Finite Group Databasebeta

← C2×C3×#63703 #13164885

Subgroups — C2×C3×#63703 #13164885

Order 1728.

1106 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C3×#63703C211yesmaximal, characteristic, direct product factor
2C2×C32×H(5,𝔽2)C311yesmaximal, characteristic
3C2×#63703C313yesmaximal, direct product factor
4C22×C3×C22⋊C9—43nomaximal
5C2×C3×Q8⋊C9—42nomaximal
6C32×H(5,𝔽2)—32no—
7C32×H(5,𝔽2)—33no—
8C32×H(5,𝔽2)C2×C311yescharacteristic
9C2×C32×#41—32no—
10C22×C32×D4—33no—
11#63703C2×C313yesdirect product factor
12C2×C3×C22⋊C9—46no—
13C2×C3×C22⋊C9—43no—
14C3×Q8⋊C9—42no—
15C2×C3×H(5,𝔽2)C913yescomplemented
16C2×C3×H(5,𝔽2)C3211yescharacteristic
17C24×C32—31no—
18C24×C32A413yes—
19C32×#41—36no—
20C32×#41—36no—
21C32×#41—32no—
22C32×#41—32no—
23C2×C32×Q8A412yes—
24C2×C32×D4—36no—
25C2×C32×D4—33no—
26C2×C32×D4—312no—
27C2×C32×D4—33no—
28C2×C32×D4—33no—
29C2×C32×D4—33no—
30C22×C32×C4—33no—
31C22×C22⋊C9—49no—
32C2×Q8⋊C9—46no—
33C22×C3×C9—161no—
34C3×C22⋊C9—43no—
35C3×H(5,𝔽2)—36no—
36C3×H(5,𝔽2)—32no—
37C3×H(5,𝔽2)—39no—
38C3×H(5,𝔽2)—33no—
39C3×H(5,𝔽2)C2×C913yescomplemented
40C3×H(5,𝔽2)C2×C3211yescharacteristic
41C2×C3×#41—36no—
42C2×C3×#41—32no—
43C22×C3×D4—39no—
44C22×C3×D4—33no—
45C23×C32—33no—
46C23×C32—31no—
47C23×C32—33no—
48C23×C32—33no—
49C23×C32—46no—
50C23×C32—122no—
51C23×C32C2×A413yes—
52C32×Q8—32no—
53C32×Q8C2×A412yes—
54C32×D4—312no—
55C32×D4—36no—
56C32×D4—36no—
57C2×C32×C4—33no—
58C2×C32×C4—36no—
59C2×C32×C4—33no—
60C2×C32×C4—32no—
61C2×C22⋊C9—418no—
62C2×C22⋊C9—49no—
63Q8⋊C9—46no—
64C2×H(5,𝔽2)C3×C911yescharacteristic, complemented, Sylow
65C2×C3×C9—162no—
66C2×C3×C9—161no—
67C24×C3—31no—
68C24×C3—33no—
69C24×C3C22⋊C919yescomplemented
70C24×C3C3×A413yes—
71C3×#41—318no—
72C3×#41—318no—
73C3×#41—36no—
74C3×#41—36no—
75C3×#41—36no—
76C3×#41—36no—
77C3×#41—32no—
78C3×#41—32no—
79C2×C3×Q8C22⋊C916yescomplemented
80C2×C3×Q8C3×A412yes—
81C2×C3×D4—36no—
82C2×C3×D4—318no—
83C2×C3×D4—39no—
84C2×C3×D4—336no—
85C2×C3×D4—39no—
86C2×C3×D4—33no—
87C2×C3×D4—312no—
88C2×C3×D4—33no—
89C2×C3×D4—39no—
90C2×C3×D4—39no—
91C2×C3×D4—33no—
92C2×C3×D4—33no—
93C22×C3×C4—33no—
94C22×C3×C4—39no—
95C22×C32—33no—
96C22×C32—33no—
97C22×C32—43no—
98C22×C32—66no—
99C22×C32—123no—
100C22×C32—121no—
101C22×C32—123no—
102C22×C32C22⋊A411yescharacteristic
103C32×C4—32no—
104C32×C4—32no—
105C22×C9—163no—
106C22⋊C9—49no—
107H(5,𝔽2)—32no—
108H(5,𝔽2)—33no—
109H(5,𝔽2)C2×C3×C911yescharacteristic, complemented
110C2×#41—32no—
111C22×D4—33no—
112C3×C9—161noSylow
113C23×C3—31no—
114C23×C3—33no—
115C23×C3—33no—
116C23×C3—39no—
117C23×C3—33no—
118C23×C3—39no—
119C23×C3—39no—
120C23×C3—33no—
121C23×C3—46no—
122C23×C3—418no—
123C23×C3—122no—
124C23×C3—126no—
125C23×C3C2×C22⋊C919yescomplemented
126C23×C3C2×C3×A413yes—
127C3×Q8—36no—
128C3×Q8—32no—
129C3×Q8C2×C22⋊C916yescomplemented
130C3×Q8C2×C3×A412yes—
131C3×D4—36no—
132C3×D4—36no—
133C3×D4—312no—
134C3×D4—318no—
135C3×D4—318no—
136C3×D4—336no—
137C2×C3×C4—33no—
138C2×C3×C4—36no—
139C2×C3×C4—33no—
140C2×C3×C4—39no—
141C2×C3×C4—318no—
142C2×C3×C4—39no—
143C2×C3×C4—32no—
144C2×C3×C4—36no—
145C2×C32—63no—
146C2×C32—63no—
147C2×C32#99712yes—
148C2×C32C2×C22⋊A411yescharacteristic
149C2×C9—166no—
150C2×C9—163no—
151C24—31no—
152C24C3×C22⋊C913yescomplemented
153#41—36no—
154#41—36no—
155#41—32no—
156#41—32no—
157C2×Q8C3×C22⋊C912yescomplemented
158C2×D4—36no—
159C2×D4—33no—
160C2×D4—312no—
161C2×D4—33no—
162C2×D4—33no—
163C2×D4—33no—
164C22×C4—33no—
165C22×C3—33no—
166C22×C3—33no—
167C22×C3—39no—
168C22×C3—39no—
169C22×C3—49no—
170C22×C3—43no—
171C22×C3—66no—
172C22×C3—618no—
173C22×C3—129no—
174C22×C3—123no—
175C22×C3—123no—
176C22×C3—129no—
177C22×C3—121no—
178C22×C3—123no—
179C22×C3C22⋊(C22⋊C9)13yes—
180C22×C3C3×C22⋊A411yescharacteristic
181C3×C4—32no—
182C3×C4—32no—
183C3×C4—36no—
184C3×C4—36no—
185C32C2×#99711yescharacteristic
186C9—163no—
187C23—31no—
188C23—33no—
189C23—33no—
190C23—33no—
191C23—46no—
192C23—122no—
193C23C2×C3×C22⋊C913yescomplemented
194Q8—32no—
195Q8C2×C3×C22⋊C912yescomplemented
196D4—36no—
197D4—36no—
198D4—312no—
199C2×C4—33no—
200C2×C4—36no—
201C2×C4—33no—
202C2×C4—32no—
203C2×C3—69no—
204C2×C3—69no—
205C2×C3—63no—
206C2×C3—63no—
207C2×C3#6370316yesdirect product factor
208C2×C3C2×C22⋊(C22⋊C9)13yes—
209C2×C3C3×#99712yes—
210C2×C3C2×C3×C22⋊A411yescharacteristic
211C22—33no—
212C22—33no—
213C22—43no—
214C22—66no—
215C22—123no—
216C22—121no—
217C22—123no—
218C22C3×C22⋊(C22⋊C9)11yescharacteristic
219C4—32no—
220C4—32no—
221C3C2×#6370313yesdirect product factor
222C3C2×C3×#99711yescharacteristic
223C2—63no—
224C2—63no—
225C2C3×#6370312yesdirect product factor
226C2C2×C3×C22⋊(C22⋊C9)11yescharacteristic
subgroupG/H
center, socle, hypercenterC22×C32C22⋊A4
commutator, hypocenterH(5,𝔽2)C2×C3×C9
FrattiniC2×C3C2×C3×C22⋊A4
Fitting, prime radicalC2×C32×H(5,𝔽2)C3
cosocleC3×H(5,𝔽2)C2×C32
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C3×#63703C2direct (N × H)
3. C2×#63703C3direct (N × H)
11. #63703C2×C3direct (N × H)
15. C2×C3×H(5,𝔽2)C9semidirect
39. C3×H(5,𝔽2)C2×C9semidirect
64. C2×H(5,𝔽2)C3×C9semidirect
69. C24×C3C22⋊C9semidirect
79. C2×C3×Q8C22⋊C9semidirect
109. H(5,𝔽2)C2×C3×C9semidirect
125. C23×C3C2×C22⋊C9semidirect
129. C3×Q8C2×C22⋊C9semidirect
152. C24C3×C22⋊C9semidirect
157. C2×Q8C3×C22⋊C9semidirect
193. C23C2×C3×C22⋊C9semidirect
195. Q8C2×C3×C22⋊C9semidirect
207. C2×C3#63703direct (N × H)
221. C3C2×#63703direct (N × H)
225. C2C3×#63703direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637032
C2×C32×H(5,𝔽2)3
C2×#637033
C22×C3×C22⋊C94
C2×C3×Q8⋊C94