Finite Group Databasebeta

← C3×#5051 #10592148

Subgroups — C3×#5051 #10592148

Order 576.

265 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C32×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#216C211yesmaximal, characteristic, complemented
3C3×#924C211yesmaximal, characteristic
4C22×C3×C3⋊C8C211yesmaximal, characteristic
5C3×#835C211yesmaximal, characteristic, complemented
6C3×#810C211yesmaximal, characteristic, complemented
7C3×#807C211yesmaximal, characteristic, complemented
8C3×#467—31nomaximal
9#5051C311yesmaximal, characteristic, direct product factor
10C2×C32×Q8C2211yescharacteristic
11C22×C32×C4C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C32×C4⋊C4—21no—
14C32×C4⋊C4C2211yescharacteristic
15C32×C22⋊C4—21no—
16C3×#217—21no—
17C3×#216—22no—
18C3×C4⋊Dic3C2211yescharacteristic
19C3×Dic3⋊C4—21no—
20C2×C3×C3⋊C8—22no—
21C2×C3×C3⋊C8C2211yescharacteristic
22C2×C3×C3⋊C8C2211yescharacteristic
23C2×C3×Q16—31no—
24C22×C3×C8—31no—
25C3×C22⋊Q8—21no—
26C3×C22⋊Q8—31no—
27C3×C22⋊Q8D311yescharacteristic
28C3×C22⋊Q8C2×C311yescharacteristic
29C2×#216C2×C311yescharacteristic, complemented
30#924C2×C311yescharacteristic
31C22×C3⋊C8C2×C311yescharacteristic
32C3×#107—31no—
33C3×#103—31no—
34C3×#103—31no—
35#835C2×C311yescharacteristic, complemented
36#810C2×C311yescharacteristic, complemented
37#807C2×C311yescharacteristic, complemented
38C23×C32D411yescharacteristic
39C32×Q8—21no—
40C32×Q8—21no—
41C2×C32×C4—21no—
42C2×C32×C4—21no—
43C2×C32×C4—21no—
44C2×C32×C4—21no—
45C2×C32×C4D411yescharacteristic
46C2×C32×C4C2311yescharacteristic
47C2×C3×Dic3—21no—
48C2×C3×Dic3—21no—
49C3×Dic6—21no—
50C3×Dic6—21no—
51C3×C3⋊C8—21no—
52C3×C3⋊C8D412yes—
53#467—31noSylow
54C2×C3×Q8—21no—
55C2×C3×Q8—31no—
56C2×C3×Q8C2×D311yescharacteristic
57C2×C3×Q8C22×C311yescharacteristic
58C22×C3×C4—21no—
59C22×C3×C4C2×D311yescharacteristic
60C22×C3×C4C22×C311yescharacteristic
61C2×Dic6C22×C311yescharacteristic
62C3×Q16—62no—
63C2×C3×C8—31no—
64C2×C3×C8—31no—
65C2×C3×C8—62no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—21no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4—31no—
71C3×C4⋊C4—61no—
72C3×C4⋊C4C2×D311yescharacteristic
73C3×C4⋊C4C22×C311yescharacteristic
74C3×C22⋊C4—22no—
75C3×C22⋊C4—21no—
76C3×C22⋊C4—21no—
77C3×C22⋊C4—61no—
78#217—21no—
79#216—22no—
80C4⋊Dic3C22×C311yescharacteristic
81Dic3⋊C4—21no—
82C2×C3⋊C8—22no—
83C2×C3⋊C8C22×C311yescharacteristic
84C2×C3⋊C8C22×C311yescharacteristic
85C22×C32—21no—
86C22×C32—21no—
87C22×C32Q1612yescomplemented
88C22×C32C2×D411yescharacteristic
89C32×C4—21no—
90C32×C4—41no—
91C32×C4—41no—
92C32×C4C2×D411yescharacteristic
93C32×C4#4111yescharacteristic
94C3×Dic3—41no—
95C3×Dic3—41no—
96C2×Q16—31no—
97C22×C8—31no—
98C22⋊Q8—31no—
99C22⋊Q8C3×D311yescharacteristic
100#107—31no—
101#103—31no—
102#103—31no—
103C23×C3—21no—
104C23×C3#6711yescharacteristic
105C23×C3C3×D411yescharacteristic
106C3×Q8—21no—
107C3×Q8—21no—
108C3×Q8—21no—
109C3×Q8—21no—
110C3×Q8—41no—
111C3×Q8—41no—
112C3×Q8—61no—
113C3×Q8—61no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—41no—
127C2×C3×C4—41no—
128C2×C3×C4—61no—
129C2×C3×C4—61no—
130C2×C3×C4#6711yescharacteristic
131C2×C3×C4C3×D411yescharacteristic
132C2×C3×C4C22×D311yescharacteristic
133C2×C3×C4C23×C311yescharacteristic
134C2×Dic3—21no—
135C2×Dic3—21no—
136Dic6—21no—
137Dic6—21no—
138C3×C8—32no—
139C3×C8—61no—
140C3⋊C8—21no—
141C3⋊C8C3×D412yes—
142C2×C32—22no—
143C2×C32#12111yescharacteristic
144C2×C32C2×Q1611yescharacteristic
145C2×C32#13511yescharacteristic
146C2×Q8—31no—
147C2×Q8C2×C3×D311yescharacteristic
148C22×C4C2×C3×D311yescharacteristic
149Q16—62no—
150C2×C8—31no—
151C2×C8—31no—
152C2×C8—62no—
153C4⋊C4—21no—
154C4⋊C4—31no—
155C4⋊C4—61no—
156C4⋊C4C2×C3×D311yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—61no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—22no—
164C22×C3—21no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3#21612yescomplemented
168C22×C3C3×Q1612yescomplemented
169C22×C3C2×#6711yescharacteristic
170C22×C3C2×C3×D411yescharacteristic
171C3×C4—21no—
172C3×C4—21no—
173C3×C4—21no—
174C3×C4—21no—
175C3×C4—41no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—41no—
179C3×C4—41no—
180C3×C4—41no—
181C3×C4—41no—
182C3×C4—121no—
183C3×C4—121no—
184C3×C4D3×D411yescharacteristic
185C3×C4#23711yescharacteristic
186C3×C4C2×C3×D411yescharacteristic
187C3×C4C3×#4111yescharacteristic
188Dic3—41no—
189Dic3—41no—
190C32#46711yescharacteristic, complemented, Sylow
191C23C3×#6711yescharacteristic
192Q8—21no—
193Q8—21no—
194Q8—61no—
195Q8—61no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4C3×#6711yescharacteristic
203C2×C4C22×C3×D311yescharacteristic
204C8—32no—
205C8—61no—
206C2×C3—22no—
207C2×C3—22no—
208C2×C3—21no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—42no—
212C2×C3#93911yescharacteristic
213C2×C3C2×#21611yescharacteristic
214C2×C3#95011yescharacteristic
215C2×C3C3×#12111yescharacteristic
216C2×C3C2×C3×Q1611yescharacteristic
217C2×C3C3×#13511yescharacteristic
218C22—21no—
219C22—21no—
220C22C3×#21612yescomplemented
221C22C2×C3×#6711yescharacteristic
222C4—21no—
223C4—41no—
224C4—41no—
225C4—121no—
226C4—121no—
227C4C3×D3×D411yescharacteristic
228C4C3×#23711yescharacteristic
229C3—21no—
230C3#505111yescharacteristic, direct product factor
231C3C3×#46711yescharacteristic, complemented
232C2—22no—
233C2C3×#93911yescharacteristic
234C2C2×C3×#21611yescharacteristic
235C2C3×#95011yescharacteristic
subgroupG/H
centerC22×C3C2×#67
commutator, cosocleC2×C3×C4C23×C3
FrattiniC2×C4C22×C3×D3
FittingC32×C22⋊Q8C2
socleC22×C32C2×D4
prime radicalC23×C32D4
hypercenterC3×C22⋊Q8D3
hypocenterC3C3×#467
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#216C2semidirect
5. C3×#835C2semidirect
6. C3×#810C2semidirect
7. C3×#807C2semidirect
9. #5051C3direct (N × H)
29. C2×#216C2×C3semidirect
35. #835C2×C3semidirect
36. #810C2×C3semidirect
37. #807C2×C3semidirect
87. C22×C32Q16semidirect
167. C22×C3#216semidirect
168. C22×C3C3×Q16semidirect
190. C32#467semidirect
220. C22C3×#216semidirect
230. C3#5051direct (N × H)
231. C3C3×#467semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
C2×C3×#2162
C3×#9242
C22×C3×C3⋊C82
C3×#8352
C3×#8102
C3×#8072
C3×#4673
#50513