Finite Group Databasebeta

← C3×#12582 #10631189

Subgroups — C3×#12582 #10631189

Order 768.

1026 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#99C212yesmaximal
2C3×#2309C211yesmaximal, characteristic
3C3×#1393C214yesmaximal, complemented
4#12582C311yesmaximal, characteristic, direct product factor, Sylow
5C23×C3×D4C2211yescharacteristic
6C2×C3×#117C411yescharacteristic, complemented
7C2×C3×#117C411yescharacteristic, complemented
8C22×C3×C22⋊C4—22no—
9C2×C3×#99—22no—
10C2×C3×#99—24no—
11C2×C3×#99C418yescomplemented
12C2×C3×#99C2214yes—
13C3×#390—22no—
14C3×#390C412yescomplemented
15C3×#390C2212yes—
16C3×#379—22no—
17C22×#99C2×C312yes—
18#2309C2×C311yescharacteristic
19#1393C2×C314yescomplemented
20C25×C3D412yes—
21C22×C3×D4—42no—
22C22×C3×D4—42no—
23C22×C3×D4C2×C414yes—
24C22×C3×D4C2×C411yescharacteristic
25C22×C3×D4D412yes—
26C22×C3×D4D412yes—
27C22×C3×D4Q812yes—
28C22×C3×D4C2311yescharacteristic
29C23×C3×C4—42no—
30C23×C3×C4C2×C411yescharacteristic
31C3×#117—42no—
32C3×#117—42no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4C2×C412yescomplemented
35C2×C3×C22⋊C4—28no—
36C2×C3×C22⋊C4—24no—
37C2×C3×C22⋊C4—21no—
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—24no—
41C2×C3×C22⋊C4—42no—
42C2×C3×C22⋊C4—44no—
43C2×C3×C42—21no—
44C2×C3×C42C2×C412yescomplemented
45C3×#99—28no—
46C3×#99—24no—
47C3×#99C2×C418yescomplemented
48C23×D4C22×C311yescharacteristic
49C2×#117C3×C411yescharacteristic, complemented
50C2×#117C3×C411yescharacteristic, complemented
51C22×C22⋊C4—22no—
52C2×#99—22no—
53C2×#99—24no—
54C2×#99C3×C418yescomplemented
55C2×#99C22×C314yes—
56#390—22no—
57#390C3×C412yescomplemented
58#390C22×C312yes—
59#379—22no—
60C24×C3—22no—
61C24×C3—24no—
62C24×C3—22no—
63C24×C3—84no—
64C24×C3C22⋊C412yes—
65C24×C3C22⋊C414yes—
66C24×C3C22⋊C411yescharacteristic
67C24×C3C4⋊C414yes—
68C24×C3C2×D412yes—
69C2×C3×D4—44no—
70C2×C3×D4—44no—
71C2×C3×D4—41no—
72C2×C3×D4—41no—
73C2×C3×D4—44no—
74C2×C3×D4—42no—
75C2×C3×D4—42no—
76C2×C3×D4—44no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C2×C3×D4C4212yes—
80C2×C3×D4C22⋊C414yes—
81C2×C3×D4C4⋊C414yes—
82C2×C3×D4C4⋊C412yes—
83C2×C3×D4C22×C412yes—
84C2×C3×D4C2×D411yescharacteristic
85C2×C3×D4C2×Q811yescharacteristic
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—48no—
91C22×C3×C4—44no—
92C22×C3×C4—88no—
93C22×C3×C4C4212yes—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C4⋊C412yes—
96C22×C3×C4C22×C411yescharacteristic
97C3×C4⋊C4—42no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—28no—
100C3×C22⋊C4—48no—
101C3×C22⋊C4—44no—
102C3×C22⋊C4—42no—
103C3×C22⋊C4—42no—
104C3×C22⋊C4—84no—
105C3×C22⋊C4—84no—
106C3×C42—42no—
107C3×C42—42no—
108C25C3×D412yes—
109C22×D4—42no—
110C22×D4—42no—
111C22×D4C2×C3×C414yes—
112C22×D4C2×C3×C411yescharacteristic
113C22×D4C3×D412yes—
114C22×D4C3×D412yes—
115C22×D4C3×Q812yes—
116C22×D4C23×C311yescharacteristic
117C23×C4—42no—
118C23×C4C2×C3×C411yescharacteristic
119#117—42no—
120#117—42no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×C3×C412yescomplemented
123C2×C22⋊C4—28no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4—42no—
131C2×C42—21no—
132C2×C42C2×C3×C412yescomplemented
133#99—28no—
134#99—24no—
135#99C2×C3×C418yescomplemented
136C23×C3—24no—
137C23×C3—24no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—24no—
141C23×C3—24no—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—42no—
145C23×C3—48no—
146C23×C3—44no—
147C23×C3—44no—
148C23×C3—84no—
149C23×C3—84no—
150C23×C3—84no—
151C23×C3—88no—
152C23×C3#9514yes—
153C23×C3#9512yes—
154C23×C3#9912yes—
155C23×C3#9912yes—
156C23×C3C2×C22⋊C411yescharacteristic
157C23×C3C2×C22⋊C412yes—
158C23×C3C2×C4⋊C412yes—
159C3×D4—42no—
160C3×D4—44no—
161C3×D4—44no—
162C3×D4—42no—
163C3×D4—42no—
164C3×D4—41no—
165C3×D4—41no—
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—48no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—44no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—816no—
179C2×C3×C4—88no—
180C2×C3×C4#9512yes—
181C2×C3×C4C2×C4211yescharacteristic
182C2×C3×C4C2×C4⋊C411yescharacteristic
183C24—22no—
184C24—22no—
185C24—24no—
186C24—84no—
187C24C3×C22⋊C412yes—
188C24C3×C22⋊C414yes—
189C24C3×C22⋊C411yescharacteristic
190C24C3×C4⋊C414yes—
191C24C2×C3×D412yes—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4C3×C4212yes—
203C2×D4C3×C22⋊C414yes—
204C2×D4C3×C4⋊C414yes—
205C2×D4C3×C4⋊C412yes—
206C2×D4C22×C3×C412yes—
207C2×D4C2×C3×D411yescharacteristic
208C2×D4C2×C3×Q811yescharacteristic
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—44no—
214C22×C4—48no—
215C22×C4—88no—
216C22×C4C3×C4212yes—
217C22×C4C3×C22⋊C411yescharacteristic
218C22×C4C3×C4⋊C412yes—
219C22×C4C22×C3×C411yescharacteristic
220C4⋊C4—42no—
221C4⋊C4—42no—
222C22⋊C4—28no—
223C22⋊C4—44no—
224C22⋊C4—48no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—84no—
228C22⋊C4—84no—
229C42—42no—
230C42—42no—
231C22×C3—22no—
232C22×C3—24no—
233C22×C3—22no—
234C22×C3—24no—
235C22×C3—22no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—44no—
239C22×C3—42no—
240C22×C3—42no—
241C22×C3—48no—
242C22×C3—48no—
243C22×C3—44no—
244C22×C3—44no—
245C22×C3—82no—
246C22×C3—82no—
247C22×C3—84no—
248C22×C3—84no—
249C22×C3#34214yes—
250C22×C3C2×#9511yescharacteristic
251C22×C3C2×#9911yescharacteristic
252C22×C3C2×#9911yescharacteristic
253C3×C4—22no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—82no—
257C3×C4—88no—
258C3×C4—82no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—24no—
263C23—22no—
264C23—22no—
265C23—24no—
266C23—24no—
267C23—42no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23—84no—
272C23—84no—
273C23—84no—
274C23—88no—
275C23C3×#9514yes—
276C23C3×#9512yes—
277C23C3×#9912yes—
278C23C3×#9912yes—
279C23C2×C3×C22⋊C411yescharacteristic
280C23C2×C3×C22⋊C412yes—
281C23C2×C3×C4⋊C412yes—
282D4—42no—
283D4—44no—
284D4—44no—
285D4—42no—
286D4—42no—
287D4—41no—
288D4—41no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—816no—
302C2×C4—88no—
303C2×C4C3×#9512yes—
304C2×C4C2×C3×C4211yescharacteristic
305C2×C4C2×C3×C4⋊C411yescharacteristic
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—24no—
309C2×C3—42no—
310C2×C3—44no—
311C2×C3—42no—
312C2×C3C2×#34211yescharacteristic
313C2×C3#212712yes—
314C22—22no—
315C22—24no—
316C22—22no—
317C22—24no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—44no—
322C22—42no—
323C22—42no—
324C22—48no—
325C22—48no—
326C22—44no—
327C22—44no—
328C22—82no—
329C22—82no—
330C22—84no—
331C22—84no—
332C22C3×#34214yes—
333C22C2×C3×#9511yescharacteristic
334C22C2×C3×#9911yescharacteristic
335C22C2×C3×#9911yescharacteristic
336C4—22no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—88no—
341C4—82no—
342C3#1258211yescharacteristic, direct product factor, Sylow
343C2—21no—
344C2—21no—
345C2—24no—
346C2—42no—
347C2—44no—
348C2—42no—
349C2C2×C3×#34211yescharacteristic
350C2C3×#212712yes—
subgroupG/H
center, socleC22×C3C2×#99
commutatorC2×C4C2×C3×C42
Frattini, cosocleC22×D4C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1393C2semidirect
4. #12582C3direct (N × H)
6. C2×C3×#117C4semidirect
7. C2×C3×#117C4semidirect
11. C2×C3×#99C4semidirect
14. C3×#390C4semidirect
19. #1393C2×C3semidirect
34. C2×C3×C4⋊C4C2×C4semidirect
44. C2×C3×C42C2×C4semidirect
47. C3×#99C2×C4semidirect
49. C2×#117C3×C4semidirect
50. C2×#117C3×C4semidirect
54. C2×#99C3×C4semidirect
57. #390C3×C4semidirect
122. C2×C4⋊C4C2×C3×C4semidirect
132. C2×C42C2×C3×C4semidirect
135. #99C2×C3×C4semidirect
342. C3#12582direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#992
C3×#23092
C3×#13932
#125823