Finite Group Databasebeta

← C3×#29656 #10648263

Subgroups — C3×#29656 #10648263

Order 768.

1428 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 242 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×#543C211yesmaximal, characteristic, complemented
2C3×#2405C213yesmaximal
3C3×#2062C214yesmaximal, complemented
4C3×#2029C214yesmaximal, complemented
5C3×#1819C213yesmaximal, complemented
6#29656C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4C2211yescharacteristic
8C3×#543—26no—
9C3×#543C2214yes—
10C2×C3×#123—23no—
11C2×C3×C22⋊Q8—23no—
12C2×C3×C22⋊Q8—23no—
13C22×C3×C22⋊C4C2213yes—
14C3×#394—26no—
15C3×#386—23no—
16C3×#385C2216yes—
17C3×#379C2216yes—
18C3×#379C2213yes—
19C3×#342C22112yes—
20C2×#543C2×C311yescharacteristic, complemented
21#2405C2×C313yes—
22#2062C2×C314yescomplemented
23#2029C2×C314yescomplemented
24#1819C2×C313yescomplemented
25C25×C3C2311yescharacteristic
26C25×C3C2311yescharacteristic
27C22×C3×Q8—21no—
28C22×C3×Q8—21no—
29C22×C3×D4—23no—
30C22×C3×D4—23no—
31C22×C3×D4—44no—
32C23×C3×C4D413yes—
33C23×C3×C4D413yes—
34C23×C3×C4C2311yescharacteristic
35C3×#123—412no—
36C3×C22⋊Q8—212no—
37C3×C22⋊Q8—212no—
38C3×C22⋊Q8—212no—
39C3×C22⋊Q8—212no—
40C2×C3×C4⋊C4—23no—
41C2×C3×C4⋊C4—23no—
42C2×C3×C4⋊C4—26no—
43C2×C3×C22⋊C4—26no—
44C2×C3×C22⋊C4—26no—
45C2×C3×C22⋊C4—23no—
46C2×C3×C22⋊C4—26no—
47C2×C3×C22⋊C4—23no—
48C2×C3×C22⋊C4—26no—
49C2×C3×C22⋊C4D416yes—
50C2×C3×C22⋊C4C2316yes—
51C2×C3×C22⋊C4C2316yes—
52C3×#95—26no—
53C23×D4C22×C311yescharacteristic
54#543—26no—
55#543C22×C314yes—
56C2×#123—23no—
57C2×C22⋊Q8—23no—
58C2×C22⋊Q8—23no—
59C22×C22⋊C4C22×C313yes—
60#394—26no—
61#386—23no—
62#385C22×C316yes—
63#379C22×C313yes—
64#379C22×C316yes—
65#342C22×C3112yes—
66C24×C3—24no—
67C24×C3—23no—
68C24×C3—24no—
69C24×C3—42no—
70C24×C3—42no—
71C24×C3—81no—
72C24×C3—81no—
73C24×C3C2×D413yes—
74C24×C3C2×D413yes—
75C24×C3C2411yescharacteristic
76C2×C3×Q8—24no—
77C2×C3×Q8—24no—
78C2×C3×Q8—82no—
79C2×C3×Q8—82no—
80C2×C3×D4—212no—
81C2×C3×D4—46no—
82C2×C3×D4—46no—
83C2×C3×D4—46no—
84C2×C3×D4—43no—
85C2×C3×D4—41no—
86C22×C3×C4—23no—
87C22×C3×C4—23no—
88C22×C3×C4—23no—
89C22×C3×C4—23no—
90C22×C3×C4—23no—
91C22×C3×C4—26no—
92C22×C3×C4—42no—
93C22×C3×C4—46no—
94C22×C3×C4—83no—
95C22×C3×C4C2×D416yes—
96C22×C3×C4C2×D416yes—
97C3×C4⋊C4—212no—
98C3×C4⋊C4—212no—
99C3×C4⋊C4—412no—
100C3×C22⋊C4—212no—
101C3×C22⋊C4—212no—
102C3×C22⋊C4—212no—
103C3×C22⋊C4—412no—
104C3×C22⋊C4—46no—
105C3×C22⋊C4—86no—
106C3×C22⋊C4—83no—
107C25C23×C311yescharacteristic
108C25C23×C311yescharacteristic
109C22×Q8—21no—
110C22×Q8—21no—
111C22×D4—23no—
112C22×D4—23no—
113C22×D4—44no—
114C23×C4C3×D413yes—
115C23×C4C3×D413yes—
116C23×C4C23×C311yescharacteristic
117#123—412no—
118C22⋊Q8—212no—
119C22⋊Q8—212no—
120C22⋊Q8—212no—
121C22⋊Q8—212no—
122C2×C4⋊C4—23no—
123C2×C4⋊C4—23no—
124C2×C4⋊C4—26no—
125C2×C22⋊C4—26no—
126C2×C22⋊C4—26no—
127C2×C22⋊C4—23no—
128C2×C22⋊C4—26no—
129C2×C22⋊C4—23no—
130C2×C22⋊C4—26no—
131C2×C22⋊C4C3×D416yes—
132C2×C22⋊C4C23×C316yes—
133C2×C22⋊C4C23×C316yes—
134#95—26no—
135C23×C3—26no—
136C23×C3—26no—
137C23×C3—26no—
138C23×C3—26no—
139C23×C3—41no—
140C23×C3—43no—
141C23×C3—41no—
142C23×C3—43no—
143C23×C3—43no—
144C23×C3—46no—
145C23×C3—43no—
146C23×C3—41no—
147C23×C3—46no—
148C23×C3—88no—
149C23×C3—88no—
150C23×C3#12013yes—
151C23×C3#12011yescharacteristic
152C23×C3C22×D413yes—
153C23×C32−1+414yes—
154C3×Q8—84no—
155C3×Q8—84no—
156C3×D4—412no—
157C3×D4—44no—
158C2×C3×C4—26no—
159C2×C3×C4—26no—
160C2×C3×C4—26no—
161C2×C3×C4—43no—
162C2×C3×C4—46no—
163C2×C3×C4—41no—
164C2×C3×C4—46no—
165C2×C3×C4—46no—
166C2×C3×C4—46no—
167C2×C3×C4—46no—
168C2×C3×C4—46no—
169C2×C3×C4—46no—
170C2×C3×C4—83no—
171C2×C3×C4—83no—
172C24—24no—
173C24—23no—
174C24—24no—
175C24—42no—
176C24—42no—
177C24—81no—
178C24—81no—
179C24C2×C3×D413yes—
180C24C2×C3×D413yes—
181C24C24×C311yescharacteristic
182C2×Q8—24no—
183C2×Q8—24no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×D4—212no—
187C2×D4—46no—
188C2×D4—46no—
189C2×D4—46no—
190C2×D4—43no—
191C2×D4—41no—
192C22×C4—23no—
193C22×C4—23no—
194C22×C4—23no—
195C22×C4—23no—
196C22×C4—23no—
197C22×C4—26no—
198C22×C4—42no—
199C22×C4—46no—
200C22×C4—83no—
201C22×C4C2×C3×D416yes—
202C22×C4C2×C3×D416yes—
203C4⋊C4—212no—
204C4⋊C4—212no—
205C4⋊C4—412no—
206C22⋊C4—212no—
207C22⋊C4—212no—
208C22⋊C4—212no—
209C22⋊C4—412no—
210C22⋊C4—46no—
211C22⋊C4—86no—
212C22⋊C4—83no—
213C22×C3—21no—
214C22×C3—23no—
215C22×C3—21no—
216C22×C3—23no—
217C22×C3—23no—
218C22×C3—23no—
219C22×C3—23no—
220C22×C3—44no—
221C22×C3—412no—
222C22×C3—412no—
223C22×C3—42no—
224C22×C3—86no—
225C22×C3—86no—
226C22×C3—81no—
227C22×C3—81no—
228C22×C3B(4,𝔽2)12yes—
229C22×C3C2×#12011yescharacteristic
230C22×C3#53616yes—
231C3×C4—42no—
232C3×C4—46no—
233C3×C4—46no—
234C3×C4—86no—
235C23—26no—
236C23—26no—
237C23—26no—
238C23—26no—
239C23—41no—
240C23—43no—
241C23—41no—
242C23—43no—
243C23—43no—
244C23—46no—
245C23—43no—
246C23—41no—
247C23—46no—
248C23—88no—
249C23—88no—
250C23C3×#12013yes—
251C23C3×#12011yescharacteristic
252C23C22×C3×D413yes—
253C23C3×2−1+414yes—
254Q8—84no—
255Q8—84no—
256D4—412no—
257D4—44no—
258C2×C4—26no—
259C2×C4—26no—
260C2×C4—26no—
261C2×C4—43no—
262C2×C4—46no—
263C2×C4—46no—
264C2×C4—41no—
265C2×C4—46no—
266C2×C4—46no—
267C2×C4—46no—
268C2×C4—46no—
269C2×C4—46no—
270C2×C4—83no—
271C2×C4—83no—
272C2×C3—26no—
273C2×C3—22no—
274C2×C3—43no—
275C2×C3—82no—
276C2×C3#261411yescharacteristic
277C2×C3C2×B(4,𝔽2)11yescharacteristic
278C2×C3#302611yescharacteristic
279C22—23no—
280C22—21no—
281C22—23no—
282C22—21no—
283C22—23no—
284C22—23no—
285C22—23no—
286C22—44no—
287C22—412no—
288C22—412no—
289C22—42no—
290C22—86no—
291C22—86no—
292C22—81no—
293C22—81no—
294C22C3×B(4,𝔽2)12yes—
295C22C2×C3×#12011yescharacteristic
296C22C3×#53616yes—
297C4—42no—
298C4—46no—
299C4—46no—
300C4—86no—
301C3#2965611yescharacteristic, direct product factor, Sylow
302C2—26no—
303C2—22no—
304C2—43no—
305C2—82no—
306C2C3×#261411yescharacteristic
307C2C2×C3×B(4,𝔽2)11yescharacteristic
308C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#543C2semidirect
3. C3×#2062C2semidirect
4. C3×#2029C2semidirect
5. C3×#1819C2semidirect
6. #29656C3direct (N × H)
20. C2×#543C2×C3semidirect
22. #2062C2×C3semidirect
23. #2029C2×C3semidirect
24. #1819C2×C3semidirect
301. C3#29656direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5432
C3×#24052
C3×#20622
C3×#20292
C3×#18192
#296563