Finite Group Databasebeta

← C3×#5030 #10592127

Subgroups — C3×#5030 #10592127

Order 576.

251 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 102 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×C4×Q8C211yesmaximal, characteristic
2C2×C3×#216C211yesmaximal, characteristic
3C3×C4×Dic6C211yesmaximal, characteristic
4C3×#835C211yesmaximal, characteristic
5C3×#810C211yesmaximal, characteristic
6C3×#808C211yesmaximal, characteristic
7C3×#803C211yesmaximal, characteristic
8C3×#441—31nomaximal
9#5030C311yesmaximal, characteristic, direct product factor
10C2×C32×Q8C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C32×C4⋊C4—21no—
13C32×C4⋊C4C2211yescharacteristic
14C32×C42—21no—
15C32×C42C2211yescharacteristic
16C3×#216C414yescomplemented
17C3×C4⋊Dic3C2211yescharacteristic
18C3×Dic3⋊C4—21no—
19C3×C4×Dic3—21no—
20C2×C3×C3⋊C8C2211yescharacteristic
21C2×C3×C3⋊C8C2211yescharacteristic
22C2×C3×Q16—31no—
23C3×C4×Q8—21no—
24C3×C4×Q8—31no—
25C3×C4×Q8D311yescharacteristic
26C3×C4×Q8C2×C311yescharacteristic
27C2×#216C2×C311yescharacteristic
28C4×Dic6C2×C311yescharacteristic
29C3×#106—31no—
30C3×#103—31no—
31C3×#103—31no—
32C3×#97—31no—
33#835C2×C311yescharacteristic
34#810C2×C311yescharacteristic
35#808C2×C311yescharacteristic
36#803C2×C311yescharacteristic
37C32×Q8—21no—
38C32×Q8C2×C412yes—
39C2×C32×C4—21no—
40C2×C32×C4—21no—
41C2×C32×C4D411yescharacteristic
42C2×C32×C4D411yescharacteristic
43C2×C32×C4C2311yescharacteristic
44C2×C3×Dic3—21no—
45C2×C3×Dic3—21no—
46C3×Dic6—21no—
47C3×Dic6C2×C412yes—
48C3×C3⋊C8—21no—
49C3×C3⋊C8C2×C412yes—
50#441—31noSylow
51C2×C3×Q8—21no—
52C2×C3×Q8—31no—
53C2×C3×Q8C2×D311yescharacteristic
54C2×C3×Q8C22×C311yescharacteristic
55C2×Dic6C22×C311yescharacteristic
56C3×Q16—34no—
57C2×C3×C8—31no—
58C2×C3×C8—31no—
59C3×C4⋊C4—22no—
60C3×C4⋊C4—21no—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—31no—
64C3×C4⋊C4—61no—
65C3×C4⋊C4C2×D311yescharacteristic
66C3×C4⋊C4C22×C311yescharacteristic
67C3×C42—22no—
68C3×C42—21no—
69C3×C42—21no—
70C3×C42—21no—
71C3×C42—61no—
72C3×C42C2×D311yescharacteristic
73C3×C42C22×C311yescharacteristic
74#216C3×C414yescomplemented
75C4⋊Dic3C22×C311yescharacteristic
76Dic3⋊C4—21no—
77C4×Dic3—21no—
78C2×C3⋊C8C22×C311yescharacteristic
79C2×C3⋊C8C22×C311yescharacteristic
80C22×C32C2×D411yescharacteristic
81C32×C4—22no—
82C32×C4—21no—
83C32×C4—21no—
84C32×C4—41no—
85C32×C4C22×C411yescharacteristic
86C32×C4#4111yescharacteristic
87C3×Dic3—22no—
88C3×Dic3—41no—
89C2×Q16—31no—
90C4×Q8—31no—
91C4×Q8C3×D311yescharacteristic
92#106—31no—
93#103—31no—
94#103—31no—
95#97—31no—
96C3×Q8—22no—
97C3×Q8—21no—
98C3×Q8—21no—
99C3×Q8—22no—
100C3×Q8—32no—
101C3×Q8—61no—
102C3×Q8C4×D312yes—
103C3×Q8C2×C3×C412yes—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—61no—
114C2×C3×C4—61no—
115C2×C3×C4#6711yescharacteristic
116C2×C3×C4#6711yescharacteristic
117C2×C3×C4C3×D411yescharacteristic
118C2×C3×C4C3×D411yescharacteristic
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C23×C311yescharacteristic
121C2×Dic3—21no—
122C2×Dic3—21no—
123Dic6—21no—
124Dic6C2×C3×C412yes—
125C3×C8—32no—
126C3×C8—61no—
127C3⋊C8—21no—
128C3⋊C8C2×C3×C412yes—
129C2×C32C4×D411yescharacteristic
130C2×C32#13711yescharacteristic
131C2×C32#13711yescharacteristic
132C2×Q8—31no—
133C2×Q8C2×C3×D311yescharacteristic
134Q16—34no—
135C2×C8—31no—
136C2×C8—31no—
137C4⋊C4—21no—
138C4⋊C4—31no—
139C4⋊C4—61no—
140C4⋊C4C2×C3×D311yescharacteristic
141C42—21no—
142C42—61no—
143C42C2×C3×D311yescharacteristic
144C22×C3—21no—
145C22×C3C2×#6711yescharacteristic
146C22×C3C2×C3×D411yescharacteristic
147C3×C4—24no—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—22no—
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—41no—
158C3×C4—42no—
159C3×C4—41no—
160C3×C4—41no—
161C3×C4—62no—
162C3×C4—121no—
163C3×C4C2×C4×D311yescharacteristic
164C3×C4#23511yescharacteristic
165C3×C4C22×C3×C411yescharacteristic
166C3×C4C3×#4111yescharacteristic
167Dic3—22no—
168Dic3—41no—
169C32#44111yescharacteristic, complemented, Sylow
170Q8—21no—
171Q8—32no—
172Q8—61no—
173Q8C3×C4×D312yes—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—61no—
177C2×C4—61no—
178C2×C4C3×#6711yescharacteristic
179C2×C4C3×#6711yescharacteristic
180C2×C4C22×C3×D311yescharacteristic
181C8—32no—
182C8—61no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—21no—
186C2×C3C4×#6711yescharacteristic
187C2×C3#94211yescharacteristic
188C2×C3#95111yescharacteristic
189C2×C3C3×C4×D411yescharacteristic
190C2×C3C3×#13711yescharacteristic
191C2×C3C3×#13711yescharacteristic
192C22C2×C3×#6711yescharacteristic
193C4—22no—
194C4—21no—
195C4—21no—
196C4—41no—
197C4—62no—
198C4—121no—
199C4C2×C3×C4×D311yescharacteristic
200C4C3×#23511yescharacteristic
201C3—21no—
202C3#503011yescharacteristic, direct product factor
203C3C3×#44111yescharacteristic, complemented
204C2C3×C4×#6711yescharacteristic
205C2C3×#94211yescharacteristic
206C2C3×#95111yescharacteristic
subgroupG/H
centerC22×C3C2×#67
commutatorC3×C4C22×C3×C4
FrattiniC2×C4C22×C3×D3
FittingC32×C4×Q8C2
socle, prime radicalC22×C32C2×D4
cosocleC2×C3×C4C23×C3
hypercenterC3×C4×Q8D3
hypocenterC3C3×#441
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
9. #5030C3direct (N × H)
16. C3×#216C4semidirect
74. #216C3×C4semidirect
169. C32#441semidirect
202. C3#5030direct (N × H)
203. C3C3×#441semidirect

Maximal subgroups

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

subgroupindex
C32×C4×Q82
C2×C3×#2162
C3×C4×Dic62
C3×#8352
C3×#8102
C3×#8082
C3×#8032
C3×#4413
#50303