Finite Group Databasebeta

← C22×C3×#339 #10638948

Subgroups — C22×C3×#339 #10638948

Order 768.

2830 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C211yesmaximal, characteristic
2C22×C3×#117C211yesmaximal, characteristic
3C23×C3×C42C211yesmaximal, characteristic
4C2×C3×#339C2112yesmaximal, direct product factor
5C22×#339C311yesmaximal, characteristic, direct product factor, Sylow
6C24×C3×C4—21no—
7C24×C3×C4C2211yescharacteristic
8C22×C3×M16—24no—
9C22×C3×M16—23no—
10C22×C3×M16C418yescomplemented
11C22×C3×M16C2216yes—
12C23×C3×C8—21no—
13C2×C3×#117—23no—
14C2×C3×#117—23no—
15C2×C3×#117C416yescomplemented
16C2×C3×#117C2216yes—
17C22×C3×C4⋊C4C412yescomplemented
18C22×C3×C22⋊C4—21no—
19C22×C3×C42—23no—
20C22×C3×C42—24no—
21C22×C3×C42—21no—
22C22×C3×C42C418yescomplemented
23C22×C3×C42C2216yes—
24C3×#339C22116yesdirect product factor
25C23×M16C2×C311yescharacteristic
26C22×#117C2×C311yescharacteristic
27C23×C42C2×C311yescharacteristic
28C2×#339C2×C3112yesdirect product factor
29C25×C3D411yescharacteristic
30C23×C3×C4—216no—
31C23×C3×C4—28no—
32C23×C3×C4—26no—
33C23×C3×C4—22no—
34C23×C3×C4C2×C414yes—
35C23×C3×C4C2×C414yes—
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C413yes—
38C23×C3×C4D413yes—
39C23×C3×C4D414yes—
40C23×C3×C4D413yes—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4Q814yes—
43C23×C3×C4C2313yes—
44C2×C3×M16—224no—
45C2×C3×M16—212no—
46C2×C3×M16—26no—
47C2×C3×M16C2×C4124yescomplemented
48C2×C3×M16C2314yes—
49C22×C3×C8—28no—
50C22×C3×C8—26no—
51C3×#117—212no—
52C3×#117—26no—
53C3×#117—26no—
54C3×#117C2×C4112yescomplemented
55C3×#117C2314yes—
56C2×C3×C4⋊C4—26no—
57C2×C3×C4⋊C4C2×C4112yescomplemented
58C2×C3×C22⋊C4—26no—
59C2×C3×C22⋊C4—26no—
60C2×C3×C42—224no—
61C2×C3×C42—26no—
62C2×C3×C42—26no—
63C2×C3×C42—212no—
64C2×C3×C42—26no—
65C2×C3×C42C2×C4124yescomplemented
66C2×C3×C42C2314yes—
67C24×C4—21no—
68C24×C4C22×C311yescharacteristic
69C22×M16—24no—
70C22×M16—23no—
71C22×M16C3×C418yescomplemented
72C22×M16C22×C316yes—
73C23×C8—21no—
74C2×#117—23no—
75C2×#117—23no—
76C2×#117C3×C416yescomplemented
77C2×#117C22×C316yes—
78C22×C4⋊C4C3×C412yescomplemented
79C22×C22⋊C4—21no—
80C22×C42—23no—
81C22×C42—24no—
82C22×C42—21no—
83C22×C42C3×C418yescomplemented
84C22×C42C22×C316yes—
85#339C22×C3116yesdirect product factor
86C24×C3—28no—
87C24×C3C22⋊C413yes—
88C24×C3C22⋊C414yes—
89C24×C3C22⋊C411yescharacteristic
90C24×C3C4⋊C414yes—
91C24×C3C2×D413yes—
92C22×C3×C4—264no—
93C22×C3×C4—248no—
94C22×C3×C4—224no—
95C22×C3×C4—24no—
96C22×C3×C4—212no—
97C22×C3×C4—48no—
98C22×C3×C4C42112yes—
99C22×C3×C4C4214yes—
100C22×C3×C4C22⋊C4112yes—
101C22×C3×C4C22⋊C413yes—
102C22×C3×C4C22⋊C413yes—
103C22×C3×C4C22⋊C416yes—
104C22×C3×C4C22⋊C414yes—
105C22×C3×C4C22⋊C4112yes—
106C22×C3×C4C4⋊C4112yes—
107C22×C3×C4C4⋊C4112yes—
108C22×C3×C4C4⋊C4112yes—
109C22×C3×C4C4⋊C414yes—
110C22×C3×C4C4⋊C414yes—
111C22×C3×C4C22×C416yes—
112C22×C3×C4C22×C416yes—
113C22×C3×C4C22×C413yes—
114C22×C3×C4C22×C413yes—
115C22×C3×C4C2×D413yes—
116C22×C3×C4C2×D413yes—
117C22×C3×C4C2×D416yes—
118C22×C3×C4C2×D413yes—
119C22×C3×C4C2×Q816yes—
120C22×C3×C4C2411yescharacteristic
121C3×M16—224no—
122C3×M16—24no—
123C3×M16C22×C418yescomplemented
124C2×C3×C8—224no—
125C2×C3×C8—24no—
126C3×C4⋊C4—212no—
127C3×C4⋊C4C22×C418yescomplemented
128C3×C22⋊C4—212no—
129C3×C22⋊C4—24no—
130C3×C42—224no—
131C3×C42—212no—
132C3×C42—24no—
133C3×C42—24no—
134C3×C42C22×C418yescomplemented
135C25C3×D411yescharacteristic
136C23×C4—216no—
137C23×C4—28no—
138C23×C4—26no—
139C23×C4—22no—
140C23×C4C2×C3×C414yes—
141C23×C4C2×C3×C411yescharacteristic
142C23×C4C2×C3×C413yes—
143C23×C4C2×C3×C414yes—
144C23×C4C3×D413yes—
145C23×C4C3×D413yes—
146C23×C4C3×D411yescharacteristic
147C23×C4C3×D414yes—
148C23×C4C3×Q814yes—
149C23×C4C23×C313yes—
150C2×M16—224no—
151C2×M16—212no—
152C2×M16—26no—
153C2×M16C2×C3×C4124yescomplemented
154C2×M16C23×C314yes—
155C22×C8—28no—
156C22×C8—26no—
157#117—212no—
158#117—26no—
159#117—26no—
160#117C2×C3×C4112yescomplemented
161#117C23×C314yes—
162C2×C4⋊C4—26no—
163C2×C4⋊C4C2×C3×C4112yescomplemented
164C2×C22⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C42—26no—
167C2×C42—224no—
168C2×C42—26no—
169C2×C42—26no—
170C2×C42—212no—
171C2×C42C2×C3×C4124yescomplemented
172C2×C42C23×C314yes—
173C23×C3—216no—
174C23×C3—216no—
175C23×C3—212no—
176C23×C3—212no—
177C23×C3#95112yes—
178C23×C3#9514yes—
179C23×C3#10418yes—
180C23×C3C2×C22⋊C413yes—
181C23×C3C2×C22⋊C416yes—
182C23×C3C2×C22⋊C413yes—
183C23×C3C2×C4⋊C416yes—
184C23×C3C22×D411yescharacteristic
185C2×C3×C4—216no—
186C2×C3×C4—296no—
187C2×C3×C4—28no—
188C2×C3×C4—28no—
189C2×C3×C4—424no—
190C2×C3×C4#95124yes—
191C2×C3×C4#95112yes—
192C2×C3×C4#95112yes—
193C2×C3×C4C2×C4216yes—
194C2×C3×C4C2×C4216yes—
195C2×C3×C4C2×C22⋊C416yes—
196C2×C3×C4C2×C22⋊C416yes—
197C2×C3×C4C2×C22⋊C416yes—
198C2×C3×C4C2×C22⋊C413yes—
199C2×C3×C4C2×C22⋊C413yes—
200C2×C3×C4C2×C4⋊C416yes—
201C2×C3×C4C2×C4⋊C416yes—
202C2×C3×C4C2×C4⋊C416yes—
203C2×C3×C4C2×C4⋊C416yes—
204C2×C3×C4C2×C4⋊C416yes—
205C2×C3×C4C23×C411yescharacteristic
206C2×C3×C4C23×C411yescharacteristic
207C2×C3×C4C23×C411yescharacteristic
208C2×C3×C4C22×D411yescharacteristic
209C2×C3×C4C22×D411yescharacteristic
210C2×C3×C4C22×Q811yescharacteristic
211C3×C8—28no—
212C24—28no—
213C24C3×C22⋊C413yes—
214C24C3×C22⋊C414yes—
215C24C3×C22⋊C411yescharacteristic
216C24C3×C4⋊C414yes—
217C24C2×C3×D413yes—
218C22×C4—248no—
219C22×C4—264no—
220C22×C4—224no—
221C22×C4—24no—
222C22×C4—212no—
223C22×C4—48no—
224C22×C4C3×C42112yes—
225C22×C4C3×C4214yes—
226C22×C4C3×C22⋊C4112yes—
227C22×C4C3×C22⋊C413yes—
228C22×C4C3×C22⋊C413yes—
229C22×C4C3×C22⋊C416yes—
230C22×C4C3×C22⋊C414yes—
231C22×C4C3×C22⋊C4112yes—
232C22×C4C3×C4⋊C4112yes—
233C22×C4C3×C4⋊C4112yes—
234C22×C4C3×C4⋊C4112yes—
235C22×C4C3×C4⋊C414yes—
236C22×C4C3×C4⋊C414yes—
237C22×C4C22×C3×C416yes—
238C22×C4C22×C3×C416yes—
239C22×C4C22×C3×C413yes—
240C22×C4C22×C3×C413yes—
241C22×C4C2×C3×D413yes—
242C22×C4C2×C3×D413yes—
243C22×C4C2×C3×D416yes—
244C22×C4C2×C3×D413yes—
245C22×C4C2×C3×Q816yes—
246C22×C4C24×C311yescharacteristic
247M16—224no—
248M16—24no—
249M16C22×C3×C418yescomplemented
250C2×C8—224no—
251C2×C8—24no—
252C4⋊C4—212no—
253C4⋊C4C22×C3×C418yescomplemented
254C22⋊C4—212no—
255C22⋊C4—24no—
256C42—224no—
257C42—212no—
258C42—24no—
259C42—24no—
260C42C22×C3×C418yescomplemented
261C22×C3—22no—
262C22×C3—26no—
263C22×C3—212no—
264C22×C3—224no—
265C22×C3—212no—
266C22×C3#339116yesdirect product factor
267C22×C3C2×#9516yes—
268C22×C3C2×#9516yes—
269C22×C3C2×#104112yes—
270C22×C3C22×C22⋊C411yescharacteristic
271C22×C3C22×C22⋊C411yescharacteristic
272C22×C3C22×C4⋊C411yescharacteristic
273C3×C4—216no—
274C3×C4—48no—
275C3×C4C2×#9516yes—
276C3×C4C2×#9516yes—
277C3×C4C22×C4211yescharacteristic
278C3×C4C22×C22⋊C411yescharacteristic
279C3×C4C22×C4⋊C411yescharacteristic
280C3×C4C22×C4⋊C411yescharacteristic
281C23—212no—
282C23—212no—
283C23—216no—
284C23—216no—
285C23C3×#95112yes—
286C23C3×#9514yes—
287C23C3×#10418yes—
288C23C2×C3×C22⋊C413yes—
289C23C2×C3×C22⋊C416yes—
290C23C2×C3×C22⋊C413yes—
291C23C2×C3×C4⋊C416yes—
292C23C22×C3×D411yescharacteristic
293C2×C4—216no—
294C2×C4—296no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—424no—
298C2×C4C3×#95124yes—
299C2×C4C3×#95112yes—
300C2×C4C3×#95112yes—
301C2×C4C2×C3×C4216yes—
302C2×C4C2×C3×C4216yes—
303C2×C4C2×C3×C22⋊C416yes—
304C2×C4C2×C3×C22⋊C413yes—
305C2×C4C2×C3×C22⋊C413yes—
306C2×C4C2×C3×C22⋊C416yes—
307C2×C4C2×C3×C22⋊C416yes—
308C2×C4C2×C3×C4⋊C416yes—
309C2×C4C2×C3×C4⋊C416yes—
310C2×C4C2×C3×C4⋊C416yes—
311C2×C4C2×C3×C4⋊C416yes—
312C2×C4C2×C3×C4⋊C416yes—
313C2×C4C23×C3×C411yescharacteristic
314C2×C4C23×C3×C411yescharacteristic
315C2×C4C23×C3×C411yescharacteristic
316C2×C4C22×C3×D411yescharacteristic
317C2×C4C22×C3×D411yescharacteristic
318C2×C4C22×C3×Q811yescharacteristic
319C8—28no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—26no—
323C2×C3C2×#339112yesdirect product factor
324C2×C3C22×#9511yescharacteristic
325C2×C3C22×#10412yes—
326C22—22no—
327C22—26no—
328C22—212no—
329C22—224no—
330C22—212no—
331C22C3×#339116yesdirect product factor
332C22C2×C3×#9516yes—
333C22C2×C3×#9516yes—
334C22C2×C3×#104112yes—
335C22C22×C3×C22⋊C411yescharacteristic
336C22C22×C3×C22⋊C411yescharacteristic
337C22C22×C3×C4⋊C411yescharacteristic
338C4—216no—
339C4—48no—
340C4C2×C3×#9516yes—
341C4C2×C3×#9516yes—
342C4C22×C3×C4211yescharacteristic
343C4C22×C3×C22⋊C411yescharacteristic
344C4C22×C3×C4⋊C411yescharacteristic
345C4C22×C3×C4⋊C411yescharacteristic
346C3C22×#33911yescharacteristic, direct product factor, Sylow
347C2—21no—
348C2—26no—
349C2—21no—
350C2C2×C3×#339112yesdirect product factor
351C2C22×C3×#9511yescharacteristic
352C2C22×C3×#10412yes—
subgroupG/H
centerC23×C3×C4D4
commutatorC4C22×C3×C42
Frattini, cosocleC22×C4C24×C3
socleC24×C3C22⋊C4
prime radicalC25×C3D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3×C4 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×C3×#339C2direct (N × H)
5. C22×#339C3direct (N × H)
10. C22×C3×M16C4semidirect
15. C2×C3×#117C4semidirect
17. C22×C3×C4⋊C4C4semidirect
22. C22×C3×C42C4semidirect
24. C3×#339C22direct (N × H)
28. C2×#339C2×C3direct (N × H)
47. C2×C3×M16C2×C4semidirect
54. C3×#117C2×C4semidirect
57. C2×C3×C4⋊C4C2×C4semidirect
65. C2×C3×C42C2×C4semidirect
71. C22×M16C3×C4semidirect
76. C2×#117C3×C4semidirect
78. C22×C4⋊C4C3×C4semidirect
83. C22×C42C3×C4semidirect
85. #339C22×C3direct (N × H)
123. C3×M16C22×C4semidirect
127. C3×C4⋊C4C22×C4semidirect
134. C3×C42C22×C4semidirect
153. C2×M16C2×C3×C4semidirect
160. #117C2×C3×C4semidirect
163. C2×C4⋊C4C2×C3×C4semidirect
171. C2×C42C2×C3×C4semidirect
249. M16C22×C3×C4semidirect
253. C4⋊C4C22×C3×C4semidirect
260. C42C22×C3×C4semidirect
266. C22×C3#339direct (N × H)
323. C2×C3C2×#339direct (N × H)
331. C22C3×#339direct (N × H)
346. C3C22×#339direct (N × H)
350. C2C2×C3×#339direct (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
C22×C3×#1172
C23×C3×C422
C2×C3×#3392
C22×#3393