Finite Group Databasebeta

← C3×#4734 #10591831

Subgroups — C3×#4734 #10591831

Order 576.

349 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 86 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×#235C211yesmaximal, characteristic, complemented
2C3×#930C211yesmaximal, characteristic, complemented
3C2×C3×D24C211yesmaximal, characteristic, complemented
4C2×C3×#204C211yesmaximal, characteristic, complemented
5C32×C22⋊C8C211yesmaximal, characteristic, complemented
6C3×#821C211yesmaximal, characteristic, complemented
7C3×#816C211yesmaximal, characteristic, complemented
8C3×#449—31nomaximal
9#4734C311yesmaximal, characteristic, direct product factor
10C22×C32×C4C2211yescharacteristic
11C2×C3×#67—21no—
12C2×C3×#67—21no—
13C3×#235—22no—
14C3×#235—22no—
15C2×C3×D12C2211yescharacteristic
16C2×C3×D12C2211yescharacteristic
17C2×C3×C4×D3—21no—
18C2×C3×Dic6C2211yescharacteristic
19C2×C32×C8C2211yescharacteristic
20C2×C32×C8C2211yescharacteristic
21C3×(C2×D3)⋊C4—21no—
22C3×C4⋊Dic3C2211yescharacteristic
23C3×D24—22no—
24C3×#204—22no—
25C2×C3×#41—31no—
26C2×#235C2×C311yescharacteristic, complemented
27C2×C3×SD16—31no—
28C2×C3×D8—31no—
29C3×#121—31no—
30#930C2×C311yescharacteristic, complemented
31C2×D24C2×C311yescharacteristic, complemented
32C2×#204C2×C311yescharacteristic, complemented
33C3×#103—31no—
34C3×#102—31no—
35C3×C22⋊C8—21no—
36C3×C22⋊C8D311yescharacteristic, complemented
37C3×C22⋊C8C2×C311yescharacteristic, complemented
38#821C2×C311yescharacteristic, complemented
39#816C2×C311yescharacteristic, complemented
40C23×C32D411yescharacteristic
41C22×C3×D3—21no—
42C22×C3×D3—21no—
43C2×C32×C4—21no—
44C2×C32×C4—21no—
45C2×C32×C4D411yescharacteristic
46C2×C32×C4C2311yescharacteristic
47C3×#67—24no—
48C3×#67—42no—
49C2×C3×Dic3—21no—
50C2×C3×Dic3—21no—
51C3×D12—21no—
52C3×D12—21no—
53C3×D12—21no—
54C3×D12D412yescomplemented
55C3×C4×D3—24no—
56C3×Dic6—21no—
57C3×Dic6D412yescomplemented
58C32×C8—21no—
59C32×C8—21no—
60#449—31noSylow
61C3×#41—62no—
62C3×#41—62no—
63C2×C3×Q8—31no—
64C2×C3×D4—31no—
65C2×C3×D4—31no—
66C2×C3×D4—61no—
67C2×C3×D4—61no—
68C22×C3×C4—21no—
69C22×C3×C4—61no—
70C22×C3×C4C2×D311yescharacteristic
71C22×C3×C4C22×C311yescharacteristic
72C2×#67—21no—
73C2×#67—21no—
74#235—22no—
75#235—22no—
76C2×D12C22×C311yescharacteristic
77C2×D12C22×C311yescharacteristic
78C2×C4×D3—21no—
79C2×Dic6C22×C311yescharacteristic
80C3×SD16—62no—
81C3×D8—62no—
82C2×C3×C8—21no—
83C2×C3×C8—21no—
84C2×C3×C8C2×D311yescharacteristic
85C2×C3×C8C2×D311yescharacteristic
86C2×C3×C8C22×C311yescharacteristic
87C2×C3×C8C22×C311yescharacteristic
88C3×C4⋊C4—31no—
89C3×C22⋊C4—61no—
90(C2×D3)⋊C4—21no—
91C4⋊Dic3C22×C311yescharacteristic
92D24—22no—
93#204—22no—
94C22×C32—21no—
95C22×C32—21no—
96C22×C32—21no—
97C22×C32C2×D411yescharacteristic
98C2×C3×D3—22no—
99C2×C3×D3—41no—
100C2×C3×D3—41no—
101C2×C3×D3—41no—
102C2×C3×D3—41no—
103C2×C3×D3—41no—
104C32×C4—21no—
105C32×C4C2×D411yescharacteristic
106C32×C4C2×D411yescharacteristic
107C3×Dic3—22no—
108C3×Dic3—41no—
109C2×#41—31no—
110C2×SD16—31no—
111C2×D8—31no—
112#121—31no—
113#103—31no—
114#102—31no—
115C22⋊C8C3×D311yescharacteristic, complemented
116C23×C3—21no—
117C23×C3—61no—
118C23×C3—61no—
119C23×C3D1211yescharacteristic
120C23×C3C3×D411yescharacteristic
121C22×D3—21no—
122C22×D3—21no—
123C3×Q8—32no—
124C3×Q8—61no—
125C3×D4—32no—
126C3×D4—64no—
127C3×D4—61no—
128C3×D4—61no—
129C3×D4—61no—
130C3×D4—122no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4—64no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4D1211yescharacteristic
143C2×C3×C4C3×D411yescharacteristic
144C2×C3×C4C22×D311yescharacteristic
145C2×C3×C4C23×C311yescharacteristic
146#67—24no—
147#67—42no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150D12—21no—
151D12—21no—
152D12—21no—
153D12C3×D412yescomplemented
154C4×D3—24no—
155Dic6—21no—
156Dic6C3×D412yescomplemented
157C3×C8—21no—
158C3×C8—21no—
159C3×C8—21no—
160C3×C8—21no—
161C3×C8—41no—
162C3×C8—41no—
163C2×C32—41no—
164C2×C32#12011yescharacteristic
165C2×C32#13511yescharacteristic
166C2×C32AGL(1,ℤ/8)11yescharacteristic
167C3×D3—42no—
168C3×D3—81no—
169#41—62no—
170#41—62no—
171C2×Q8—31no—
172C2×D4—31no—
173C2×D4—31no—
174C2×D4—61no—
175C2×D4—61no—
176C22×C4—61no—
177C22×C4C2×C3×D311yescharacteristic
178SD16—62no—
179D8—62no—
180C2×C8C2×C3×D311yescharacteristic
181C2×C8C2×C3×D311yescharacteristic
182C4⋊C4—31no—
183C22⋊C4—61no—
184C22×C3—22no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—62no—
195C22×C3—121no—
196C22×C3—121no—
197C22×C3—121no—
198C22×C3—121no—
199C22×C3—121no—
200C22×C3C2×D1211yescharacteristic
201C22×C3C2×C3×D411yescharacteristic
202C2×D3—22no—
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—41no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—41no—
213C3×C4—62no—
214C3×C4—121no—
215C3×C4D3×D411yescharacteristic
216C3×C4D3×D411yescharacteristic
217C3×C4C2×C3×D411yescharacteristic
218C3×C4C2×C3×D411yescharacteristic
219Dic3—22no—
220Dic3—41no—
221C32#44911yescharacteristic, complemented, Sylow
222C23—61no—
223C23—61no—
224C23C3×D1211yescharacteristic
225Q8—32no—
226Q8—61no—
227D4—32no—
228D4—64no—
229D4—61no—
230D4—61no—
231D4—61no—
232D4—122no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—64no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4C3×D1211yescharacteristic
239C2×C4C22×C3×D311yescharacteristic
240C8—21no—
241C8—21no—
242C2×C3—21no—
243C2×C3—21no—
244C2×C3—21no—
245C2×C3—42no—
246C2×C3—41no—
247C2×C3—41no—
248C2×C3—122no—
249C2×C3—241no—
250C2×C3#88211yescharacteristic
251C2×C3#90411yescharacteristic
252C2×C3#90811yescharacteristic
253C2×C3C3×#12011yescharacteristic
254C2×C3C3×#13511yescharacteristic
255C2×C3C3×AGL(1,ℤ/8)11yescharacteristic
256D3—42no—
257D3—81no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—62no—
262C22—121no—
263C22—121no—
264C22—121no—
265C22—121no—
266C22—121no—
267C22C2×C3×D1211yescharacteristic
268C4—21no—
269C4—62no—
270C4—121no—
271C4C3×D3×D411yescharacteristic
272C4C3×D3×D411yescharacteristic
273C3—21no—
274C3#473411yescharacteristic, direct product factor
275C3C3×#44911yescharacteristic, complemented
276C2—41no—
277C2—122no—
278C2—241no—
279C2C3×#88211yescharacteristic
280C2C3×#90411yescharacteristic
281C2C3×#90811yescharacteristic
subgroupG/H
centerC22×C3C2×D12
commutator, cosocleC2×C3×C4C23×C3
FrattiniC2×C4C22×C3×D3
FittingC32×C22⋊C8C2
socleC22×C32C2×D4
hypercenterC3×C22⋊C8D3
hypocenterC3C3×#449
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#235C2semidirect
2. C3×#930C2semidirect
3. C2×C3×D24C2semidirect
4. C2×C3×#204C2semidirect
5. C32×C22⋊C8C2semidirect
6. C3×#821C2semidirect
7. C3×#816C2semidirect
9. #4734C3direct (N × H)
26. C2×#235C2×C3semidirect
30. #930C2×C3semidirect
31. C2×D24C2×C3semidirect
32. C2×#204C2×C3semidirect
36. C3×C22⋊C8D3semidirect
37. C3×C22⋊C8C2×C3semidirect
38. #821C2×C3semidirect
39. #816C2×C3semidirect
54. C3×D12D4semidirect
57. C3×Dic6D4semidirect
115. C22⋊C8C3×D3semidirect
153. D12C3×D4semidirect
156. Dic6C3×D4semidirect
221. C32#449semidirect
274. C3#4734direct (N × H)
275. C3C3×#449semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2352
C3×#9302
C2×C3×D242
C2×C3×#2042
C32×C22⋊C82
C3×#8212
C3×#8162
C3×#4493
#47343