Finite Group Databasebeta

← C22×C3×#342 #10638959

Subgroups — C22×C3×#342 #10638959

Order 768.

4334 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C22⋊C4C212yesmaximal
2C23×C3×C22⋊C4C211yesmaximal, characteristic
3C2×C3×#342C2112yesmaximal, direct product factor
4C22×#342C311yesmaximal, characteristic, direct product factor, Sylow
5C26×C3C2211yescharacteristic
6C24×C3×C4—22no—
7C24×C3×C4C412yescomplemented
8C22×C3×C22⋊C4—22no—
9C22×C3×C22⋊C4—26no—
10C22×C3×C22⋊C4—28no—
11C22×C3×C22⋊C4—26no—
12C22×C3×C22⋊C4C416yescomplemented
13C22×C3×C22⋊C4C4116yescomplemented
14C22×C3×C22⋊C4C2216yes—
15C22×C3×C22⋊C4C22112yes—
16C3×#342C22116yesdirect product factor
17C23×C22⋊C4C2×C311yescharacteristic
18C23×C22⋊C4C2×C312yes—
19C2×#342C2×C3112yesdirect product factor
20C25×C3—48no—
21C25×C3C2×C411yescharacteristic
22C25×C3C2×C418yes—
23C25×C3C2×C413yes—
24C25×C3D412yes—
25C25×C3D416yes—
26C25×C3D414yes—
27C25×C3Q814yes—
28C25×C3C2313yes—
29C23×C3×C4—216no—
30C23×C3×C4—26no—
31C23×C3×C4—22no—
32C23×C3×C4—212no—
33C23×C3×C4—416no—
34C23×C3×C4—48no—
35C23×C3×C4C2×C4112yescomplemented
36C2×C3×C22⋊C4—212no—
37C2×C3×C22⋊C4—212no—
38C2×C3×C22⋊C4—248no—
39C2×C3×C22⋊C4—212no—
40C2×C3×C22⋊C4—212no—
41C2×C3×C22⋊C4—212no—
42C2×C3×C22⋊C4—224no—
43C2×C3×C22⋊C4C2×C4112yescomplemented
44C2×C3×C22⋊C4C2×C4148yescomplemented
45C2×C3×C22⋊C4C2314yes—
46C2×C3×C22⋊C4C2318yes—
47C26C22×C311yescharacteristic
48C24×C4—22no—
49C24×C4C3×C412yescomplemented
50C22×C22⋊C4—26no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—26no—
53C22×C22⋊C4—28no—
54C22×C22⋊C4C3×C416yescomplemented
55C22×C22⋊C4C3×C4116yescomplemented
56C22×C22⋊C4C22×C316yes—
57C22×C22⋊C4C22×C3112yes—
58#342C22×C3116yesdirect product factor
59C24×C3—28no—
60C24×C3—216no—
61C24×C3—432no—
62C24×C3—412no—
63C24×C3—416no—
64C24×C3—424no—
65C24×C3—416no—
66C24×C3—412no—
67C24×C3C4214yes—
68C24×C3C42112yes—
69C24×C3C22⋊C416yes—
70C24×C3C22⋊C418yes—
71C24×C3C22⋊C413yes—
72C24×C3C22⋊C4124yes—
73C24×C3C22⋊C416yes—
74C24×C3C22⋊C411yescharacteristic
75C24×C3C4⋊C414yes—
76C24×C3C4⋊C418yes—
77C24×C3C4⋊C4124yes—
78C24×C3C4⋊C4112yes—
79C24×C3C22×C413yes—
80C24×C3C22×C4112yes—
81C24×C3C22×C413yes—
82C24×C3C2×D416yes—
83C24×C3C2×D416yes—
84C24×C3C2×D416yes—
85C24×C3C2×Q816yes—
86C24×C3C2411yescharacteristic
87C22×C3×C4—212no—
88C22×C3×C4—212no—
89C22×C3×C4—248no—
90C22×C3×C4—28no—
91C22×C3×C4—464no—
92C22×C3×C4—448no—
93C22×C3×C4—424no—
94C22×C3×C4—48no—
95C22×C3×C4—424no—
96C22×C3×C4C22×C418yescomplemented
97C3×C22⋊C4—224no—
98C3×C22⋊C4—248no—
99C3×C22⋊C4—28no—
100C3×C22⋊C4—28no—
101C3×C22⋊C4C22×C4116yescomplemented
102C25—48no—
103C25C2×C3×C411yescharacteristic
104C25C2×C3×C418yes—
105C25C2×C3×C413yes—
106C25C3×D412yes—
107C25C3×D416yes—
108C25C3×D414yes—
109C25C3×Q814yes—
110C25C23×C313yes—
111C23×C4—26no—
112C23×C4—22no—
113C23×C4—216no—
114C23×C4—212no—
115C23×C4—416no—
116C23×C4—48no—
117C23×C4C2×C3×C4112yescomplemented
118C2×C22⋊C4—212no—
119C2×C22⋊C4—212no—
120C2×C22⋊C4—248no—
121C2×C22⋊C4—212no—
122C2×C22⋊C4—212no—
123C2×C22⋊C4—212no—
124C2×C22⋊C4—224no—
125C2×C22⋊C4C2×C3×C4112yescomplemented
126C2×C22⋊C4C2×C3×C4148yescomplemented
127C2×C22⋊C4C23×C314yes—
128C2×C22⋊C4C23×C318yes—
129C23×C3—232no—
130C23×C3—216no—
131C23×C3—216no—
132C23×C3—232no—
133C23×C3—212no—
134C23×C3—212no—
135C23×C3—224no—
136C23×C3—224no—
137C23×C3—424no—
138C23×C3—412no—
139C23×C3—448no—
140C23×C3—424no—
141C23×C3—448no—
142C23×C3—46no—
143C23×C3—424no—
144C23×C3—412no—
145C23×C3—412no—
146C23×C3—412no—
147C23×C3—42no—
148C23×C3#95112yes—
149C23×C3#95124yes—
150C23×C3#95124yes—
151C23×C3#9514yes—
152C23×C3#9918yes—
153C23×C3C2×C4216yes—
154C23×C3C2×C4216yes—
155C23×C3C2×C22⋊C416yes—
156C23×C3C2×C22⋊C413yes—
157C23×C3C2×C22⋊C4112yes—
158C23×C3C2×C22⋊C4112yes—
159C23×C3C2×C22⋊C413yes—
160C23×C3C2×C4⋊C416yes—
161C23×C3C2×C4⋊C4112yes—
162C23×C3C2×C4⋊C416yes—
163C23×C3C2×C4⋊C4112yes—
164C23×C3C23×C411yescharacteristic
165C23×C3C23×C412yes—
166C23×C3C22×D412yes—
167C23×C3C22×D411yescharacteristic
168C23×C3C22×Q811yescharacteristic
169C2×C3×C4—28no—
170C2×C3×C4—216no—
171C2×C3×C4—416no—
172C2×C3×C4—496no—
173C2×C3×C4—424no—
174C2×C3×C4—424no—
175C2×C3×C4—48no—
176C24—28no—
177C24—216no—
178C24—412no—
179C24—432no—
180C24—424no—
181C24—416no—
182C24—416no—
183C24—412no—
184C24C3×C4214yes—
185C24C3×C42112yes—
186C24C3×C22⋊C416yes—
187C24C3×C22⋊C413yes—
188C24C3×C22⋊C418yes—
189C24C3×C22⋊C4124yes—
190C24C3×C22⋊C416yes—
191C24C3×C22⋊C411yescharacteristic
192C24C3×C4⋊C414yes—
193C24C3×C4⋊C418yes—
194C24C3×C4⋊C4124yes—
195C24C3×C4⋊C4112yes—
196C24C22×C3×C413yes—
197C24C22×C3×C4112yes—
198C24C22×C3×C413yes—
199C24C2×C3×D416yes—
200C24C2×C3×D416yes—
201C24C2×C3×D416yes—
202C24C2×C3×Q816yes—
203C24C24×C311yescharacteristic
204C22×C4—212no—
205C22×C4—212no—
206C22×C4—248no—
207C22×C4—28no—
208C22×C4—448no—
209C22×C4—464no—
210C22×C4—424no—
211C22×C4—48no—
212C22×C4—424no—
213C22×C4C22×C3×C418yescomplemented
214C22⋊C4—224no—
215C22⋊C4—248no—
216C22⋊C4—28no—
217C22⋊C4—28no—
218C22⋊C4C22×C3×C4116yescomplemented
219C22×C3—26no—
220C22×C3—22no—
221C22×C3—24no—
222C22×C3—212no—
223C22×C3—224no—
224C22×C3—248no—
225C22×C3—224no—
226C22×C3—212no—
227C22×C3—212no—
228C22×C3—224no—
229C22×C3—424no—
230C22×C3—46no—
231C22×C3—412no—
232C22×C3—46no—
233C22×C3—412no—
234C22×C3—42no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3#342116yesdirect product factor
238C22×C3C2×#9516yes—
239C22×C3C2×#95112yes—
240C22×C3C2×#9516yes—
241C22×C3C2×#99112yes—
242C22×C3C22×C4211yescharacteristic
243C22×C3C22×C22⋊C412yes—
244C22×C3C22×C22⋊C411yescharacteristic
245C22×C3C22×C4⋊C411yescharacteristic
246C22×C3C22×C4⋊C412yes—
247C3×C4—416no—
248C3×C4—48no—
249C23—212no—
250C23—212no—
251C23—224no—
252C23—224no—
253C23—232no—
254C23—216no—
255C23—216no—
256C23—232no—
257C23—424no—
258C23—448no—
259C23—412no—
260C23—448no—
261C23—424no—
262C23—424no—
263C23—46no—
264C23—412no—
265C23—412no—
266C23—412no—
267C23—42no—
268C23C3×#95112yes—
269C23C3×#95124yes—
270C23C3×#95124yes—
271C23C3×#9514yes—
272C23C3×#9918yes—
273C23C2×C3×C4216yes—
274C23C2×C3×C4216yes—
275C23C2×C3×C22⋊C416yes—
276C23C2×C3×C22⋊C4112yes—
277C23C2×C3×C22⋊C413yes—
278C23C2×C3×C22⋊C4112yes—
279C23C2×C3×C22⋊C413yes—
280C23C2×C3×C4⋊C416yes—
281C23C2×C3×C4⋊C4112yes—
282C23C2×C3×C4⋊C4112yes—
283C23C2×C3×C4⋊C416yes—
284C23C23×C3×C411yescharacteristic
285C23C23×C3×C412yes—
286C23C22×C3×D412yes—
287C23C22×C3×D411yescharacteristic
288C23C22×C3×Q811yescharacteristic
289C2×C4—28no—
290C2×C4—216no—
291C2×C4—496no—
292C2×C4—416no—
293C2×C4—424no—
294C2×C4—424no—
295C2×C4—48no—
296C2×C3—22no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—26no—
300C2×C3—212no—
301C2×C3—21no—
302C2×C3C2×#342112yesdirect product factor
303C2×C3C22×#9511yescharacteristic
304C2×C3C22×#9912yes—
305C22—224no—
306C22—224no—
307C22—248no—
308C22—212no—
309C22—212no—
310C22—224no—
311C22—26no—
312C22—22no—
313C22—24no—
314C22—212no—
315C22—424no—
316C22—46no—
317C22—412no—
318C22—46no—
319C22—412no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22C3×#342116yesdirect product factor
324C22C2×C3×#9516yes—
325C22C2×C3×#95112yes—
326C22C2×C3×#9516yes—
327C22C2×C3×#99112yes—
328C22C22×C3×C4211yescharacteristic
329C22C22×C3×C22⋊C412yes—
330C22C22×C3×C22⋊C411yescharacteristic
331C22C22×C3×C4⋊C411yescharacteristic
332C22C22×C3×C4⋊C412yes—
333C4—416no—
334C4—48no—
335C3C22×#34211yescharacteristic, direct product factor, Sylow
336C2—22no—
337C2—212no—
338C2—26no—
339C2—21no—
340C2—21no—
341C2—22no—
342C2C2×C3×#342112yesdirect product factor
343C2C22×C3×#9511yescharacteristic
344C2C22×C3×#9912yes—
subgroupG/H
center, socleC24×C3C22⋊C4
commutatorC22C22×C3×C42
Frattini, cosocleC24C24×C3
prime radicalC26×C3C22
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C24×C3 ◁ C26×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#342C2direct (N × H)
4. C22×#342C3direct (N × H)
7. C24×C3×C4C4semidirect
12. C22×C3×C22⋊C4C4semidirect
13. C22×C3×C22⋊C4C4semidirect
16. C3×#342C22direct (N × H)
19. C2×#342C2×C3direct (N × H)
35. C23×C3×C4C2×C4semidirect
43. C2×C3×C22⋊C4C2×C4semidirect
44. C2×C3×C22⋊C4C2×C4semidirect
49. C24×C4C3×C4semidirect
54. C22×C22⋊C4C3×C4semidirect
55. C22×C22⋊C4C3×C4semidirect
58. #342C22×C3direct (N × H)
96. C22×C3×C4C22×C4semidirect
101. C3×C22⋊C4C22×C4semidirect
117. C23×C4C2×C3×C4semidirect
125. C2×C22⋊C4C2×C3×C4semidirect
126. C2×C22⋊C4C2×C3×C4semidirect
213. C22×C4C22×C3×C4semidirect
218. C22⋊C4C22×C3×C4semidirect
237. C22×C3#342direct (N × H)
302. C2×C3C2×#342direct (N × H)
323. C22C3×#342direct (N × H)
335. C3C22×#342direct (N × H)
342. C2C2×C3×#342direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C42
C2×C3×#3422
C22×#3423