107C2×M16C2×C3×C4148yescomplemented
108C2×M16C23×C318yes—
109C22×C8—216no—
110C22×C8—212no—
111C2×C4⋊C4—212no—
112C2×C4⋊C4—212no—
113C2×C4⋊C4—212no—
114C2×C4⋊C4—212no—
115C2×C4⋊C4C2×C3×C4112yes—
116C2×C4⋊C4C23×C314yes—
117C23×C3—216no—
118C23×C3—216no—
119C23×C3—26no—
120C23×C3—26no—
121C23×C3—26no—
122C23×C3—26no—
123C23×C3#95112yes—
124C23×C3#9514yes—
125C23×C3#10014yes—
126C23×C3#10114yes—
127C23×C3C2×C4216yes—
128C23×C3C2×C22⋊C413yes—
129C23×C3C2×C22⋊C413yes—
130C23×C3C2×C4⋊C416yes—
131C23×C3C23×C411yescharacteristic
132C2×C3×C4—28no—
133C2×C3×C4—424no—
134C2×C3×C4—424no—
135C2×C3×C4—48no—
136C2×C3×C4#95124yes—
137C2×C3×C4#95124yes—
138C2×C3×C4C2×C4216yes—
139C2×C3×C4C2×C22⋊C4112yes—
140C2×C3×C4C2×C22⋊C4112yes—
141C2×C3×C4C2×C22⋊C416yes—
142C2×C3×C4C2×C4⋊C4112yes—
143C2×C3×C4C2×C4⋊C4112yes—
144C2×C3×C4C2×C4⋊C416yes—
145C2×C3×C4C23×C412yes—
146C2×C3×C4C22×D411yescharacteristic
147C2×C3×C4C22×D412yes—
148C2×C3×C4C22×Q811yescharacteristic
149C3×C8—216no—
150C24—24no—
151C24—24no—
152C24C3×C4214yes—
153C24C3×C22⋊C413yes—
154C24C3×C22⋊C411yescharacteristic
155C24C3×C4⋊C414yes—
156C24C22×C3×C413yes—
157C22×C4—212no—
158C22×C4—212no—
159C22×C4—424no—
160C22×C4—48no—
161C22×C4—424no—
162C22×C4C3×C42112yes—
163C22×C4C3×C22⋊C416yes—
164C22×C4C3×C22⋊C416yes—
165C22×C4C3×C22⋊C418yes—
166C22×C4C3×C22⋊C4124yes—
167C22×C4C3×C4⋊C4112yes—
168C22×C4C3×C4⋊C4124yes—
169C22×C4C3×C4⋊C418yes—
170C22×C4C22×C3×C418yes—
171C22×C4C22×C3×C413yes—
172C22×C4C22×C3×C4112yes—
173C22×C4C2×C3×D416yes—
174C22×C4C2×C3×D416yes—
175C22×C4C2×C3×D416yes—
176C22×C4C2×C3×Q816yes—
177C22×C4C24×C311yescharacteristic
178M16—248no—
179M16—28no—
180M16C22×C3×C4116yescomplemented
181C2×C8—248no—
182C2×C8—28no—
183C4⋊C4—224no—
184C4⋊C4—28no—
185C22×C3—23no—
186C22×C3—21no—
187C22×C3—23no—
188C22×C3—21no—
189C22×C3—224no—
190C22×C3—212no—
191C22×C3—212no—
192C22×C3#343116yesdirect product factor
193C22×C3C2×#9516yes—
194C22×C3C2×#9516yes—
195C22×C3C2×#10016yes—
196C22×C3C2×#10116yes—
197C22×C3C22×C4211yescharacteristic
198C22×C3C22×C22⋊C411yescharacteristic
199C22×C3C22×C4⋊C411yescharacteristic
200C3×C4—48no—
201C3×C4C2×#95112yes—
202C3×C4C22×C22⋊C412yes—
203C3×C4C22×C4⋊C412yes—
204C23—26no—
205C23—26no—
206C23—26no—
207C23—26no—
208C23—216no—
209C23—216no—
210C23C3×#95112yes—
211C23C3×#9514yes—
212C23C3×#10014yes—
213C23C3×#10114yes—
214C23C2×C3×C4216yes—
215C23C2×C3×C22⋊C413yes—
216C23C2×C3×C22⋊C413yes—
217C23C2×C3×C4⋊C416yes—
218C23C23×C3×C411yescharacteristic
219C2×C4—28no—
220C2×C4—424no—
221C2×C4—424no—
222C2×C4—48no—
223C2×C4C3×#95124yes—
224C2×C4C3×#95124yes—
225C2×C4C2×C3×C4216yes—
226C2×C4C2×C3×C22⋊C416yes—
227C2×C4C2×C3×C22⋊C4112yes—
228C2×C4C2×C3×C22⋊C4112yes—
229C2×C4C2×C3×C4⋊C416yes—
230C2×C4C2×C3×C4⋊C4112yes—
231C2×C4C2×C3×C4⋊C4112yes—
232C2×C4C23×C3×C412yes—
233C2×C4C22×C3×D411yescharacteristic
234C2×C4C22×C3×D412yes—
235C2×C4C22×C3×Q811yescharacteristic
236C8—216no—
237C2×C3—21no—
238C2×C3—26no—
239C2×C3—21no—
240C2×C3C2×#343112yesdirect product factor
241C2×C3C22×#9511yescharacteristic
242C2×C3C22×#10011yescharacteristic
243C2×C3C22×#10111yescharacteristic
244C22—224no—
245C22—212no—
246C22—212no—
247C22—23no—
248C22—21no—
249C22—23no—
250C22—21no—
251C22C3×#343116yesdirect product factor
252C22C2×C3×#9516yes—
253C22C2×C3×#9516yes—
254C22C2×C3×#10016yes—
255C22C2×C3×#10116yes—
256C22C22×C3×C4211yescharacteristic
257C22C22×C3×C22⋊C411yescharacteristic
258C22C22×C3×C4⋊C411yescharacteristic
259C4—48no—
260C4C2×C3×#95112yes—
261C4C22×C3×C22⋊C412yes—
262C4C22×C3×C4⋊C412yes—
263C3C22×#34311yescharacteristic, direct product factor, Sylow
264C2—26no—
265C2—21no—
266C2—21no—
267C2C2×C3×#343112yesdirect product factor
268C2C22×C3×#9511yescharacteristic
269C2C22×C3×#10011yescharacteristic
270C2C22×C3×#10111yescharacteristic
subgroupG/H
center, socleC24×C3C22⋊C4
commutatorC22C22×C3×C42
Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3C2×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C24×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#343C2direct (N × H)
4. C22×#343C3direct (N × H)
9. C22×C3×M16C4semidirect
16. C3×#343C22direct (N × H)
19. C2×#343C2×C3direct (N × H)
35. C2×C3×M16C2×C4semidirect
49. C22×M16C3×C4semidirect
56. #343C22×C3direct (N × H)
87. C3×M16C22×C4semidirect
107. C2×M16C2×C3×C4semidirect
180. M16C22×C3×C4semidirect
192. C22×C3#343direct (N × H)
240. C2×C3C2×#343direct (N × H)
251. C22C3×#343direct (N × H)
263. C3C22×#343direct (N × H)
267. C2C2×C3×#343direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×M162
C23×C3×C4⋊C42
C2×C3×#3432
C22×#3433