Finite Group Databasebeta

← C2×C3×#2931 #10678742

Subgroups — C2×C3×#2931 #10678742

Order 768.

2894 conjugacy classes of subgroups.

The subgroup lattice has 360 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×#2931C2116yesmaximal, direct product factor
4C2×C3×#436C214yesmaximal, complemented
5C2×C3×C8×D4C214yesmaximal, complemented
6C2×C3×#423C211yesmaximal, characteristic, complemented
7C2×C3×#408C212yesmaximal, complemented
8C22×C3×C4×C8C211yesmaximal, characteristic, complemented
9C2×#2931C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41C2212yes—
11C2×C3×#131—216no—
12C2×C3×#131C22116yescomplemented
13C22×C3×M16C2212yes—
14C22×C3×M16C2214yes—
15C23×C3×C8C2211yescharacteristic
16C23×C3×C8C2214yes—
17C23×C3×C8C2211yescharacteristic
18C3×#532C418yes—
19C3×#532C2218yes—
20C2×C3×C4⋊Q8C411yescharacteristic
21C2×C3×#127C411yescharacteristic
22C2×C3×#124C412yes—
23C2×C3×#121C414yes—
24C2×C3×C4×D4C2214yes—
25C22×C3×C42C2211yescharacteristic
26C3×#436C22132yescomplemented
27C3×C8×D4C22132yescomplemented
28C3×#423C2218yescomplemented
29C2×C3×C4⋊C8C2214yes—
30C3×#408C22116yescomplemented
31C2×C3×C22⋊C8C2218yes—
32C2×C3×C4×C8C2212yes—
33C2×C3×C4×C8C2212yes—
34C2×C3×C4×C8C2218yescomplemented
35C22×#131C2×C312yescomplemented
36C2×#532C2×C311yescharacteristic
37#2931C2×C3116yesdirect product factor
38C2×#436C2×C314yescomplemented
39C2×C8×D4C2×C314yescomplemented
40C2×#423C2×C311yescharacteristic, complemented
41C2×#408C2×C312yescomplemented
42C22×C4×C8C2×C311yescharacteristic, complemented
43C2×C3×#41—28no—
44C2×C3×#41—28no—
45C2×C3×#41C2×C418yes—
46C2×C3×#41C2318yes—
47C22×C3×Q8C2×C412yes—
48C22×C3×D4C2×C414yes—
49C22×C3×D4C2×C412yes—
50C22×C3×D4C2×C414yes—
51C23×C3×C4C2×C412yes—
52C23×C3×C4C2314yes—
53C23×C3×C4C2311yescharacteristic
54C3×#131—264no—
55C2×C3×M16—28no—
56C2×C3×M16—216no—
57C2×C3×M16C2318yes—
58C2×C3×M16C23116yes—
59C22×C3×C8—24no—
60C22×C3×C8—216no—
61C22×C3×C8D418yescomplemented
62C22×C3×C8C2314yes—
63C22×C3×C8C23116yes—
64C22×C3×C8C2312yes—
65C22×C3×C8C2314yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2314yes—
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8C2×C418yes—
70C3×#127C2×C418yes—
71C3×#124C2×C4116yes—
72C3×#121C2×C4132yes—
73C3×C4×D4C2×C4116yes—
74C3×C4×D4C23116yes—
75C2×C3×C4⋊C4C2×C414yes—
76C2×C3×C22⋊C4C2×C418yes—
77C2×C3×C42C2×C411yescharacteristic
78C2×C3×C42C2×C411yescharacteristic
79C2×C3×C42C2×C414yes—
80C2×C3×C42C2312yes—
81C2×C3×C42C2314yes—
82C3×C4⋊C8C23116yes—
83C3×C22⋊C8C23132yes—
84C3×C4×C8C2318yes—
85C3×C4×C8C2318yes—
86C22×#41C22×C312yes—
87C2×#131—216no—
88C2×#131C22×C3116yescomplemented
89C22×M16C22×C312yes—
90C22×M16C22×C314yes—
91C23×C8C22×C311yescharacteristic
92C23×C8C22×C314yes—
93C23×C8C22×C311yescharacteristic
94#532C3×C418yes—
95#532C22×C318yes—
96C2×C4⋊Q8C3×C411yescharacteristic
97C2×#127C3×C411yescharacteristic
98C2×#124C3×C412yes—
99C2×#121C3×C414yes—
100C2×C4×D4C22×C314yes—
101C22×C42C22×C311yescharacteristic
102#436C22×C3132yescomplemented
103C8×D4C22×C3132yescomplemented
104#423C22×C318yescomplemented
105C2×C4⋊C8C22×C314yes—
106#408C22×C3116yescomplemented
107C2×C22⋊C8C22×C318yes—
108C2×C4×C8C22×C312yes—
109C2×C4×C8C22×C312yes—
110C2×C4×C8C22×C318yescomplemented
111C24×C3C22×C414yes—
112C24×C3C22×C411yescharacteristic
113C3×#41—232no—
114C3×#41—216no—
115C3×#41—216no—
116C2×C3×Q8—28no—
117C2×C3×Q8C22×C418yes—
118C2×C3×D4—216no—
119C2×C3×D4—28no—
120C2×C3×D4—216no—
121C2×C3×D4C22×C4116yes—
122C2×C3×D4C22×C4116yes—
123C2×C3×D4C22×C418yes—
124C22×C3×C4—28no—
125C22×C3×C4—28no—
126C22×C3×C4—28no—
127C22×C3×C4C22×C412yes—
128C22×C3×C4C22×C418yes—
129C22×C3×C4C22×C414yes—
130C22×C3×C4C22×C418yes—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C22×C412yes—
133C22×C3×C4C22×C412yes—
134C22×C3×C4C2×D414yes—
135C22×C3×C4#4114yes—
136C22×C3×C4C2418yes—
137C22×C3×C4C2411yescharacteristic
138C22×C3×C4C2412yes—
139C3×M16—216no—
140C3×M16—232no—
141C2×C3×C8—28no—
142C2×C3×C8—24no—
143C2×C3×C8—28no—
144C2×C3×C8—232no—
145C2×C3×C8C2×D414yes—
146C2×C3×C8C2×D414yes—
147C2×C3×C8C2×D4116yescomplemented
148C2×C3×C8C2414yes—
149C2×C3×C8C2418yes—
150C2×C3×C8C2412yes—
151C2×C3×C8C2412yes—
152C3×C4⋊C4C22×C4116yes—
153C3×C22⋊C4C22×C4132yes—
154C3×C42C22×C414yes—
155C3×C42C22×C414yes—
156C3×C42C22×C414yes—
157C3×C42C2414yes—
158C2×#41—28no—
159C2×#41—28no—
160C2×#41C2×C3×C418yes—
161C2×#41C23×C318yes—
162C22×Q8C2×C3×C412yes—
163C22×D4C2×C3×C414yes—
164C22×D4C2×C3×C414yes—
165C22×D4C2×C3×C412yes—
166C23×C4C2×C3×C412yes—
167C23×C4C23×C314yes—
168C23×C4C23×C311yescharacteristic
169#131—264no—
170C2×M16—28no—
171C2×M16—216no—
172C2×M16C23×C318yes—
173C2×M16C23×C3116yes—
174C22×C8—24no—
175C22×C8—216no—
176C22×C8C3×D418yescomplemented
177C22×C8C23×C314yes—
178C22×C8C23×C3116yes—
179C22×C8C23×C312yes—
180C22×C8C23×C314yes—
181C22×C8C23×C311yescharacteristic
182C22×C8C23×C311yescharacteristic
183C22×C8C23×C314yes—
184C4⋊Q8C2×C3×C418yes—
185#127C2×C3×C418yes—
186#124C2×C3×C4116yes—
187#121C2×C3×C4132yes—
188C4×D4C2×C3×C4116yes—
189C4×D4C23×C3116yes—
190C2×C4⋊C4C2×C3×C414yes—
191C2×C22⋊C4C2×C3×C418yes—
192C2×C42C2×C3×C411yescharacteristic
193C2×C42C2×C3×C411yescharacteristic
194C2×C42C2×C3×C414yes—
195C2×C42C23×C312yes—
196C2×C42C23×C314yes—
197C4⋊C8C23×C3116yes—
198C22⋊C8C23×C3132yes—
199C4×C8C23×C318yes—
200C4×C8C23×C318yes—
201C23×C3—24no—
202C23×C3—28no—
203C23×C3—216no—
204C23×C3C4×D414yes—
205C23×C3C23×C418yes—
206C23×C3C23×C412yes—
207C23×C3C23×C411yescharacteristic
208C3×Q8—216no—
209C3×D4—232no—
210C3×D4—216no—
211C3×D4—232no—
212C2×C3×C4—28no—
213C2×C3×C4—24no—
214C2×C3×C4—216no—
215C2×C3×C4—216no—
216C2×C3×C4—28no—
217C2×C3×C4—28no—
218C2×C3×C4C4×D414yes—
219C2×C3×C4C4×D418yes—
220C2×C3×C4#13118yescomplemented
221C2×C3×C4C23×C414yes—
222C2×C3×C4C23×C418yes—
223C2×C3×C4C23×C411yescharacteristic
224C2×C3×C4C23×C412yes—
225C2×C3×C4C23×C414yes—
226C2×C3×C4C22×D412yes—
227C2×C3×C4C22×D414yes—
228C2×C3×C4C2×#4112yes—
229C2×C3×C4C2×#4114yes—
230C2×C3×C4C2511yescharacteristic
231C3×C8—28no—
232C3×C8—24no—
233C3×C8C22×D414yes—
234C3×C8C22×D414yes—
235C24C22×C3×C414yes—
236C24C22×C3×C411yescharacteristic
237#41—232no—
238#41—216no—
239#41—216no—
240C2×Q8—28no—
241C2×Q8C22×C3×C418yes—
242C2×D4—216no—
243C2×D4—28no—
244C2×D4—216no—
245C2×D4C22×C3×C4116yes—
246C2×D4C22×C3×C4116yes—
247C2×D4C22×C3×C418yes—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4C22×C3×C412yes—
252C22×C4C22×C3×C418yes—
253C22×C4C22×C3×C414yes—
254C22×C4C22×C3×C418yes—
255C22×C4C22×C3×C411yescharacteristic
256C22×C4C22×C3×C412yes—
257C22×C4C22×C3×C412yes—
258C22×C4C2×C3×D414yes—
259C22×C4C3×#4114yes—
260C22×C4C24×C318yes—
261C22×C4C24×C311yescharacteristic
262C22×C4C24×C312yes—
263M16—216no—
264M16—232no—
265C2×C8—28no—
266C2×C8—24no—
267C2×C8—28no—
268C2×C8—232no—
269C2×C8C2×C3×D414yes—
270C2×C8C2×C3×D414yes—
271C2×C8C2×C3×D4116yescomplemented
272C2×C8C24×C314yes—
273C2×C8C24×C318yes—
274C2×C8C24×C312yes—
275C2×C8C24×C312yes—
276C4⋊C4C22×C3×C4116yes—
277C22⋊C4C22×C3×C4132yes—
278C42C22×C3×C414yes—
279C42C22×C3×C414yes—
280C42C22×C3×C414yes—
281C42C24×C314yes—
282C22×C3—28no—
283C22×C3—24no—
284C22×C3—28no—
285C22×C3—216no—
286C22×C3—416no—
287C22×C3C2×C4×D414yes—
288C22×C3C2×C4×D412yes—
289C22×C3C2×#13114yes—
290C22×C3C24×C411yescharacteristic
291C3×C4—28no—
292C3×C4—24no—
293C3×C4C2×C4×D412yes—
294C3×C4C2×C4×D414yes—
295C3×C4C2×#13118yescomplemented
296C3×C4C23×D411yescharacteristic
297C3×C4C22×#4111yescharacteristic
298C23—24no—
299C23—216no—
300C23—28no—
301C23C3×C4×D414yes—
302C23C23×C3×C418yes—
303C23C23×C3×C412yes—
304C23C23×C3×C411yescharacteristic
305Q8—216no—
306D4—232no—
307D4—232no—
308D4—216no—
309C2×C4—24no—
310C2×C4—28no—
311C2×C4—216no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—216no—
315C2×C4C3×C4×D414yes—
316C2×C4C3×C4×D418yes—
317C2×C4C3×#13118yescomplemented
318C2×C4C23×C3×C414yes—
319C2×C4C23×C3×C418yes—
320C2×C4C23×C3×C414yes—
321C2×C4C23×C3×C411yescharacteristic
322C2×C4C23×C3×C412yes—
323C2×C4C22×C3×D412yes—
324C2×C4C22×C3×D414yes—
325C2×C4C2×C3×#4112yes—
326C2×C4C2×C3×#4114yes—
327C2×C4C25×C311yescharacteristic
328C8—28no—
329C8—24no—
330C8C22×C3×D414yes—
331C8C22×C3×D414yes—
332C2×C3—24no—
333C2×C3—48no—
334C2×C3#293114yesdirect product factor
335C2×C3C22×C4×D411yescharacteristic
336C2×C3C22×#13112yes—
337C22—28no—
338C22—24no—
339C22—28no—
340C22—216no—
341C22—416no—
342C22C2×C3×C4×D414yes—
343C22C2×C3×C4×D412yes—
344C22C2×C3×#13114yes—
345C22C24×C3×C411yescharacteristic
346C4—28no—
347C4—24no—
348C4C2×C3×C4×D412yes—
349C4C2×C3×C4×D414yes—
350C4C2×C3×#13118yescomplemented
351C4C23×C3×D411yescharacteristic
352C4C22×C3×#4111yescharacteristic
353C3C2×#293111yescharacteristic, direct product factor, Sylow
354C2—24no—
355C2—48no—
356C2C3×#293114yesdirect product factor
357C2C22×C3×C4×D411yescharacteristic
358C2C22×C3×#13112yes—
subgroupG/H
centerC22×C3×C8C23
commutatorC22C24×C3×C4
Frattini, cosocleC2×C4C25×C3
socleC23×C3C23×C4
prime radicalC2×C3×#532C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#131C2semidirect
3. C3×#2931C2direct (N × H)
4. C2×C3×#436C2semidirect
5. C2×C3×C8×D4C2semidirect
6. C2×C3×#423C2semidirect
7. C2×C3×#408C2semidirect
8. C22×C3×C4×C8C2semidirect
9. C2×#2931C3direct (N × H)
12. C2×C3×#131C22semidirect
26. C3×#436C22semidirect
27. C3×C8×D4C22semidirect
28. C3×#423C22semidirect
30. C3×#408C22semidirect
34. C2×C3×C4×C8C22semidirect
35. C22×#131C2×C3semidirect
37. #2931C2×C3direct (N × H)
38. C2×#436C2×C3semidirect
39. C2×C8×D4C2×C3semidirect
40. C2×#423C2×C3semidirect
41. C2×#408C2×C3semidirect
42. C22×C4×C8C2×C3semidirect
61. C22×C3×C8D4semidirect
88. C2×#131C22×C3semidirect
102. #436C22×C3semidirect
103. C8×D4C22×C3semidirect
104. #423C22×C3semidirect
106. #408C22×C3semidirect
110. C2×C4×C8C22×C3semidirect
147. C2×C3×C8C2×D4semidirect
176. C22×C8C3×D4semidirect
220. C2×C3×C4#131semidirect
271. C2×C8C2×C3×D4semidirect
295. C3×C4C2×#131semidirect
317. C2×C4C3×#131semidirect
334. C2×C3#2931direct (N × H)
350. C4C2×C3×#131semidirect
353. C3C2×#2931direct (N × H)
356. C2C3×#2931direct (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×#29312
C2×C3×#4362
C2×C3×C8×D42
C2×C3×#4232
C2×C3×#4082
C22×C3×C4×C82
C2×#29313