Finite Group Databasebeta

← C3×#10590309 #13157319

Subgroups — C3×#10590309 #13157319

Order 1728.

602 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 168 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×#869C212yesmaximal
2C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
3C3×#63574C214yesmaximal
4C3×#5080—32nomaximal
5#10590309C311yesmaximal, characteristic, direct product factor
6C2×C32×Dic6—22no—
7C33×C42C2211yescharacteristic
8C2×C3×C3⋊(C3⋊C8)C2212yes—
9C32×C4⋊Dic3—22no—
10C32×C4⋊Dic3C2214yes—
11C32×C4⋊Q8—32no—
12C3×#869—22no—
13C3×#869D312yes—
14C3×#869C2×C312yes—
15C4×C3⋊(C3⋊C8)C2×C311yescharacteristic
16C3×#808—38no—
17C3×C4×C3⋊C8—32no—
18C3×C4×C3⋊C8—61no—
19#63574C2×C314yes—
20C2×C33×C4D412yes—
21C2×C33×C4C2311yescharacteristic
22C2×C32×Dic3—24no—
23C32×Dic6—44no—
24C3×C3⋊(C3⋊C8)Q814yescomplemented
25C3×#498—91no—
26#5080—32no—
27C2×C32×Q8—62no—
28C2×C3×Dic6—24no—
29C2×C3×Dic6—22no—
30C2×C3×Dic6—22no—
31C32×C4⋊C4—34no—
32C32×C4⋊C4—62no—
33C32×C42—21no—
34C32×C42—22no—
35C32×C42—41no—
36C32×C42C2×D312yes—
37C32×C42C22×C311yescharacteristic
38C2×C3⋊(C3⋊C8)C22×C312yes—
39C3×C4⋊Dic3—24no—
40C3×C4⋊Dic3—24no—
41C3×C4⋊Dic3—22no—
42C3×C4⋊Dic3—22no—
43C3×C4⋊Dic3C2×D314yes—
44C3×C4⋊Dic3C22×C314yes—
45C2×C3×C3⋊C8—34no—
46C2×C3×C3⋊C8—62no—
47C22×C33C2×D411yescharacteristic
48C33×C4SD1614yes—
49C33×C4C2×Q812yes—
50C32×Dic3—44no—
51C3×C4⋊Q8—32no—
52C3×C4⋊Q8—32no—
53C3×C4⋊Q8—62no—
54#869C3×D312yes—
55C3×#106—94no—
56C3×C4×C8—91no—
57#808—38no—
58C4×C3⋊C8—32no—
59C4×C3⋊C8—61no—
60C32×Q8—124no—
61C2×C32×C4—22no—
62C2×C32×C4—22no—
63C2×C32×C4—22no—
64C2×C32×C4—21no—
65C2×C32×C4—22no—
66C2×C32×C4—42no—
67C2×C32×C4—41no—
68C2×C32×C4—64no—
69C2×C32×C4#6712yes—
70C2×C32×C4#6712yes—
71C2×C32×C4C3×D412yes—
72C2×C32×C4C22×D312yes—
73C2×C32×C4C23×C311yescharacteristic
74C2×C3×Dic3—28no—
75C2×C3×Dic3—24no—
76C2×C3×Dic3—24no—
77C3×Dic6—48no—
78C3×Dic6—44no—
79C3×Dic6—44no—
80C3⋊(C3⋊C8)C3×Q814yescomplemented
81C3×C3⋊C8—38no—
82C3×C3⋊C8—64no—
83#498—91noSylow
84C2×C33C4⋊Q811yescharacteristic
85C2×C33C2×SD1612yes—
86C2×C3×Q8—64no—
87C2×C3×Q8—62no—
88C2×C3×Q8—62no—
89C2×Dic6—22no—
90C2×C3×C8—92no—
91C3×C4⋊C4—34no—
92C3×C4⋊C4—34no—
93C3×C4⋊C4—64no—
94C3×C4⋊C4—64no—
95C3×C4⋊C4—62no—
96C3×C4⋊C4—62no—
97C3×C42—21no—
98C3×C42—22no—
99C3×C42—41no—
100C3×C42D3211yescharacteristic
101C3×C42C2×C3×D312yes—
102C4⋊Dic3—22no—
103C4⋊Dic3C2×C3×D314yes—
104C2×C3⋊C8—34no—
105C2×C3⋊C8—62no—
106C22×C32—21no—
107C22×C32—22no—
108C22×C32—41no—
109C22×C32C2×#6712yes—
110C22×C32C2×C3×D411yescharacteristic
111C32×C4—24no—
112C32×C4—24no—
113C32×C4—24no—
114C32×C4—22no—
115C32×C4—24no—
116C32×C4—44no—
117C32×C4—42no—
118C32×C4—124no—
119C32×C4#21414yes—
120C32×C4#21514yes—
121C32×C4C3×SD1614yes—
122C32×C4D3×Q814yes—
123C32×C4C2×C3×Q812yes—
124C3×Dic3—48no—
125C3×Dic3—44no—
126C3×Dic3—44no—
127C4⋊Q8—32no—
128#106—94no—
129C4×C8—91no—
130C33#49811yescharacteristic, complemented, Sylow
131C3×Q8—128no—
132C3×Q8—124no—
133C3×Q8—124no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—42no—
140C2×C3×C4—41no—
141C2×C3×C4—68no—
142C2×C3×C4—64no—
143C2×C3×C4—64no—
144C2×C3×C4#62512yes—
145C2×C3×C4C3×#6712yes—
146C2×C3×C4C3×#6712yes—
147C2×C3×C4C2×D3211yescharacteristic
148C2×C3×C4C22×C3×D312yes—
149C2×Dic3—24no—
150Dic6—44no—
151C3×C8—94no—
152C3⋊C8—38no—
153C3⋊C8—64no—
154C2×C32—22no—
155C2×C32—22no—
156C2×C32—22no—
157C2×C32—22no—
158C2×C32—21no—
159C2×C32—41no—
160C2×C32—42no—
161C2×C32C2×#21412yes—
162C2×C32C2×#21512yes—
163C2×C32#94412yes—
164C2×C32C3×C4⋊Q811yescharacteristic
165C2×C32C2×C3×SD1612yes—
166C2×Q8—62no—
167C2×C8—92no—
168C4⋊C4—34no—
169C4⋊C4—62no—
170C42C3×D3211yescharacteristic
171C22×C3—22no—
172C22×C3—21no—
173C22×C3—41no—
174C22×C3C2×#62511yescharacteristic
175C22×C3C2×C3×#6712yes—
176C3×C4—22no—
177C3×C4—24no—
178C3×C4—24no—
179C3×C4—24no—
180C3×C4—24no—
181C3×C4—42no—
182C3×C4—44no—
183C3×C4—128no—
184C3×C4—124no—
185C3×C4—124no—
186C3×C4#371514yes—
187C3×C4C3×#21414yes—
188C3×C4C3×#21514yes—
189C3×C4#379712yes—
190C3×C4C3×D3×Q814yes—
191Dic3—44no—
192C32—21no—
193C32—22no—
194C32—41no—
195C32#508012yescomplemented
196C32C3×#49811yescharacteristic, complemented
197Q8—124no—
198C2×C4—64no—
199C2×C4C3×#62512yes—
200C2×C4C2×C3×D3211yescharacteristic
201C8—94no—
202C2×C3—22no—
203C2×C3—22no—
204C2×C3—22no—
205C2×C3—22no—
206C2×C3—21no—
207C2×C3—42no—
208C2×C3—41no—
209C2×C3C2×#371512yes—
210C2×C3#6387311yescharacteristic
211C2×C3C2×C3×#21412yes—
212C2×C3C2×C3×#21512yes—
213C2×C3C3×#94412yes—
214C22C2×C3×#62511yescharacteristic
215C4—124no—
216C4C3×#371514yes—
217C4C3×#379712yes—
218C3—22no—
219C3—21no—
220C3—41no—
221C3#1059030911yescharacteristic, direct product factor
222C3C3×#508012yescomplemented
223C2C2×C3×#371512yes—
224C2C3×#6387311yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#498
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
5. #10590309C3direct (N × H)
24. C3×C3⋊(C3⋊C8)Q8semidirect
80. C3⋊(C3⋊C8)C3×Q8semidirect
130. C33#498semidirect
195. C32#5080semidirect
196. C32C3×#498semidirect
221. C3#10590309direct (N × H)
222. C3C3×#5080semidirect

Maximal subgroups

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

subgroupindex
C32×#8692
C3×C4×C3⋊(C3⋊C8)2
C3×#635742
C3×#50803
#105903093