Finite Group Databasebeta

← C3×#32901 #10651508

Subgroups — C3×#32901 #10651508

Order 768.

3706 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 3706 conjugacy classes of subgroups — too many to draw. Only its 366 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×#585C211yesmaximal, characteristic
2C2×C3×#458C214yesmaximal, complemented
3C3×#2406C213yesmaximal, complemented
4C3×#1790C214yesmaximal, complemented
5C3×#1782C213yesmaximal, complemented
6#32901C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#585—216no—
8C2×C3×2−1+4—23no—
9C2×C3×H(5,𝔽2)—23no—
10C2×C3×H(5,𝔽2)C2214yes—
11C22×C3×#41—26no—
12C22×C3×#41C2213yes—
13C2×C3×#123C2216yescomplemented
14C22×C3×C4⋊C4C2211yescharacteristic, complemented
15C3×#458—216no—
16C2×C3×#99C22112yescomplemented
17C3×#395—23no—
18C3×#394—23no—
19C3×#386C2216yescomplemented
20C3×#385—26no—
21C3×#381C2213yescomplemented
22C2×#585C2×C311yescharacteristic
23C2×#458C2×C314yescomplemented
24#2406C2×C313yescomplemented
25#1790C2×C314yescomplemented
26#1782C2×C313yescomplemented
27C3×2−1+4—248no—
28C3×H(5,𝔽2)—248no—
29C3×H(5,𝔽2)—224no—
30C3×H(5,𝔽2)D4116yes—
31C2×C3×#41—24no—
32C2×C3×#41—212no—
33C2×C3×#41—248no—
34C2×C3×#41—248no—
35C2×C3×#41—224no—
36C2×C3×#41—26no—
37C2×C3×#41—212no—
38C2×C3×#41—26no—
39C22×C3×Q8—26no—
40C22×C3×Q8D413yes—
41C22×C3×D4—26no—
42C22×C3×D4—212no—
43C22×C3×D4D413yes—
44C22×C3×D4C2316yes—
45C23×C3×C4—24no—
46C23×C3×C4D416yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C3×#123—224no—
51C2×C3×C4⋊C4—23no—
52C2×C3×C4⋊C4—26no—
53C2×C3×C4⋊C4—23no—
54C2×C3×C22⋊C4—26no—
55C2×C3×C22⋊C4—26no—
56C2×C3×C22⋊C4C2316yes—
57C3×#99—224no—
58C3×#95—23no—
59C3×#95—23no—
60C3×#95—26no—
61#585—216no—
62C2×2−1+4—23no—
63C2×H(5,𝔽2)—23no—
64C2×H(5,𝔽2)C22×C314yes—
65C22×#41—26no—
66C22×#41C22×C313yes—
67C2×#123C22×C316yescomplemented
68C22×C4⋊C4C22×C311yescharacteristic, complemented
69#458—216no—
70C2×#99C22×C3112yescomplemented
71#395—23no—
72#394—23no—
73#386C22×C316yescomplemented
74#385—26no—
75#381C22×C313yescomplemented
76C24×C3—24no—
77C24×C3C2×D416yes—
78C24×C3C2411yescharacteristic
79C3×#41—232no—
80C3×#41—248no—
81C3×#41—296no—
82C3×#41—248no—
83C3×#41—248no—
84C3×#41—248no—
85C2×C3×Q8—24no—
86C2×C3×Q8—248no—
87C2×C3×Q8—26no—
88C2×C3×Q8—212no—
89C2×C3×D4—212no—
90C2×C3×D4—248no—
91C2×C3×D4—248no—
92C2×C3×D4—248no—
93C2×C3×D4—212no—
94C2×C3×D4—212no—
95C2×C3×D4—212no—
96C2×C3×D4—26no—
97C2×C3×D4C2×D4124yes—
98C22×C3×C4—212no—
99C22×C3×C4—232no—
100C22×C3×C4—212no—
101C22×C3×C4—24no—
102C22×C3×C4—26no—
103C22×C3×C4—26no—
104C22×C3×C4—212no—
105C22×C3×C4—23no—
106C22×C3×C4—26no—
107C22×C3×C4—23no—
108C22×C3×C4—24no—
109C22×C3×C4—42no—
110C22×C3×C4C2×D413yes—
111C22×C3×C4C2×D416yes—
112C22×C3×C4C2×D413yes—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—43no—
115C3×C4⋊C4—412no—
116C3×C22⋊C4—212no—
117C3×C22⋊C4—412no—
118C3×C22⋊C4—412no—
1192−1+4—248no—
120H(5,𝔽2)—248no—
121H(5,𝔽2)—224no—
122H(5,𝔽2)C3×D4116yes—
123C2×#41—24no—
124C2×#41—212no—
125C2×#41—248no—
126C2×#41—248no—
127C2×#41—224no—
128C2×#41—26no—
129C2×#41—212no—
130C2×#41—26no—
131C22×Q8—26no—
132C22×Q8C3×D413yes—
133C22×D4—26no—
134C22×D4—212no—
135C22×D4C3×D413yes—
136C22×D4C23×C316yes—
137C23×C4—24no—
138C23×C4C3×D416yes—
139C23×C4C23×C311yescharacteristic
140C23×C4C23×C311yescharacteristic
141C23×C4C23×C311yescharacteristic
142#123—224no—
143C2×C4⋊C4—23no—
144C2×C4⋊C4—23no—
145C2×C4⋊C4—26no—
146C2×C22⋊C4—26no—
147C2×C22⋊C4—26no—
148C2×C22⋊C4C23×C316yes—
149#99—224no—
150#95—23no—
151#95—23no—
152#95—26no—
153C23×C3—216no—
154C23×C3—26no—
155C23×C3—26no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—46no—
159C23×C3—46no—
160C23×C3—84no—
161C23×C3—84no—
162C23×C3#120112yes—
163C23×C3C22×D413yes—
164C23×C3C22×D414yes—
165C3×Q8—216no—
166C3×Q8—224no—
167C3×D4—248no—
168C3×D4—224no—
169C3×D4—248no—
170C2×C3×C4—224no—
171C2×C3×C4—248no—
172C2×C3×C4—224no—
173C2×C3×C4—26no—
174C2×C3×C4—26no—
175C2×C3×C4—26no—
176C2×C3×C4—26no—
177C2×C3×C4—26no—
178C2×C3×C4—43no—
179C2×C3×C4—41no—
180C2×C3×C4—46no—
181C2×C3×C4—46no—
182C2×C3×C4—46no—
183C2×C3×C4—46no—
184C2×C3×C4—41no—
185C2×C3×C4#120112yes—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12013yes—
188C24—24no—
189C24C2×C3×D416yes—
190C24C24×C311yescharacteristic
191#41—232no—
192#41—248no—
193#41—296no—
194#41—248no—
195#41—248no—
196#41—248no—
197C2×Q8—24no—
198C2×Q8—248no—
199C2×Q8—26no—
200C2×Q8—212no—
201C2×D4—212no—
202C2×D4—248no—
203C2×D4—248no—
204C2×D4—248no—
205C2×D4—212no—
206C2×D4—212no—
207C2×D4—212no—
208C2×D4—26no—
209C2×D4C2×C3×D4124yes—
210C22×C4—212no—
211C22×C4—232no—
212C22×C4—212no—
213C22×C4—24no—
214C22×C4—26no—
215C22×C4—26no—
216C22×C4—212no—
217C22×C4—23no—
218C22×C4—26no—
219C22×C4—23no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4C2×C3×D413yes—
223C22×C4C2×C3×D416yes—
224C22×C4C2×C3×D413yes—
225C4⋊C4—41no—
226C4⋊C4—43no—
227C4⋊C4—412no—
228C22⋊C4—212no—
229C22⋊C4—412no—
230C22⋊C4—412no—
231C22×C3—23no—
232C22×C3—23no—
233C22×C3—212no—
234C22×C3—44no—
235C22×C3—412no—
236C22×C3—46no—
237C22×C3—46no—
238C22×C3—86no—
239C22×C3—816no—
240C22×C3C2×#12016yes—
241C22×C3C2×#12011yescharacteristic
242C3×C4—21no—
243C3×C4—212no—
244C3×C4—23no—
245C3×C4—42no—
246C3×C4—86no—
247C3×C4—81no—
248C23—216no—
249C23—26no—
250C23—26no—
251C23—41no—
252C23—41no—
253C23—46no—
254C23—46no—
255C23—84no—
256C23—84no—
257C23C3×#120112yes—
258C23C22×C3×D413yes—
259C23C22×C3×D414yes—
260Q8—216no—
261Q8—224no—
262D4—248no—
263D4—224no—
264D4—248no—
265C2×C4—224no—
266C2×C4—248no—
267C2×C4—224no—
268C2×C4—26no—
269C2×C4—26no—
270C2×C4—26no—
271C2×C4—26no—
272C2×C4—26no—
273C2×C4—43no—
274C2×C4—41no—
275C2×C4—46no—
276C2×C4—46no—
277C2×C4—46no—
278C2×C4—46no—
279C2×C4—41no—
280C2×C4C3×#12011yescharacteristic
281C2×C4C3×#12013yes—
282C2×C4C3×#120112yes—
283C2×C3—26no—
284C2×C3—412no—
285C2×C3#284611yescharacteristic
286C2×C3#302511yescharacteristic
287C2×C3#302811yescharacteristic
288C22—23no—
289C22—23no—
290C22—212no—
291C22—44no—
292C22—412no—
293C22—46no—
294C22—46no—
295C22—86no—
296C22—816no—
297C22C2×C3×#12016yes—
298C22C2×C3×#12011yescharacteristic
299C4—21no—
300C4—212no—
301C4—23no—
302C4—42no—
303C4—86no—
304C4—81no—
305C3#3290111yescharacteristic, direct product factor, Sylow
306C2—26no—
307C2—412no—
308C2C3×#284611yescharacteristic
309C2C3×#302511yescharacteristic
310C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC2×C3×#585C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#458C2semidirect
3. C3×#2406C2semidirect
4. C3×#1790C2semidirect
5. C3×#1782C2semidirect
6. #32901C3direct (N × H)
13. C2×C3×#123C22semidirect
14. C22×C3×C4⋊C4C22semidirect
16. C2×C3×#99C22semidirect
19. C3×#386C22semidirect
21. C3×#381C22semidirect
23. C2×#458C2×C3semidirect
24. #2406C2×C3semidirect
25. #1790C2×C3semidirect
26. #1782C2×C3semidirect
67. C2×#123C22×C3semidirect
68. C22×C4⋊C4C22×C3semidirect
70. C2×#99C22×C3semidirect
73. #386C22×C3semidirect
75. #381C22×C3semidirect
305. C3#32901direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5852
C2×C3×#4582
C3×#24062
C3×#17902
C3×#17822
#329013