Finite Group Databasebeta

← C3×#32899 #10651506

Subgroups — C3×#32899 #10651506

Order 768.

3866 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 3866 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, complemented
2C2×C3×B(4,𝔽2)C214yesmaximal, complemented
3C3×#2404C213yesmaximal, complemented
4C3×#1790C214yesmaximal, complemented
5C3×#1782C213yesmaximal, complemented
6#32899C311yesmaximal, 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×#120C2216yescomplemented
14C22×C3×C22⋊C4C2211yescharacteristic, complemented
15C3×B(4,𝔽2)—216no—
16C2×C3×#99C22112yescomplemented
17C3×#393—23no—
18C3×#392—23no—
19C3×#386—26no—
20C3×#386C2216yescomplemented
21C3×#381C2213yescomplemented
22C2×#585C2×C311yescharacteristic, complemented
23C2×B(4,𝔽2)C2×C314yescomplemented
24#2404C2×C313yescomplemented
25#1790C2×C314yescomplemented
26#1782C2×C313yescomplemented
27C25×C3C2311yescharacteristic
28C3×2−1+4—248no—
29C3×H(5,𝔽2)—248no—
30C3×H(5,𝔽2)—224no—
31C3×H(5,𝔽2)D4116yescomplemented
32C2×C3×#41—24no—
33C2×C3×#41—212no—
34C2×C3×#41—248no—
35C2×C3×#41—248no—
36C2×C3×#41—224no—
37C2×C3×#41—26no—
38C2×C3×#41—212no—
39C2×C3×#41—26no—
40C22×C3×Q8—26no—
41C22×C3×Q8D413yescomplemented
42C22×C3×D4—26no—
43C22×C3×D4—212no—
44C22×C3×D4—26no—
45C22×C3×D4D413yescomplemented
46C22×C3×D4C2316yes—
47C23×C3×C4—24no—
48C23×C3×C4D416yescomplemented
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C3×#120—224no—
52C2×C3×C22⋊C4—23no—
53C2×C3×C22⋊C4—26no—
54C2×C3×C22⋊C4—23no—
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×#120C22×C316yescomplemented
68C22×C22⋊C4C22×C311yescharacteristic, complemented
69B(4,𝔽2)—216no—
70C2×#99C22×C3112yescomplemented
71#393—23no—
72#392—23no—
73#386—26no—
74#386C22×C316yescomplemented
75#381C22×C313yescomplemented
76C24×C3—24no—
77C24×C3—23no—
78C24×C3—24no—
79C24×C3—42no—
80C24×C3—81no—
81C24×C3C2×D416yes—
82C24×C3C2411yescharacteristic
83C3×#41—232no—
84C3×#41—248no—
85C3×#41—296no—
86C3×#41—248no—
87C3×#41—248no—
88C3×#41—248no—
89C2×C3×Q8—24no—
90C2×C3×Q8—248no—
91C2×C3×Q8—26no—
92C2×C3×Q8—212no—
93C2×C3×D4—212no—
94C2×C3×D4—248no—
95C2×C3×D4—248no—
96C2×C3×D4—248no—
97C2×C3×D4—212no—
98C2×C3×D4—212no—
99C2×C3×D4—212no—
100C2×C3×D4—26no—
101C2×C3×D4—412no—
102C2×C3×D4—412no—
103C2×C3×D4—86no—
104C2×C3×D4C2×D4124yes—
105C22×C3×C4—212no—
106C22×C3×C4—232no—
107C22×C3×C4—212no—
108C22×C3×C4—24no—
109C22×C3×C4—26no—
110C22×C3×C4—26no—
111C22×C3×C4—212no—
112C22×C3×C4—26no—
113C22×C3×C4—23no—
114C22×C3×C4—81no—
115C22×C3×C4C2×D413yes—
116C22×C3×C4C2×D416yes—
117C22×C3×C4C2×D413yes—
118C3×C22⋊C4—212no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—43no—
121C3×C22⋊C4—412no—
122C3×C22⋊C4—86no—
123C25C23×C311yescharacteristic
1242−1+4—248no—
125H(5,𝔽2)—248no—
126H(5,𝔽2)—224no—
127H(5,𝔽2)C3×D4116yescomplemented
128C2×#41—24no—
129C2×#41—212no—
130C2×#41—248no—
131C2×#41—248no—
132C2×#41—224no—
133C2×#41—26no—
134C2×#41—212no—
135C2×#41—26no—
136C22×Q8—26no—
137C22×Q8C3×D413yescomplemented
138C22×D4—26no—
139C22×D4—212no—
140C22×D4—26no—
141C22×D4C3×D413yescomplemented
142C22×D4C23×C316yes—
143C23×C4—24no—
144C23×C4C3×D416yescomplemented
145C23×C4C23×C311yescharacteristic
146C23×C4C23×C311yescharacteristic
147#120—224no—
148C2×C22⋊C4—23no—
149C2×C22⋊C4—23no—
150C2×C22⋊C4—26no—
151C2×C22⋊C4—26no—
152C2×C22⋊C4C23×C316yes—
153#99—224no—
154#95—23no—
155#95—23no—
156#95—26no—
157C23×C3—216no—
158C23×C3—26no—
159C23×C3—26no—
160C23×C3—26no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—46no—
164C23×C3—46no—
165C23×C3—46no—
166C23×C3—43no—
167C23×C3—46no—
168C23×C3—41no—
169C23×C3—88no—
170C23×C3—84no—
171C23×C3—84no—
172C23×C3#120112yes—
173C23×C3C22×D413yes—
174C23×C3C22×D414yes—
175C3×Q8—216no—
176C3×Q8—224no—
177C3×D4—248no—
178C3×D4—224no—
179C3×D4—248no—
180C3×D4—824no—
181C2×C3×C4—224no—
182C2×C3×C4—248no—
183C2×C3×C4—224no—
184C2×C3×C4—26no—
185C2×C3×C4—26no—
186C2×C3×C4—26no—
187C2×C3×C4—26no—
188C2×C3×C4—46no—
189C2×C3×C4—46no—
190C2×C3×C4—46no—
191C2×C3×C4—46no—
192C2×C3×C4—41no—
193C2×C3×C4#120112yes—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12013yes—
196C24—24no—
197C24—23no—
198C24—24no—
199C24—42no—
200C24—81no—
201C24C2×C3×D416yes—
202C24C24×C311yescharacteristic
203#41—232no—
204#41—248no—
205#41—296no—
206#41—248no—
207#41—248no—
208#41—248no—
209C2×Q8—24no—
210C2×Q8—248no—
211C2×Q8—26no—
212C2×Q8—212no—
213C2×D4—212no—
214C2×D4—248no—
215C2×D4—248no—
216C2×D4—248no—
217C2×D4—212no—
218C2×D4—212no—
219C2×D4—212no—
220C2×D4—26no—
221C2×D4—412no—
222C2×D4—412no—
223C2×D4—86no—
224C2×D4C2×C3×D4124yes—
225C22×C4—212no—
226C22×C4—232no—
227C22×C4—212no—
228C22×C4—24no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—212no—
232C22×C4—26no—
233C22×C4—23no—
234C22×C4—81no—
235C22×C4C2×C3×D413yes—
236C22×C4C2×C3×D416yes—
237C22×C4C2×C3×D413yes—
238C22⋊C4—212no—
239C22⋊C4—41no—
240C22⋊C4—43no—
241C22⋊C4—412no—
242C22⋊C4—86no—
243C22×C3—23no—
244C22×C3—23no—
245C22×C3—212no—
246C22×C3—44no—
247C22×C3—412no—
248C22×C3—412no—
249C22×C3—46no—
250C22×C3—46no—
251C22×C3—42no—
252C22×C3—86no—
253C22×C3—86no—
254C22×C3—816no—
255C22×C3—81no—
256C22×C3—81no—
257C22×C3C2×#12016yes—
258C22×C3C2×#12011yescharacteristic
259C3×C4—21no—
260C3×C4—212no—
261C3×C4—23no—
262C3×C4—86no—
263C3×C4—81no—
264C23—216no—
265C23—26no—
266C23—26no—
267C23—26no—
268C23—41no—
269C23—41no—
270C23—46no—
271C23—46no—
272C23—46no—
273C23—43no—
274C23—41no—
275C23—46no—
276C23—88no—
277C23—84no—
278C23—84no—
279C23C3×#120112yes—
280C23C22×C3×D413yes—
281C23C22×C3×D414yes—
282Q8—216no—
283Q8—224no—
284D4—248no—
285D4—224no—
286D4—248no—
287D4—824no—
288C2×C4—224no—
289C2×C4—248no—
290C2×C4—224no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—46no—
296C2×C4—46no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—41no—
300C2×C4C3×#12011yescharacteristic
301C2×C4C3×#12013yes—
302C2×C4C3×#120112yes—
303C2×C3—26no—
304C2×C3—412no—
305C2×C3—82no—
306C2×C3#284611yescharacteristic
307C2×C3#302511yescharacteristic
308C2×C3#302711yescharacteristic
309C22—23no—
310C22—23no—
311C22—212no—
312C22—44no—
313C22—412no—
314C22—412no—
315C22—46no—
316C22—46no—
317C22—42no—
318C22—86no—
319C22—86no—
320C22—816no—
321C22—81no—
322C22—81no—
323C22C2×C3×#12016yes—
324C22C2×C3×#12011yescharacteristic
325C4—21no—
326C4—212no—
327C4—23no—
328C4—86no—
329C4—81no—
330C3#3289911yescharacteristic, direct product factor, Sylow
331C2—26no—
332C2—412no—
333C2—82no—
334C2C3×#284611yescharacteristic
335C2C3×#302511yescharacteristic
336C2C3×#302711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
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
1. C2×C3×#585C2semidirect
2. C2×C3×B(4,𝔽2)C2semidirect
3. C3×#2404C2semidirect
4. C3×#1790C2semidirect
5. C3×#1782C2semidirect
6. #32899C3direct (N × H)
13. C2×C3×#120C22semidirect
14. C22×C3×C22⋊C4C22semidirect
16. C2×C3×#99C22semidirect
20. C3×#386C22semidirect
21. C3×#381C22semidirect
22. C2×#585C2×C3semidirect
23. C2×B(4,𝔽2)C2×C3semidirect
24. #2404C2×C3semidirect
25. #1790C2×C3semidirect
26. #1782C2×C3semidirect
31. C3×H(5,𝔽2)D4semidirect
41. C22×C3×Q8D4semidirect
45. C22×C3×D4D4semidirect
48. C23×C3×C4D4semidirect
67. C2×#120C22×C3semidirect
68. C22×C22⋊C4C22×C3semidirect
70. C2×#99C22×C3semidirect
74. #386C22×C3semidirect
75. #381C22×C3semidirect
127. H(5,𝔽2)C3×D4semidirect
137. C22×Q8C3×D4semidirect
141. C22×D4C3×D4semidirect
144. C23×C4C3×D4semidirect
330. C3#32899direct (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×B(4,𝔽2)2
C3×#24042
C3×#17902
C3×#17822
#328993