Finite Group Databasebeta

← C2×C3×#2922 #10678726

Subgroups — C2×C3×#2922 #10678726

Order 768.

2222 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#41C211yesmaximal, characteristic
2C3×#2922C2116yesmaximal, direct product factor
3C2×C3×#433C214yesmaximal, complemented
4C2×C3×#431C214yesmaximal, complemented
5C2×C3×#424C212yesmaximal, complemented
6C2×C3×#408C212yesmaximal
7C2×C3×C4×M16C211yesmaximal, characteristic, complemented
8C22×C3×C4×C8C211yesmaximal, characteristic, complemented
9C2×#2922C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41C2211yescharacteristic
11C22×C3×M16C2212yes—
12C23×C3×C8C2212yes—
13C3×C4×#41C418yes—
14C3×C4×#41C2218yes—
15C2×C3×C4×Q8C412yes—
16C2×C3×C4×D4C414yes—
17C2×C3×C4×D4C412yes—
18C2×C3×#117C2211yescharacteristic
19C2×C3×#117C2212yes—
20C22×C3×C42C2212yes—
21C22×C3×C42C2211yescharacteristic
22C3×#433C22132yescomplemented
23C3×#431C22132yescomplemented
24C3×#424C22116yescomplemented
25C2×C3×C4⋊C8C2218yes—
26C3×#408C22116yes—
27C2×C3×C22⋊C8C2218yes—
28C3×C4×M16C2218yescomplemented
29C2×C3×#97C2212yes—
30C2×C3×C4×C8C2212yes—
31C2×C3×C4×C8C2212yes—
32C2×C3×C4×C8C2212yes—
33C2×C3×C4×C8C2218yescomplemented
34C2×C4×#41C2×C311yescharacteristic
35#2922C2×C3116yesdirect product factor
36C2×#433C2×C314yescomplemented
37C2×#431C2×C314yescomplemented
38C2×#424C2×C312yescomplemented
39C2×#408C2×C312yes—
40C2×C4×M16C2×C311yescharacteristic, complemented
41C22×C4×C8C2×C311yescharacteristic, complemented
42C2×C3×#41—28no—
43C2×C3×#41C2×C414yes—
44C2×C3×#41C2314yes—
45C22×C3×Q8C2×C411yescharacteristic
46C22×C3×D4C2×C412yes—
47C22×C3×D4C2×C411yescharacteristic
48C23×C3×C4C2×C414yes—
49C23×C3×C4C2×C412yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C2×C3×M16—28no—
53C2×C3×M16C2318yes—
54C22×C3×C8—28no—
55C22×C3×C8C2314yes—
56C22×C3×C8C2318yes—
57C22×C3×C8C2314yes—
58C3×C4×Q8C2×C4116yes—
59C3×C4×D4C2×C4132yes—
60C3×C4×D4C2×C4116yes—
61C3×#117C2×C418yes—
62C3×#117C2×C414yes—
63C3×#117C2314yes—
64C3×#117C2318yes—
65C2×C3×C4⋊C4C2×C414yes—
66C2×C3×C4⋊C4C2×C412yes—
67C2×C3×C22⋊C4C2×C414yes—
68C2×C3×C22⋊C4C2×C412yes—
69C2×C3×C42C2×C414yes—
70C2×C3×C42C2×C412yes—
71C2×C3×C42C2×C418yes—
72C2×C3×C42C2×C414yes—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2314yes—
75C2×C3×C42C2312yes—
76C2×C3×C42C2318yes—
77C2×C3×C42C2311yescharacteristic
78C3×C4⋊C8C23132yes—
79C3×C22⋊C8C23132yes—
80C3×#97C2318yes—
81C3×C4×C8C2318yes—
82C3×C4×C8C2318yes—
83C3×C4×C8C2318yes—
84C22×#41C22×C311yescharacteristic
85C22×M16C22×C312yes—
86C23×C8C22×C312yes—
87C4×#41C3×C418yes—
88C4×#41C22×C318yes—
89C2×C4×Q8C3×C412yes—
90C2×C4×D4C3×C414yes—
91C2×C4×D4C3×C412yes—
92C2×#117C22×C311yescharacteristic
93C2×#117C22×C312yes—
94C22×C42C22×C311yescharacteristic
95C22×C42C22×C312yes—
96#433C22×C3132yescomplemented
97#431C22×C3132yescomplemented
98#424C22×C3116yescomplemented
99C2×C4⋊C8C22×C318yes—
100#408C22×C3116yes—
101C2×C22⋊C8C22×C318yes—
102C4×M16C22×C318yescomplemented
103C2×#97C22×C312yes—
104C2×C4×C8C22×C312yes—
105C2×C4×C8C22×C312yes—
106C2×C4×C8C22×C312yes—
107C2×C4×C8C22×C318yescomplemented
108C24×C3C22×C412yes—
109C24×C3C22×C411yescharacteristic
110C3×#41—216no—
111C3×#41—216no—
112C2×C3×Q8—24no—
113C2×C3×Q8C22×C414yes—
114C2×C3×D4—28no—
115C2×C3×D4—24no—
116C2×C3×D4C22×C418yes—
117C2×C3×D4C22×C414yes—
118C22×C3×C4—28no—
119C22×C3×C4—216no—
120C22×C3×C4—28no—
121C22×C3×C4C22×C412yes—
122C22×C3×C4C22×C414yes—
123C22×C3×C4C22×C412yes—
124C22×C3×C4C22×C414yes—
125C22×C3×C4C22×C4116yes—
126C22×C3×C4C22×C418yes—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C22×C412yes—
129C22×C3×C4C22×C412yes—
130C22×C3×C4#4118yes—
131C22×C3×C4C2414yes—
132C22×C3×C4C2411yescharacteristic
133C22×C3×C4C2412yes—
134C3×M16—216no—
135C2×C3×C8—28no—
136C2×C3×C8—216no—
137C2×C3×C8—28no—
138C2×C3×C8C2418yes—
139C2×C3×C8C2418yes—
140C3×C4⋊C4C22×C4116yes—
141C3×C4⋊C4C22×C418yes—
142C3×C22⋊C4C22×C4116yes—
143C3×C22⋊C4C22×C418yes—
144C3×C42C22×C412yes—
145C3×C42C22×C412yes—
146C3×C42C22×C414yes—
147C3×C42C22×C4116yes—
148C3×C42C22×C418yes—
149C3×C42C2412yes—
150C3×C42C2412yes—
151C3×C42C2414yes—
152C2×#41—28no—
153C2×#41C2×C3×C414yes—
154C2×#41C23×C314yes—
155C22×Q8C2×C3×C411yescharacteristic
156C22×D4C2×C3×C412yes—
157C22×D4C2×C3×C411yescharacteristic
158C23×C4C2×C3×C414yes—
159C23×C4C2×C3×C412yes—
160C23×C4C23×C312yes—
161C23×C4C23×C311yescharacteristic
162C2×M16—28no—
163C2×M16C23×C318yes—
164C22×C8—28no—
165C22×C8C23×C314yes—
166C22×C8C23×C318yes—
167C22×C8C23×C314yes—
168C4×Q8C2×C3×C4116yes—
169C4×D4C2×C3×C4132yes—
170C4×D4C2×C3×C4116yes—
171#117C2×C3×C418yes—
172#117C2×C3×C414yes—
173#117C23×C314yes—
174#117C23×C318yes—
175C2×C4⋊C4C2×C3×C414yes—
176C2×C4⋊C4C2×C3×C412yes—
177C2×C22⋊C4C2×C3×C414yes—
178C2×C22⋊C4C2×C3×C412yes—
179C2×C42C2×C3×C414yes—
180C2×C42C2×C3×C412yes—
181C2×C42C2×C3×C418yes—
182C2×C42C2×C3×C414yes—
183C2×C42C23×C311yescharacteristic
184C2×C42C23×C312yes—
185C2×C42C23×C314yes—
186C2×C42C23×C318yes—
187C2×C42C23×C311yescharacteristic
188C4⋊C8C23×C3132yes—
189C22⋊C8C23×C3132yes—
190#97C23×C318yes—
191C4×C8C23×C318yes—
192C4×C8C23×C318yes—
193C4×C8C23×C318yes—
194C23×C3—24no—
195C23×C3—24no—
196C23×C3—28no—
197C23×C3#11714yes—
198C23×C3C23×C414yes—
199C23×C3C23×C412yes—
200C23×C3C23×C411yescharacteristic
201C3×Q8—28no—
202C3×D4—216no—
203C3×D4—28no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—28no—
207C2×C3×C4—28no—
208C2×C3×C4—216no—
209C2×C3×C4—232no—
210C2×C3×C4—28no—
211C2×C3×C4#11714yes—
212C2×C3×C4#11718yes—
213C2×C3×C4#13118yes—
214C2×C3×C4C23×C414yes—
215C2×C3×C4C23×C414yes—
216C2×C3×C4C23×C418yes—
217C2×C3×C4C23×C411yescharacteristic
218C2×C3×C4C23×C412yes—
219C2×C3×C4C23×C414yes—
220C2×C3×C4C2×#4114yes—
221C2×C3×C4C2×#4118yes—
222C2×C3×C4C2511yescharacteristic
223C3×C8—28no—
224C3×C8—28no—
225C24C22×C3×C412yes—
226C24C22×C3×C411yescharacteristic
227#41—216no—
228#41—216no—
229C2×Q8—24no—
230C2×Q8C22×C3×C414yes—
231C2×D4—28no—
232C2×D4—24no—
233C2×D4C22×C3×C418yes—
234C2×D4C22×C3×C414yes—
235C22×C4—28no—
236C22×C4—216no—
237C22×C4—28no—
238C22×C4C22×C3×C412yes—
239C22×C4C22×C3×C414yes—
240C22×C4C22×C3×C412yes—
241C22×C4C22×C3×C414yes—
242C22×C4C22×C3×C4116yes—
243C22×C4C22×C3×C418yes—
244C22×C4C22×C3×C411yescharacteristic
245C22×C4C22×C3×C412yes—
246C22×C4C22×C3×C412yes—
247C22×C4C3×#4118yes—
248C22×C4C24×C314yes—
249C22×C4C24×C311yescharacteristic
250C22×C4C24×C312yes—
251M16—216no—
252C2×C8—28no—
253C2×C8—216no—
254C2×C8—28no—
255C2×C8C24×C318yes—
256C2×C8C24×C318yes—
257C4⋊C4C22×C3×C4116yes—
258C4⋊C4C22×C3×C418yes—
259C22⋊C4C22×C3×C4116yes—
260C22⋊C4C22×C3×C418yes—
261C42C22×C3×C412yes—
262C42C22×C3×C412yes—
263C42C22×C3×C414yes—
264C42C22×C3×C4116yes—
265C42C22×C3×C418yes—
266C42C24×C312yes—
267C42C24×C312yes—
268C42C24×C314yes—
269C22×C3—28no—
270C22×C3—24no—
271C22×C3—24no—
272C22×C3—28no—
273C22×C3—48no—
274C22×C3C2×#11714yes—
275C22×C3C2×#11712yes—
276C22×C3C2×#13114yes—
277C22×C3C24×C411yescharacteristic
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—28no—
281C3×C4C2×#11712yes—
282C3×C4C2×#11714yes—
283C3×C4C2×#13118yes—
284C3×C4C22×#4112yes—
285C23—24no—
286C23—28no—
287C23—24no—
288C23C3×#11714yes—
289C23C23×C3×C414yes—
290C23C23×C3×C412yes—
291C23C23×C3×C411yescharacteristic
292Q8—28no—
293D4—216no—
294D4—28no—
295C2×C4—24no—
296C2×C4—28no—
297C2×C4—24no—
298C2×C4—216no—
299C2×C4—232no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4C3×#11714yes—
303C2×C4C3×#11718yes—
304C2×C4C3×#13118yes—
305C2×C4C23×C3×C414yes—
306C2×C4C23×C3×C414yes—
307C2×C4C23×C3×C418yes—
308C2×C4C23×C3×C414yes—
309C2×C4C23×C3×C411yescharacteristic
310C2×C4C23×C3×C412yes—
311C2×C4C2×C3×#4114yes—
312C2×C4C2×C3×#4118yes—
313C2×C4C25×C311yescharacteristic
314C8—28no—
315C8—28no—
316C2×C3—24no—
317C2×C3—44no—
318C2×C3#292214yesdirect product factor
319C2×C3C22×#11711yescharacteristic
320C2×C3C22×#13112yes—
321C22—28no—
322C22—24no—
323C22—24no—
324C22—28no—
325C22—48no—
326C22C2×C3×#11714yes—
327C22C2×C3×#11712yes—
328C22C2×C3×#13114yes—
329C22C24×C3×C411yescharacteristic
330C4—24no—
331C4—24no—
332C4—28no—
333C4C2×C3×#11712yes—
334C4C2×C3×#11714yes—
335C4C2×C3×#13118yes—
336C4C22×C3×#4112yes—
337C3C2×#292211yescharacteristic, direct product factor, Sylow
338C2—24no—
339C2—44no—
340C2C3×#292214yesdirect product factor
341C2C22×C3×#11711yescharacteristic
342C2C22×C3×#13112yes—
subgroupG/H
centerC2×C3×C42C23
commutatorC22C24×C3×C4
Frattini, cosocleC2×C4C25×C3
socleC23×C3C23×C4
prime radicalC22×C3×#41C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2922C2direct (N × H)
3. C2×C3×#433C2semidirect
4. C2×C3×#431C2semidirect
5. C2×C3×#424C2semidirect
7. C2×C3×C4×M16C2semidirect
8. C22×C3×C4×C8C2semidirect
9. C2×#2922C3direct (N × H)
22. C3×#433C22semidirect
23. C3×#431C22semidirect
24. C3×#424C22semidirect
28. C3×C4×M16C22semidirect
33. C2×C3×C4×C8C22semidirect
35. #2922C2×C3direct (N × H)
36. C2×#433C2×C3semidirect
37. C2×#431C2×C3semidirect
38. C2×#424C2×C3semidirect
40. C2×C4×M16C2×C3semidirect
41. C22×C4×C8C2×C3semidirect
96. #433C22×C3semidirect
97. #431C22×C3semidirect
98. #424C22×C3semidirect
102. C4×M16C22×C3semidirect
107. C2×C4×C8C22×C3semidirect
318. C2×C3#2922direct (N × H)
337. C3C2×#2922direct (N × H)
340. C2C3×#2922direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C3×#29222
C2×C3×#4332
C2×C3×#4312
C2×C3×#4242
C2×C3×#4082
C2×C3×C4×M162
C22×C3×C4×C82
C2×#29223