Finite Group Databasebeta

← C3×#12698 #10631305

Subgroups — C3×#12698 #10631305

Order 768.

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

normal subgroups only

This group has 990 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×#536C211yesmaximal, characteristic
2C2×C3×#354C214yesmaximal, complemented
3C3×#1782C212yesmaximal
4#12698C311yesmaximal, characteristic, direct product factor, Sylow
5C22×C3×#41C2211yescharacteristic
6C23×C3×Q8C411yescharacteristic, complemented
7C3×#536—28no—
8C2×C3×C4⋊Q8C412yescomplemented
9C2×C3×#124C2212yes—
10C2×C3×#123—22no—
11C2×C3×C22⋊Q8—22no—
12C2×C3×#117C411yescharacteristic, complemented
13C2×C3×#99—22no—
14C2×C3×#99C2214yes—
15C3×#381—22no—
16C3×#354—28no—
17C2×#536C2×C311yescharacteristic
18C2×#354C2×C314yescomplemented
19#1782C2×C312yes—
20C2×C3×#41—24no—
21C2×C3×#41—42no—
22C2×C3×#41—42no—
23C22×C3×Q8—24no—
24C22×C3×Q8—21no—
25C22×C3×Q8—42no—
26C22×C3×Q8—42no—
27C22×C3×Q8C2×C412yes—
28C22×C3×Q8C2×C411yescharacteristic
29C22×C3×D4D412yes—
30C22×C3×D4C2311yescharacteristic
31C23×C3×C4—21no—
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4D412yes—
34C3×C4⋊Q8—24no—
35C3×C4⋊Q8—24no—
36C3×#124—24no—
37C3×#124D418yes—
38C3×#123—216no—
39C3×C22⋊Q8—216no—
40C3×#117—24no—
41C2×C3×C4⋊C4—24no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—24no—
45C2×C3×C22⋊C4—24no—
46C2×C3×C42C2×C412yes—
47C3×#99—28no—
48C3×#99—44no—
49C3×#95—44no—
50C22×#41C22×C311yescharacteristic
51C23×Q8C3×C411yescharacteristic, complemented
52#536—28no—
53C2×C4⋊Q8C3×C412yescomplemented
54C2×#124C22×C312yes—
55C2×#123—22no—
56C2×C22⋊Q8—22no—
57C2×#117C3×C411yescharacteristic, complemented
58C2×#99—22no—
59C2×#99C22×C314yes—
60#381—22no—
61#354—28no—
62C24×C3C22⋊C411yescharacteristic
63C24×C3C2×D412yes—
64C3×#41—416no—
65C2×C3×Q8—24no—
66C2×C3×Q8—22no—
67C2×C3×Q8—22no—
68C2×C3×Q8—41no—
69C2×C3×Q8—41no—
70C2×C3×Q8—416no—
71C2×C3×Q8—42no—
72C2×C3×Q8—42no—
73C2×C3×Q8—42no—
74C2×C3×Q8—42no—
75C2×C3×Q8C22⋊C414yes—
76C2×C3×D4—24no—
77C2×C3×D4—24no—
78C2×C3×D4—42no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4C2×D414yes—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—24no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—44no—
94C22×C3×C4C22⋊C412yes—
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C3×C4⋊C4—24no—
98C3×C4⋊C4—216no—
99C3×C22⋊C4—24no—
100C3×C22⋊C4—216no—
101C3×C22⋊C4—44no—
102C3×C22⋊C4—44no—
103C3×C42—22no—
104C3×C42C22⋊C414yes—
105C2×#41—24no—
106C2×#41—42no—
107C2×#41—42no—
108C22×Q8—24no—
109C22×Q8—21no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8C2×C3×C412yes—
113C22×Q8C2×C3×C411yescharacteristic
114C22×D4C3×D412yes—
115C22×D4C23×C311yescharacteristic
116C23×C4—21no—
117C23×C4C2×C3×C411yescharacteristic
118C23×C4C3×D412yes—
119C4⋊Q8—24no—
120C4⋊Q8—24no—
121#124—24no—
122#124C3×D418yes—
123#123—216no—
124C22⋊Q8—216no—
125#117—24no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—24no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C42C2×C3×C412yes—
132#99—28no—
133#99—44no—
134#95—44no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—42no—
140C23×C3—44no—
141C23×C3#9912yes—
142C23×C3C2×C22⋊C411yescharacteristic
143C23×C3#12014yes—
144C3×Q8—44no—
145C3×Q8—48no—
146C3×Q8—44no—
147C3×Q8—44no—
148C3×D4—44no—
149C3×D4—48no—
150C2×C3×C4—28no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—48no—
161C2×C3×C4—44no—
162C2×C3×C4—48no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—84no—
166C2×C3×C4—88no—
167C2×C3×C4#9912yes—
168C2×C3×C4C2×C22⋊C412yes—
169C24C3×C22⋊C411yescharacteristic
170C24C2×C3×D412yes—
171#41—416no—
172C2×Q8—24no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—416no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8C3×C22⋊C414yes—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4C2×C3×D414yes—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4C3×C22⋊C412yes—
202C22×C4C3×C22⋊C411yescharacteristic
203C22×C4C22×C3×C411yescharacteristic
204C4⋊C4—24no—
205C4⋊C4—216no—
206C22⋊C4—24no—
207C22⋊C4—216no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C42—22no—
211C42C3×C22⋊C414yes—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—24no—
215C22×C3—42no—
216C22×C3—44no—
217C22×C3—42no—
218C22×C3—44no—
219C22×C3—42no—
220C22×C3—82no—
221C22×C3#37911yescharacteristic
222C22×C3C2×#9911yescharacteristic
223C22×C3C2×#9911yescharacteristic
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—44no—
227C3×C4—42no—
228C3×C4—44no—
229C3×C4—164no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—42no—
235C23—44no—
236C23C3×#9912yes—
237C23C2×C3×C22⋊C411yescharacteristic
238C23C3×#12014yes—
239Q8—44no—
240Q8—48no—
241Q8—44no—
242Q8—44no—
243D4—44no—
244D4—48no—
245C2×C4—28no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—84no—
261C2×C4—88no—
262C2×C4C3×#9912yes—
263C2×C4C2×C3×C22⋊C412yes—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—44no—
267C2×C3—42no—
268C2×C3#178111yescharacteristic
269C2×C3#212912yes—
270C22—22no—
271C22—22no—
272C22—24no—
273C22—42no—
274C22—44no—
275C22—42no—
276C22—44no—
277C22—42no—
278C22—82no—
279C22C3×#37911yescharacteristic
280C22C2×C3×#9911yescharacteristic
281C22C2×C3×#9911yescharacteristic
282C4—22no—
283C4—22no—
284C4—44no—
285C4—42no—
286C4—44no—
287C4—164no—
288C3#1269811yescharacteristic, direct product factor, Sylow
289C2—21no—
290C2—21no—
291C2—44no—
292C2—42no—
293C2C3×#178111yescharacteristic
294C2C3×#212912yes—
subgroupG/H
center, socleC22×C3C2×#99
commutatorC22×C4C22×C3×C4
Frattini, cosocleC22×D4C23×C3
prime radicalC22×C3×#41C22
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×#354C2semidirect
4. #12698C3direct (N × H)
6. C23×C3×Q8C4semidirect
8. C2×C3×C4⋊Q8C4semidirect
12. C2×C3×#117C4semidirect
18. C2×#354C2×C3semidirect
51. C23×Q8C3×C4semidirect
53. C2×C4⋊Q8C3×C4semidirect
57. C2×#117C3×C4semidirect
288. C3#12698direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#3542
C3×#17822
#126983