Finite Group Databasebeta

← C3×#12699 #10631306

Subgroups — C3×#12699 #10631306

Order 768.

1146 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#124C211yesmaximal, characteristic
2C2×C3×#355C214yesmaximal, complemented
3C3×#1784C212yesmaximal, complemented
4#12699C311yesmaximal, characteristic, direct product factor, Sylow
5C23×C3×Q8C2211yescharacteristic
6C23×C3×D4C411yescharacteristic
7C2×C3×#124—22no—
8C2×C3×#124—28no—
9C2×C3×#124C412yes—
10C2×C3×#124C2212yes—
11C22×C3×C22⋊C4—22no—
12C22×C3×C42C411yescharacteristic
13C2×C3×#101—22no—
14C2×C3×#101C2214yescomplemented
15C3×#407—22no—
16C3×#355—28no—
17C22×#124C2×C311yescharacteristic
18C2×#355C2×C314yescomplemented
19#1784C2×C312yescomplemented
20C25×C3—21no—
21C22×C3×Q8—24no—
22C22×C3×Q8—42no—
23C22×C3×Q8—42no—
24C22×C3×Q8C2×C411yescharacteristic
25C22×C3×Q8D412yes—
26C22×C3×Q8C2311yescharacteristic
27C22×C3×D4—24no—
28C22×C3×D4—21no—
29C22×C3×D4—42no—
30C22×C3×D4—42no—
31C22×C3×D4C2×C412yes—
32C23×C3×C4—21no—
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4D412yes—
35C2×C3×M16—24no—
36C3×#124—216no—
37C3×#124—24no—
38C3×#124—24no—
39C3×#124—24no—
40C3×#124D418yes—
41C2×C3×C22⋊C4—216no—
42C2×C3×C22⋊C4—24no—
43C2×C3×C22⋊C4—24no—
44C2×C3×C42—21no—
45C2×C3×C42—24no—
46C2×C3×C42C2×C412yes—
47C3×#101—28no—
48C3×#101—44no—
49C3×C22⋊C8—44no—
50C23×Q8C22×C311yescharacteristic
51C23×D4C3×C411yescharacteristic
52C2×#124—22no—
53C2×#124—28no—
54C2×#124C3×C412yes—
55C2×#124C22×C312yes—
56C22×C22⋊C4—22no—
57C22×C42C3×C411yescharacteristic
58C2×#101—22no—
59C2×#101C22×C314yescomplemented
60#407—22no—
61#355—28no—
62C24×C3—24no—
63C24×C3—22no—
64C24×C3—48no—
65C24×C3—42no—
66C24×C3—42no—
67C24×C3C22⋊C411yescharacteristic
68C2×C3×Q8—24no—
69C2×C3×Q8—22no—
70C2×C3×Q8—24no—
71C2×C3×Q8—41no—
72C2×C3×Q8—41no—
73C2×C3×Q8—42no—
74C2×C3×Q8—41no—
75C2×C3×Q8—41no—
76C2×C3×Q8—416no—
77C2×C3×Q8C2×D414yes—
78C2×C3×D4—24no—
79C2×C3×D4—22no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—416no—
83C2×C3×D4—42no—
84C2×C3×D4—42no—
85C2×C3×D4C22⋊C414yes—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—22no—
92C22×C3×C4—24no—
93C22×C3×C4—42no—
94C22×C3×C4—44no—
95C22×C3×C4C22⋊C412yes—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D412yes—
99C3×M16—44no—
100C3×M16—44no—
101C2×C3×C8—44no—
102C3×C22⋊C4—216no—
103C3×C22⋊C4—216no—
104C3×C42—24no—
105C3×C42—22no—
106C3×C42C22⋊C414yes—
107C25—21no—
108C22×Q8—24no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8C2×C3×C411yescharacteristic
112C22×Q8C3×D412yes—
113C22×Q8C23×C311yescharacteristic
114C22×D4—24no—
115C22×D4—21no—
116C22×D4—42no—
117C22×D4—42no—
118C22×D4C2×C3×C412yes—
119C23×C4—21no—
120C23×C4C2×C3×C411yescharacteristic
121C23×C4C3×D412yes—
122C2×M16—24no—
123#124—216no—
124#124—24no—
125#124—24no—
126#124—24no—
127#124C3×D418yes—
128C2×C22⋊C4—216no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C42—21no—
132C2×C42—24no—
133C2×C42C2×C3×C412yes—
134#101—28no—
135#101—44no—
136C22⋊C8—44no—
137C23×C3—24no—
138C23×C3—21no—
139C23×C3—21no—
140C23×C3—24no—
141C23×C3—48no—
142C23×C3—416no—
143C23×C3—48no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—88no—
147C23×C3—84no—
148C23×C3—84no—
149C23×C3#9912yes—
150C23×C3C2×C22⋊C411yescharacteristic
151C3×Q8—44no—
152C3×Q8—48no—
153C3×Q8—44no—
154C3×D4—44no—
155C3×D4—48no—
156C3×D4—44no—
157C2×C3×C4—28no—
158C2×C3×C4—21no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—22no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—24no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—48no—
169C2×C3×C4—48no—
170C2×C3×C4—44no—
171C2×C3×C4#9912yes—
172C2×C3×C4C2×C22⋊C412yes—
173C2×C3×C4#12014yes—
174C3×C8—84no—
175C24—24no—
176C24—22no—
177C24—42no—
178C24—48no—
179C24—42no—
180C24C3×C22⋊C411yescharacteristic
181C2×Q8—24no—
182C2×Q8—22no—
183C2×Q8—24no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—416no—
190C2×Q8C2×C3×D414yes—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—416no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4C3×C22⋊C414yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4C3×C22⋊C412yes—
209C22×C4C3×C22⋊C411yescharacteristic
210C22×C4C22×C3×C411yescharacteristic
211C22×C4C2×C3×D412yes—
212M16—44no—
213M16—44no—
214C2×C8—44no—
215C22⋊C4—216no—
216C22⋊C4—216no—
217C42—24no—
218C42—22no—
219C42C3×C22⋊C414yes—
220C22×C3—24no—
221C22×C3—24no—
222C22×C3—22no—
223C22×C3—22no—
224C22×C3—22no—
225C22×C3—48no—
226C22×C3—44no—
227C22×C3—816no—
228C22×C3—84no—
229C22×C3—84no—
230C22×C3#35514yescomplemented
231C22×C3#37911yescharacteristic
232C22×C3C2×#9911yescharacteristic
233C22×C3C2×#9911yescharacteristic
234C3×C4—24no—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—44no—
238C3×C4—42no—
239C23—21no—
240C23—21no—
241C23—24no—
242C23—24no—
243C23—48no—
244C23—416no—
245C23—48no—
246C23—42no—
247C23—42no—
248C23—88no—
249C23—84no—
250C23—84no—
251C23C3×#9912yes—
252C23C2×C3×C22⋊C411yescharacteristic
253Q8—44no—
254Q8—48no—
255Q8—44no—
256D4—44no—
257D4—48no—
258D4—44no—
259C2×C4—28no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—44no—
273C2×C4C3×#9912yes—
274C2×C4C2×C3×C22⋊C412yes—
275C2×C4C3×#12014yes—
276C8—84no—
277C2×C3—21no—
278C2×C3—24no—
279C2×C3—21no—
280C2×C3—84no—
281C2×C3#178111yescharacteristic
282C2×C3C2×#35512yes—
283C22—24no—
284C22—24no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—48no—
289C22—44no—
290C22—84no—
291C22—816no—
292C22—84no—
293C22C3×#35514yescomplemented
294C22C3×#37911yescharacteristic
295C22C2×C3×#9911yescharacteristic
296C22C2×C3×#9911yescharacteristic
297C4—24no—
298C4—22no—
299C4—22no—
300C4—44no—
301C4—42no—
302C3#1269911yescharacteristic, direct product factor, Sylow
303C2—21no—
304C2—21no—
305C2—24no—
306C2—84no—
307C2C3×#178111yescharacteristic
308C2C2×C3×#35512yes—
subgroupG/H
center, socleC22×C3C2×#99
commutatorC22×C4C22×C3×C4
Frattini, cosocleC22×Q8C23×C3
prime radicalC23×C3×D4C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#355C2semidirect
3. C3×#1784C2semidirect
4. #12699C3direct (N × H)
14. C2×C3×#101C22semidirect
18. C2×#355C2×C3semidirect
19. #1784C2×C3semidirect
59. C2×#101C22×C3semidirect
230. C22×C3#355semidirect
293. C22C3×#355semidirect
302. C3#12699direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1242
C2×C3×#3552
C3×#17842
#126993