Finite Group Databasebeta

← #10592767⋊C3 #13142956

Subgroups — #10592767⋊C3 #13142956

Order 1728.

392 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic
3#63636⋊C3C212yesmaximal
4#63637⋊C3C212yesmaximal, complemented
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
6#10592767C311yesmaximal, characteristic, complemented
7C3×#5068—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×#634⋊C3C2211yescharacteristic
10C4⋊C4×H(3,𝔽3)—22no—
11C4⋊C4×H(3,𝔽3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#3752⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15C32×#125—21no—
16C32×#125D311yescharacteristic, complemented
17C32×#125C2×C311yescharacteristic, complemented
18#64084C2×C311yescharacteristic
19C3×#875—31no—
20#63637C2×C312yescomplemented
21#63636C2×C312yes—
22C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
23C3×#810—32no—
24C3×#809—32no—
25C3×C4×C3⋊C8—31no—
26C22×(C3⋊D3)⋊C3—21no—
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)D412yes—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊Dic3)⋊C3—21no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32#634⋊C3—22no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34C3×#488—91no—
35#5068—31no—
36#5068—91no—
37C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
38C2×#634C22×C311yescharacteristic
39C2×C3×D12—31no—
40C2×C3×Dic6—31no—
41C32×C4⋊C4—24no—
42C32×C4⋊C4—22no—
43C32×C4⋊C4—22no—
44C32×C4⋊C4—22no—
45C32×C4⋊C4C2×D312yes—
46C32×C4⋊C4C22×C312yes—
47C32×C42—21no—
48C32×C42C2×D311yescharacteristic
49C32×C42C22×C311yescharacteristic
50#3752—22no—
51C2×C3⋊(C3⋊C8)C22×C312yes—
52C3×(C2×D3)⋊C4—62no—
53C2×C3×C3⋊C8—32no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×(C3⋊D3)⋊C3—41no—
56C2×(C3⋊D3)⋊C3—42no—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)—42no—
59C4×H(3,𝔽3)#4112yes—
60(C3⋊Dic3)⋊C3—41no—
61C3×#125—31no—
62C3×#125—31no—
63C3×#125—61no—
64C3×#125C3×D311yescharacteristic
65C3×#124—91no—
66#875—31no—
67#875—91no—
68C3×#103—92no—
69C3×#102—92no—
70C3×C4×C8—91no—
71#810—32no—
72#810—92no—
73#809—32no—
74#809—92no—
75C4×C3⋊C8—31no—
76C4×C3⋊C8—91no—
77C22×C3⋊D3—21no—
78C22×C3×D3—61no—
79C2×C32×C4—24no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4—21no—
84C2×C32×C4#6712yes—
85C2×C32×C4C3×D412yes—
86C2×C32×C4C22×D311yescharacteristic
87C2×C32×C4C23×C311yescharacteristic
88C2×C3⋊Dic3—21no—
89C4⋊(C3⋊D3)—22no—
90#634—22no—
91C2×C3×Dic3—61no—
92C3×D12—62no—
93C3×Dic6—62no—
94C3⋊(C3⋊C8)—22no—
95C3×C3⋊C8—62no—
96#488—91noSylow
97C2×H(3,𝔽3)#12411yescharacteristic
98C2×H(3,𝔽3)#13512yes—
99(C3⋊D3)⋊C3—81no—
100C2×C3×Q8—91no—
101C2×C3×D4—91no—
102C2×D12—31no—
103C2×D12—91no—
104C2×Dic6—31no—
105C2×Dic6—91no—
106C2×C3×C8—92no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—32no—
110C3×C4⋊C4—62no—
111C3×C4⋊C4—62no—
112C3×C4⋊C4—64no—
113C3×C4⋊C4—62no—
114C3×C4⋊C4C2×C3×D312yes—
115C3×C22⋊C4—182no—
116C3×C42—31no—
117C3×C42—31no—
118C3×C42—61no—
119C3×C42C2×C3×D311yescharacteristic
120(C2×D3)⋊C4—62no—
121(C2×D3)⋊C4—182no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—92no—
124C22×C32—21no—
125C22×C32C2×#6711yescharacteristic
126C22×C32C2×C3×D411yescharacteristic
127C2×C3⋊D3—41no—
128C2×C3⋊D3—42no—
129C2×C3×D3—121no—
130C2×C3×D3—122no—
131C32×C4—22no—
132C32×C4—22no—
133C32×C4—22no—
134C32×C4—42no—
135C32×C4—44no—
136C32×C4—42no—
137C32×C4—42no—
138C32×C4#23912yes—
139C32×C4C3×#4112yes—
140C3⋊Dic3—41no—
141C3×Dic3—121no—
142#125(C3⋊D3)⋊C311yescharacteristic, complemented
143#124—91no—
144#103—92no—
145#102—92no—
146C4×C8—91no—
147H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
148C23×C3—181no—
149C22×D3—61no—
150C22×D3—181no—
151C3×Q8—182no—
152C3×D4—182no—
153C2×C3×C4—22no—
154C2×C3×C4—32no—
155C2×C3×C4—32no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—62no—
159C2×C3×C4—62no—
160C2×C3×C4—64no—
161C2×C3×C4—62no—
162C2×C3×C4—61no—
163C2×C3×C4—181no—
164C2×C3×C4C3×#6712yes—
165C2×C3×C4C22×C3×D311yescharacteristic
166C2×Dic3—61no—
167C2×Dic3—181no—
168D12—62no—
169D12—182no—
170Dic6—62no—
171Dic6—182no—
172C3×C8—182no—
173C3⋊C8—62no—
174C3⋊C8—182no—
175C2×C32—21no—
176C2×C32—22no—
177C2×C32#94711yescharacteristic
178C2×C32#95012yes—
179C2×C32C3×#12411yescharacteristic
180C2×C32C3×#13512yes—
181C3⋊D3—81no—
182C3×D3—241no—
183C2×Q8—91no—
184C2×D4—91no—
185C2×C8—92no—
186C4⋊C4—22no—
187C4⋊C4C2×(C3⋊D3)⋊C312yes—
188C22⋊C4—182no—
189C42C2×(C3⋊D3)⋊C311yescharacteristic
190C22×C3—31no—
191C22×C3—31no—
192C22×C3—61no—
193C22×C3—361no—
194C22×C3—362no—
195C22×C3C2×C3×#6711yescharacteristic
196C2×D3—121no—
197C2×D3—122no—
198C2×D3—361no—
199C2×D3—362no—
200C3×C4—22no—
201C3×C4—32no—
202C3×C4—32no—
203C3×C4—42no—
204C3×C4—62no—
205C3×C4—62no—
206C3×C4—62no—
207C3×C4—122no—
208C3×C4—122no—
209C3×C4—122no—
210C3×C4—124no—
211C3×C4—361no—
212C3×C4C3×#23912yes—
213Dic3—121no—
214Dic3—361no—
215C32—21no—
216C32#506811yescharacteristic, complemented
217C32C3×#48811yescharacteristic, complemented
218C23—181no—
219Q8—182no—
220D4—182no—
221C2×C4—22no—
222C2×C4—181no—
223C2×C4#638⋊C312yes—
224C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
225C8—182no—
226C2×C3—31no—
227C2×C3—31no—
228C2×C3—32no—
229C2×C3—32no—
230C2×C3—61no—
231C2×C3—62no—
232C2×C3—721no—
233C2×C3C3×#94711yescharacteristic
234C2×C3C3×#95012yes—
235D3—241no—
236D3—721no—
237C22—361no—
238C22—362no—
239C22C2×#638⋊C311yescharacteristic
240C4—22no—
241C4—42no—
242C4—361no—
243C4#8872212yes—
244C3—31no—
245C3—31no—
246C3—61no—
247C3C3×#506811yescharacteristic
248C2—721no—
249C2#64156⋊C311yescharacteristic
250C2#64159⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#125C2
socleC22×C3C2×C3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#488
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
4. #63637⋊C3C2semidirect
5. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
6. #10592767C3semidirect
16. C32×#125D3semidirect
17. C32×#125C2×C3semidirect
20. #63637C2×C3semidirect
22. C4×C3⋊(C3⋊C8)C2×C3semidirect
142. #125(C3⋊D3)⋊C3semidirect
147. H(3,𝔽3)#488semidirect
216. C32#5068semidirect
217. C32C3×#488semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64084⋊C32
#63636⋊C32
#63637⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#105927673
C3×#50683