Finite Group Databasebeta

← C3×#12703 #10631310

Subgroups — C3×#12703 #10631310

Order 768.

1154 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C3×#535C211yesmaximal, characteristic
2C2×C3×#355C214yesmaximal, complemented
3C3×#1785C212yesmaximal, complemented
4#12703C311yesmaximal, characteristic, direct product factor, Sylow
5C22×C3×#41C2211yescharacteristic
6C23×C3×D4C411yescharacteristic
7C3×#535—28no—
8C2×C3×#127C412yes—
9C2×C3×#124C2212yes—
10C2×C3×#121—22no—
11C2×C3×#120—22no—
12C2×C3×#117C411yescharacteristic
13C2×C3×#101C2214yescomplemented
14C2×C3×#100—22no—
15C3×#407—22no—
16C3×#355—28no—
17C2×#535C2×C311yescharacteristic
18C2×#355C2×C314yescomplemented
19#1785C2×C312yescomplemented
20C25×C3—21no—
21C2×C3×#41—24no—
22C2×C3×#41—42no—
23C2×C3×#41—42no—
24C22×C3×Q8C2311yescharacteristic
25C22×C3×D4—24no—
26C22×C3×D4—24no—
27C22×C3×D4—21no—
28C22×C3×D4—42no—
29C22×C3×D4—42no—
30C22×C3×D4C2×C411yescharacteristic
31C22×C3×D4C2×C412yes—
32C22×C3×D4D412yes—
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4D412yes—
35C2×C3×M16—24no—
36C3×#127—24no—
37C3×#127—24no—
38C3×#124—24no—
39C3×#124D418yes—
40C3×#121—216no—
41C3×#120—216no—
42C3×#117—24no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—24no—
46C2×C3×C42C2×C412yes—
47C3×#101—28no—
48C3×#100—44no—
49C3×C22⋊C8—44no—
50C22×#41C22×C311yescharacteristic
51C23×D4C3×C411yescharacteristic
52#535—28no—
53C2×#127C3×C412yes—
54C2×#124C22×C312yes—
55C2×#121—22no—
56C2×#120—22no—
57C2×#117C3×C411yescharacteristic
58C2×#101C22×C314yescomplemented
59C2×#100—22no—
60#407—22no—
61#355—28no—
62C24×C3—24no—
63C24×C3—22no—
64C24×C3—42no—
65C24×C3—42no—
66C24×C3—84no—
67C24×C3C22⋊C412yes—
68C24×C3C22⋊C411yescharacteristic
69C3×#41—416no—
70C2×C3×Q8—41no—
71C2×C3×Q8—41no—
72C2×C3×Q8C2×D414yes—
73C2×C3×D4—24no—
74C2×C3×D4—24no—
75C2×C3×D4—216no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—24no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—42no—
84C2×C3×D4—416no—
85C2×C3×D4—42no—
86C2×C3×D4—42no—
87C2×C3×D4—88no—
88C2×C3×D4C22⋊C414yes—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—24no—
94C22×C3×C4—24no—
95C22×C3×C4—42no—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D412yes—
99C3×M16—44no—
100C3×M16—44no—
101C2×C3×C8—44no—
102C3×C4⋊C4—24no—
103C3×C22⋊C4—24no—
104C3×C22⋊C4—216no—
105C3×C42—22no—
106C3×C42C22⋊C414yes—
107C25—21no—
108C2×#41—24no—
109C2×#41—42no—
110C2×#41—42no—
111C22×Q8C23×C311yescharacteristic
112C22×D4—24no—
113C22×D4—24no—
114C22×D4—21no—
115C22×D4—42no—
116C22×D4—42no—
117C22×D4C2×C3×C411yescharacteristic
118C22×D4C2×C3×C412yes—
119C22×D4C3×D412yes—
120C23×C4C2×C3×C411yescharacteristic
121C23×C4C3×D412yes—
122C2×M16—24no—
123#127—24no—
124#127—24no—
125#124—24no—
126#124C3×D418yes—
127#121—216no—
128#120—216no—
129#117—24no—
130C2×C4⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—24no—
133C2×C42C2×C3×C412yes—
134#101—28no—
135#100—44no—
136C22⋊C8—44no—
137C23×C3—21no—
138C23×C3—21no—
139C23×C3—24no—
140C23×C3—22no—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—42no—
144C23×C3—44no—
145C23×C3—48no—
146C23×C3—48no—
147C23×C3—42no—
148C23×C3—42no—
149C23×C3—88no—
150C23×C3—88no—
151C23×C3—88no—
152C23×C3#9912yes—
153C23×C3C2×C22⋊C411yescharacteristic
154C3×Q8—44no—
155C3×D4—48no—
156C3×D4—44no—
157C3×D4—48no—
158C3×D4—44no—
159C3×D4—44no—
160C3×D4—816no—
161C2×C3×C4—28no—
162C2×C3×C4—21no—
163C2×C3×C4—22no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—24no—
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—42no—
179C24—84no—
180C24C3×C22⋊C412yes—
181C24C3×C22⋊C411yescharacteristic
182#41—416no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8C2×C3×D414yes—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—216no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—24no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—416no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—88no—
201C2×D4C3×C22⋊C414yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—42no—
209C22×C4C3×C22⋊C411yescharacteristic
210C22×C4C22×C3×C411yescharacteristic
211C22×C4C2×C3×D412yes—
212M16—44no—
213M16—44no—
214C2×C8—44no—
215C4⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—216no—
218C42—22no—
219C42C3×C22⋊C414yes—
220C22×C3—22no—
221C22×C3—22no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3—44no—
225C22×C3—48no—
226C22×C3—42no—
227C22×C3—44no—
228C22×C3—42no—
229C22×C3—44no—
230C22×C3—82no—
231C22×C3—84no—
232C22×C3—816no—
233C22×C3—84no—
234C22×C3#37911yescharacteristic
235C22×C3C2×#9911yescharacteristic
236C22×C3C2×#9911yescharacteristic
237C3×C4—24no—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—44no—
241C23—21no—
242C23—21no—
243C23—24no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—42no—
248C23—44no—
249C23—48no—
250C23—48no—
251C23—42no—
252C23—42no—
253C23—88no—
254C23—88no—
255C23—88no—
256C23C3×#9912yes—
257C23C2×C3×C22⋊C411yescharacteristic
258Q8—44no—
259D4—48no—
260D4—44no—
261D4—48no—
262D4—44no—
263D4—44no—
264D4—816no—
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4C3×#9912yes—
276C2×C4C2×C3×C22⋊C412yes—
277C2×C4C3×#12014yes—
278C8—84no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3—82no—
283C2×C3—84no—
284C2×C3#178111yescharacteristic
285C2×C3#213412yes—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—44no—
291C22—48no—
292C22—42no—
293C22—42no—
294C22—44no—
295C22—44no—
296C22—82no—
297C22—84no—
298C22—816no—
299C22—84no—
300C22C3×#37911yescharacteristic
301C22C2×C3×#9911yescharacteristic
302C22C2×C3×#9911yescharacteristic
303C4—24no—
304C4—22no—
305C4—22no—
306C4—44no—
307C3#1270311yescharacteristic, direct product factor, Sylow
308C2—21no—
309C2—21no—
310C2—42no—
311C2—82no—
312C2—84no—
313C2C3×#178111yescharacteristic
314C2C3×#213412yes—
subgroupG/H
center, socleC22×C3C2×#99
commutatorC22×C4C22×C3×C4
Frattini, cosocleC22×Q8C23×C3
prime radicalC2×C3×#535C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#355C2semidirect
3. C3×#1785C2semidirect
4. #12703C3direct (N × H)
13. C2×C3×#101C22semidirect
18. C2×#355C2×C3semidirect
19. #1785C2×C3semidirect
58. C2×#101C22×C3semidirect
307. C3#12703direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×C3×#3552
C3×#17852
#127033