Finite Group Databasebeta

← C2×C3×#2932 #10678743

Subgroups — C2×C3×#2932 #10678743

Order 768.

2806 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
1C22×C3×#131C212yesmaximal, complemented
2C2×C3×#532C211yesmaximal, characteristic
3C3×#2932C2116yesmaximal, direct product factor
4C2×C3×#435C218yesmaximal, complemented
5C2×C3×#423C211yesmaximal, characteristic, complemented
6C2×C3×#408C212yesmaximal, complemented
7C22×C3×#97C211yesmaximal, characteristic, complemented
8C2×#2932C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2212yes—
10C2×C3×#131—216no—
11C2×C3×#131C22116yescomplemented
12C22×C3×M16C2212yes—
13C22×C3×M16C2214yes—
14C23×C3×C8C2211yescharacteristic
15C23×C3×C8C2214yes—
16C23×C3×C8C2211yescharacteristic
17C3×#532C418yes—
18C3×#532C2218yes—
19C2×C3×C4⋊Q8C411yescharacteristic
20C2×C3×#127C411yescharacteristic
21C2×C3×#124C412yes—
22C2×C3×#121C414yes—
23C2×C3×C4×D4C2214yes—
24C22×C3×C42C2211yescharacteristic
25C3×#435C22164yescomplemented
26C3×#423C2218yescomplemented
27C2×C3×C4⋊C8C2214yes—
28C3×#408C22116yescomplemented
29C2×C3×C22⋊C8C2218yes—
30C2×C3×#97C2214yes—
31C2×C3×#97C2218yescomplemented
32C22×#131C2×C312yescomplemented
33C2×#532C2×C311yescharacteristic
34#2932C2×C3116yesdirect product factor
35C2×#435C2×C318yescomplemented
36C2×#423C2×C311yescharacteristic, complemented
37C2×#408C2×C312yescomplemented
38C22×#97C2×C311yescharacteristic, complemented
39C2×C3×#41—28no—
40C2×C3×#41—28no—
41C2×C3×#41C2×C418yes—
42C2×C3×#41C2318yes—
43C22×C3×Q8C2×C412yes—
44C22×C3×D4C2×C414yes—
45C22×C3×D4C2×C412yes—
46C22×C3×D4C2×C414yes—
47C23×C3×C4C2×C412yes—
48C23×C3×C4C2314yes—
49C23×C3×C4C2311yescharacteristic
50C3×#131—264no—
51C2×C3×M16—28no—
52C2×C3×M16—216no—
53C2×C3×M16C2318yes—
54C2×C3×M16C23116yes—
55C22×C3×C8—24no—
56C22×C3×C8—216no—
57C22×C3×C8D418yescomplemented
58C22×C3×C8C23116yes—
59C22×C3×C8C2314yes—
60C22×C3×C8C2314yes—
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C22×C3×C8C2314yes—
64C3×C4⋊Q8C2×C418yes—
65C3×#127C2×C418yes—
66C3×#124C2×C4116yes—
67C3×#121C2×C4132yes—
68C3×C4×D4C2×C4116yes—
69C3×C4×D4C23116yes—
70C2×C3×C4⋊C4C2×C414yes—
71C2×C3×C22⋊C4C2×C418yes—
72C2×C3×C42C2×C411yescharacteristic
73C2×C3×C42C2×C411yescharacteristic
74C2×C3×C42C2×C414yes—
75C2×C3×C42C2314yes—
76C2×C3×C42C2312yes—
77C3×C4⋊C8C23116yes—
78C3×C22⋊C8C23132yes—
79C3×#97C23116yes—
80C22×#41C22×C312yes—
81C2×#131—216no—
82C2×#131C22×C3116yescomplemented
83C22×M16C22×C312yes—
84C22×M16C22×C314yes—
85C23×C8C22×C311yescharacteristic
86C23×C8C22×C314yes—
87C23×C8C22×C311yescharacteristic
88#532C3×C418yes—
89#532C22×C318yes—
90C2×C4⋊Q8C3×C411yescharacteristic
91C2×#127C3×C411yescharacteristic
92C2×#124C3×C412yes—
93C2×#121C3×C414yes—
94C2×C4×D4C22×C314yes—
95C22×C42C22×C311yescharacteristic
96#435C22×C3164yescomplemented
97#423C22×C318yescomplemented
98C2×C4⋊C8C22×C314yes—
99#408C22×C3116yescomplemented
100C2×C22⋊C8C22×C318yes—
101C2×#97C22×C314yes—
102C2×#97C22×C318yescomplemented
103C24×C3C22×C414yes—
104C24×C3C22×C411yescharacteristic
105C3×#41—232no—
106C3×#41—216no—
107C3×#41—216no—
108C2×C3×Q8—28no—
109C2×C3×Q8C22×C418yes—
110C2×C3×D4—216no—
111C2×C3×D4—28no—
112C2×C3×D4—48no—
113C2×C3×D4C22×C4116yes—
114C2×C3×D4C22×C418yes—
115C2×C3×D4C22×C4116yes—
116C22×C3×C4—28no—
117C22×C3×C4—28no—
118C22×C3×C4—44no—
119C22×C3×C4C22×C412yes—
120C22×C3×C4C22×C412yes—
121C22×C3×C4C22×C414yes—
122C22×C3×C4C22×C418yes—
123C22×C3×C4C22×C418yes—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C22×C412yes—
126C22×C3×C4C2×D414yes—
127C22×C3×C4#4114yes—
128C22×C3×C4C2418yes—
129C22×C3×C4C2412yes—
130C22×C3×C4C2411yescharacteristic
131C3×M16—216no—
132C3×M16—232no—
133C2×C3×C8—28no—
134C2×C3×C8—24no—
135C2×C3×C8—28no—
136C2×C3×C8—232no—
137C2×C3×C8C2×D418yes—
138C2×C3×C8C2×D4116yescomplemented
139C2×C3×C8C2414yes—
140C2×C3×C8C2418yes—
141C2×C3×C8C2414yes—
142C3×C4⋊C4C22×C4116yes—
143C3×C22⋊C4C22×C4132yes—
144C3×C42C22×C414yes—
145C3×C42C22×C414yes—
146C3×C42C22×C414yes—
147C3×C42C2414yes—
148C2×#41—28no—
149C2×#41—28no—
150C2×#41C2×C3×C418yes—
151C2×#41C23×C318yes—
152C22×Q8C2×C3×C412yes—
153C22×D4C2×C3×C414yes—
154C22×D4C2×C3×C412yes—
155C22×D4C2×C3×C414yes—
156C23×C4C2×C3×C412yes—
157C23×C4C23×C314yes—
158C23×C4C23×C311yescharacteristic
159#131—264no—
160C2×M16—28no—
161C2×M16—216no—
162C2×M16C23×C318yes—
163C2×M16C23×C3116yes—
164C22×C8—24no—
165C22×C8—216no—
166C22×C8C3×D418yescomplemented
167C22×C8C23×C314yes—
168C22×C8C23×C3116yes—
169C22×C8C23×C312yes—
170C22×C8C23×C314yes—
171C22×C8C23×C312yes—
172C22×C8C23×C314yes—
173C4⋊Q8C2×C3×C418yes—
174#127C2×C3×C418yes—
175#124C2×C3×C4116yes—
176#121C2×C3×C4132yes—
177C4×D4C2×C3×C4116yes—
178C4×D4C23×C3116yes—
179C2×C4⋊C4C2×C3×C414yes—
180C2×C22⋊C4C2×C3×C418yes—
181C2×C42C2×C3×C411yescharacteristic
182C2×C42C2×C3×C411yescharacteristic
183C2×C42C2×C3×C414yes—
184C2×C42C23×C312yes—
185C2×C42C23×C314yes—
186C4⋊C8C23×C3116yes—
187C22⋊C8C23×C3132yes—
188#97C23×C3116yes—
189C23×C3—24no—
190C23×C3—28no—
191C23×C3—216no—
192C23×C3C4×D414yes—
193C23×C3C23×C418yes—
194C23×C3C23×C412yes—
195C23×C3C23×C411yescharacteristic
196C3×Q8—216no—
197C3×D4—232no—
198C3×D4—216no—
199C3×D4—416no—
200C2×C3×C4—28no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—216no—
204C2×C3×C4—28no—
205C2×C3×C4—28no—
206C2×C3×C4—48no—
207C2×C3×C4C4×D414yes—
208C2×C3×C4C4×D418yes—
209C2×C3×C4C23×C414yes—
210C2×C3×C4C23×C414yes—
211C2×C3×C4C23×C418yes—
212C2×C3×C4C23×C411yescharacteristic
213C2×C3×C4C23×C412yes—
214C2×C3×C4C22×D412yes—
215C2×C3×C4C22×D414yes—
216C2×C3×C4C2×#4112yes—
217C2×C3×C4C2×#4114yes—
218C2×C3×C4C2511yescharacteristic
219C3×C8—28no—
220C3×C8—24no—
221C3×C8C22×D418yes—
222C24C22×C3×C414yes—
223C24C22×C3×C411yescharacteristic
224#41—232no—
225#41—216no—
226#41—216no—
227C2×Q8—28no—
228C2×Q8C22×C3×C418yes—
229C2×D4—216no—
230C2×D4—28no—
231C2×D4—48no—
232C2×D4C22×C3×C4116yes—
233C2×D4C22×C3×C418yes—
234C2×D4C22×C3×C4116yes—
235C22×C4—28no—
236C22×C4—28no—
237C22×C4—44no—
238C22×C4C22×C3×C412yes—
239C22×C4C22×C3×C412yes—
240C22×C4C22×C3×C414yes—
241C22×C4C22×C3×C418yes—
242C22×C4C22×C3×C418yes—
243C22×C4C22×C3×C411yescharacteristic
244C22×C4C22×C3×C412yes—
245C22×C4C2×C3×D414yes—
246C22×C4C3×#4114yes—
247C22×C4C24×C318yes—
248C22×C4C24×C311yescharacteristic
249C22×C4C24×C312yes—
250M16—216no—
251M16—232no—
252C2×C8—28no—
253C2×C8—24no—
254C2×C8—28no—
255C2×C8—232no—
256C2×C8C2×C3×D418yes—
257C2×C8C2×C3×D4116yescomplemented
258C2×C8C24×C314yes—
259C2×C8C24×C318yes—
260C2×C8C24×C314yes—
261C4⋊C4C22×C3×C4116yes—
262C22⋊C4C22×C3×C4132yes—
263C42C22×C3×C414yes—
264C42C22×C3×C414yes—
265C42C22×C3×C414yes—
266C42C24×C314yes—
267C22×C3—28no—
268C22×C3—24no—
269C22×C3—28no—
270C22×C3—216no—
271C22×C3—416no—
272C22×C3C2×C4×D414yes—
273C22×C3C2×C4×D412yes—
274C22×C3#56814yes—
275C22×C3C24×C411yescharacteristic
276C3×C4—28no—
277C3×C4—24no—
278C3×C4—24no—
279C3×C4C2×C4×D412yes—
280C3×C4C2×C4×D414yes—
281C3×C4C23×D411yescharacteristic
282C3×C4C22×#4111yescharacteristic
283C23—24no—
284C23—216no—
285C23—28no—
286C23C3×C4×D414yes—
287C23C23×C3×C418yes—
288C23C23×C3×C412yes—
289C23C23×C3×C411yescharacteristic
290Q8—216no—
291D4—232no—
292D4—216no—
293D4—416no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—216no—
300C2×C4—48no—
301C2×C4C3×C4×D414yes—
302C2×C4C3×C4×D418yes—
303C2×C4C23×C3×C414yes—
304C2×C4C23×C3×C414yes—
305C2×C4C23×C3×C418yes—
306C2×C4C23×C3×C411yescharacteristic
307C2×C4C23×C3×C412yes—
308C2×C4C22×C3×D412yes—
309C2×C4C22×C3×D414yes—
310C2×C4C2×C3×#4112yes—
311C2×C4C2×C3×#4114yes—
312C2×C4C25×C311yescharacteristic
313C8—28no—
314C8—24no—
315C8C22×C3×D418yes—
316C2×C3—24no—
317C2×C3—48no—
318C2×C3#293214yesdirect product factor
319C2×C3C22×C4×D411yescharacteristic
320C2×C3C2×#56812yes—
321C22—28no—
322C22—24no—
323C22—28no—
324C22—216no—
325C22—416no—
326C22C2×C3×C4×D414yes—
327C22C2×C3×C4×D412yes—
328C22C3×#56814yes—
329C22C24×C3×C411yescharacteristic
330C4—28no—
331C4—24no—
332C4—24no—
333C4C2×C3×C4×D412yes—
334C4C2×C3×C4×D414yes—
335C4C23×C3×D411yescharacteristic
336C4C22×C3×#4111yescharacteristic
337C3C2×#293211yescharacteristic, direct product factor, Sylow
338C2—24no—
339C2—48no—
340C2C3×#293214yesdirect product factor
341C2C22×C3×C4×D411yescharacteristic
342C2C2×C3×#56812yes—
subgroupG/H
centerC22×C3×C4C24
commutatorC22C24×C3×C4
Frattini, cosocleC2×C4C25×C3
socleC23×C3C23×C4
prime radicalC2×C3×#532C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#131C2semidirect
3. C3×#2932C2direct (N × H)
4. C2×C3×#435C2semidirect
5. C2×C3×#423C2semidirect
6. C2×C3×#408C2semidirect
7. C22×C3×#97C2semidirect
8. C2×#2932C3direct (N × H)
11. C2×C3×#131C22semidirect
25. C3×#435C22semidirect
26. C3×#423C22semidirect
28. C3×#408C22semidirect
31. C2×C3×#97C22semidirect
32. C22×#131C2×C3semidirect
34. #2932C2×C3direct (N × H)
35. C2×#435C2×C3semidirect
36. C2×#423C2×C3semidirect
37. C2×#408C2×C3semidirect
38. C22×#97C2×C3semidirect
57. C22×C3×C8D4semidirect
82. C2×#131C22×C3semidirect
96. #435C22×C3semidirect
97. #423C22×C3semidirect
99. #408C22×C3semidirect
102. C2×#97C22×C3semidirect
138. C2×C3×C8C2×D4semidirect
166. C22×C8C3×D4semidirect
257. C2×C8C2×C3×D4semidirect
318. C2×C3#2932direct (N × H)
337. C3C2×#2932direct (N × H)
340. C2C3×#2932direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1312
C2×C3×#5322
C3×#29322
C2×C3×#4352
C2×C3×#4232
C2×C3×#4082
C22×C3×#972
C2×#29323