Finite Group Databasebeta

← #13138803

Subgroups — #13138803

Order 1728.

425 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#11721144C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic, complemented
3#11720548C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#11720318C212yesmaximal, complemented
6#10590882—32nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
8C2×#6198—22no—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#88448—22no—
14#63911—32no—
15#63869—32no—
16#63648D312yescomplemented
17C3×#821—32no—
18C3×#815—32no—
19C3×#815—61no—
20#63564—32no—
21#63563—32no—
22C22×(C3⋊D3)⋊C3—22no—
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×#1084D412yes—
25C2×(C3⋊Dic3)⋊C3—22no—
26C4⋊((C3⋊D3)⋊C3)—22no—
27C4⋊((C3⋊D3)⋊C3)—22no—
28#6198—44no—
29C8×H(3,𝔽3)—21no—
30#6180—21no—
31#4764—92no—
32#4713—91no—
33C2×C4⋊(C3⋊D3)C2×D312yes—
34C2×C3×D12—32no—
35C2×#626—62no—
36C2×#626—62no—
37C2×C32×C8—21no—
38C2×C32×C8C2×D312yes—
39C4⋊(C3⋊Dic3)C2×D312yes—
40C3×C4⋊Dic3—32no—
41C3×C4×Dic3—32no—
42C3×C4×Dic3—61no—
43C2×C3×C3⋊C8—32no—
44C2×C3×C3⋊C8—61no—
45#3724—62no—
46Dic3⋊Dic3—62no—
47C22×H(3,𝔽3)C2×D411yescharacteristic
48C2×(C3⋊D3)⋊C3—42no—
49C2×(C3⋊D3)⋊C3—42no—
50C2×(C3⋊D3)⋊C3—42no—
51C4×H(3,𝔽3)#4111yescharacteristic
52C4×H(3,𝔽3)#4111yescharacteristic
53#1084—22no—
54(C3⋊Dic3)⋊C3—42no—
55#940—92no—
56#890—92no—
57#874—91no—
58#870—91no—
59C3×#102—92no—
60C3×#97—91no—
61#832—92no—
62#821—32no—
63#821—92no—
64#821—92no—
65#815—32no—
66#815—61no—
67#809—92no—
68C22×C3⋊D3—22no—
69C22×C3×D3—62no—
70C2×C32×C4—21no—
71C2×C32×C4C22×D312yes—
72C2×C3⋊Dic3—22no—
73C4⋊(C3⋊D3)—22no—
74C4⋊(C3⋊D3)—22no—
75C2×C3×Dic3—32no—
76C2×C3×Dic3—32no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C3×D12—62no—
80C3×D12—62no—
81#626—124no—
82#626—124no—
83C32×C8—22no—
84C32×C8—22no—
85C3×C3⋊C8—62no—
86C3×C3⋊C8—62no—
87#490—271noSylow
88C2×H(3,𝔽3)#12411yescharacteristic
89C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
90(C3⋊D3)⋊C3—82no—
91C2×C3×D4—92no—
92C2×#67—182no—
93C2×#67—182no—
94C2×D12—32no—
95C2×D12—92no—
96C2×D12—182no—
97C2×C3×C8—32no—
98C2×C3×C8—61no—
99C2×C3×C8—91no—
100C2×C3×C8D3211yescharacteristic
101C3×C4⋊C4—92no—
102C3×C42—91no—
103C4⋊Dic3—32no—
104C4⋊Dic3—92no—
105C4⋊Dic3—182no—
106Dic3⋊C4—182no—
107Dic3⋊C4—182no—
108C4×Dic3—32no—
109C4×Dic3—61no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—61no—
112C22×C32—21no—
113C22×C32D3×D412yes—
114C2×C3⋊D3—42no—
115C2×C3⋊D3—42no—
116C2×C3⋊D3—42no—
117C2×C3×D3—122no—
118C2×C3×D3—122no—
119C2×C3×D3—122no—
120C32×C4—21no—
121C32×C4—21no—
122C32×C4#23512yes—
123C32×C4#23712yes—
124C3⋊Dic3—42no—
125C3×Dic3—62no—
126C3×Dic3—62no—
127C3×Dic3—122no—
128C3×Dic3—122no—
129#127—271no—
130#125—271no—
131#102—92no—
132#102—272no—
133#97—91no—
134H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
135C23×C3—182no—
136C22×D3—62no—
137C22×D3—182no—
138C3×D4—182no—
139C3×D4—182no—
140C2×C3×C4—32no—
141C2×C3×C4—61no—
142C2×C3×C4—92no—
143C2×C3×C4—182no—
144C2×C3×C4C2×D3211yescharacteristic
145#67—364no—
146#67—364no—
147C2×Dic3—32no—
148C2×Dic3—32no—
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—182no—
152D12—62no—
153D12—62no—
154D12—182no—
155D12—182no—
156D12—364no—
157C3×C8—21no—
158C3×C8—62no—
159C3×C8—62no—
160C3×C8—181no—
161C3⋊C8—62no—
162C3⋊C8—62no—
163C2×C32—22no—
164C2×C32—21no—
165C2×C32#88512yes—
166C2×C32#91112yes—
167C2×C32#91412yes—
168C3⋊D3—82no—
169C3×D3—242no—
170C2×D4—92no—
171C2×D4—542no—
172C2×C8—91no—
173C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
174C4⋊C4—92no—
175C4⋊C4—542no—
176C42—91no—
177C22×C3—32no—
178C22×C3—61no—
179C22×C3—362no—
180C22×C3—362no—
181C22×C3—362no—
182C22×C3#380211yescharacteristic
183C2×D3—122no—
184C2×D3—122no—
185C2×D3—122no—
186C2×D3—362no—
187C2×D3—362no—
188C2×D3—362no—
189C3×C4—32no—
190C3×C4—32no—
191C3×C4—61no—
192C3×C4—61no—
193C3×C4—182no—
194C3×C4—362no—
195C3×C4#379811yescharacteristic
196C3×C4#380511yescharacteristic
197Dic3—62no—
198Dic3—62no—
199Dic3—122no—
200Dic3—122no—
201Dic3—362no—
202C32—21no—
203C32#476412yescomplemented
204C23—182no—
205D4—182no—
206D4—182no—
207D4—1084no—
208C2×C4—92no—
209C2×C4—182no—
210C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
211C8—21no—
212C8—181no—
213C2×C3—32no—
214C2×C3—32no—
215C2×C3—32no—
216C2×C3—62no—
217C2×C3—61no—
218C2×C3—722no—
219C2×C3#6379812yes—
220C2×C3#6391511yescharacteristic
221D3—242no—
222D3—722no—
223C22—362no—
224C22—362no—
225C22—362no—
226C22#8865511yescharacteristic
227C4—182no—
228C4—362no—
229C4#8865011yescharacteristic
230C4#8867211yescharacteristic
231C3—32no—
232C3—61no—
233C3#1059088811yescharacteristic
234C2—722no—
235C2#1172083912yes—
236C2#1172114511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radical#11721144C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#490
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721095C2semidirect
3. #11720548C2semidirect
4. #63648⋊C3C2semidirect
5. #11720318C2semidirect
16. #63648D3semidirect
134. H(3,𝔽3)#490semidirect
203. C32#4764semidirect

Maximal subgroups

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

subgroupindex
#117211442
#117210952
#117205482
#63648⋊C32
#117203182
#105908823