Finite Group Databasebeta

← #13137313

Subgroups — #13137313

Order 1728.

357 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721716C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic
3C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4#11720307C212yesmaximal
5#11720306C212yesmaximal
6#10590314—31nomaximal
7#10590304—31nomaximal
8C42×H(3,𝔽3)C2211yescharacteristic
9C4⋊#1084C2212yes—
10#88540—22no—
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#3750⋊C3—22no—
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#64079D311yescharacteristic
16C3×#870—31no—
17C3×#870—31no—
18C3×#870—31no—
19C3×#870—61no—
20C4×C3⋊(C3⋊C8)D311yescharacteristic
21C3×C4×C3⋊C8—31no—
22#63576—32no—
23#63575—32no—
24#63574—32no—
25#63573—32no—
26C2×C4×H(3,𝔽3)D412yes—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—22no—
29C2×(C3⋊Dic3)⋊C3—22no—
30(C3⋊(C3⋊C8))⋊C3Q812yes—
31(C3⋊(C3⋊C8))⋊C3Q812yes—
32#5065—91no—
33#5065—91no—
34#4683—91no—
35C32×C42—21no—
36C32×C42C2×D311yescharacteristic
37C32×C42C2×D311yescharacteristic
38C4⋊(C3⋊Dic3)C2×D312yes—
39#3750—22no—
40C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
41C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
42C3×C4⋊Dic3—32no—
43C3×C4⋊Dic3—32no—
44C3×C4⋊Dic3—32no—
45C3×C4⋊Dic3—62no—
46C3×Dic3⋊C4—64no—
47C3×Dic3⋊C4—62no—
48C3×Dic3⋊C4—62no—
49C3×Dic3⋊C4—62no—
50C2×C3×C3⋊C8—31no—
51C2×C3×C3⋊C8—31no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)—22no—
54C4×H(3,𝔽3)C2×Q811yescharacteristic
55C4×H(3,𝔽3)C2×Q811yescharacteristic
56#1084—42no—
57(C3⋊Dic3)⋊C3—42no—
58C3×#125—91no—
59C3×#125—91no—
60#870—31no—
61#870—31no—
62#870—31no—
63#870—61no—
64#870—91no—
65C3×C4×C8—91no—
66#818—92no—
67#817—92no—
68#808—92no—
69#808—92no—
70#807—92no—
71#807—92no—
72C4×C3⋊C8—31no—
73C4×C3⋊C8—91no—
74C2×C32×C4—22no—
75C2×C32×C4—21no—
76C2×C32×C4D1212yes—
77C2×C32×C4#6712yes—
78C2×C32×C4C22×D311yescharacteristic
79C2×C32×C4C22×D311yescharacteristic
80C2×C3⋊Dic3—22no—
81C2×C3×Dic3—64no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C3⋊(C3⋊C8)Dic612yes—
86C3⋊(C3⋊C8)Dic612yes—
87C3×C3⋊C8—32no—
88C3×C3⋊C8—32no—
89#499—271noSylow
90C2×H(3,𝔽3)C4⋊Q811yescharacteristic
91C2×H(3,𝔽3)#13512yes—
92C2×C3×C8—91no—
93C2×C3×C8—91no—
94C3×C4⋊C4—92no—
95C3×C4⋊C4—92no—
96C3×C4⋊C4—182no—
97C3×C4⋊C4—182no—
98C3×C42—31no—
99C3×C42—31no—
100C3×C42—61no—
101C3×C42D3211yescharacteristic
102C4⋊Dic3—32no—
103C4⋊Dic3—32no—
104C4⋊Dic3—32no—
105C4⋊Dic3—62no—
106C4⋊Dic3—92no—
107Dic3⋊C4—62no—
108Dic3⋊C4—62no—
109Dic3⋊C4—64no—
110Dic3⋊C4—62no—
111Dic3⋊C4—182no—
112C2×C3⋊C8—31no—
113C2×C3⋊C8—31no—
114C2×C3⋊C8—91no—
115C2×C3⋊C8—91no—
116C22×C32—21no—
117C22×C32C2×D1211yescharacteristic
118C22×C32C2×#6711yescharacteristic
119C32×C4—22no—
120C32×C4—22no—
121C32×C4—21no—
122C32×C4—21no—
123C32×C4—42no—
124C32×C4C2×Dic611yescharacteristic
125C32×C4C2×Dic611yescharacteristic
126C32×C4D3×Q811yescharacteristic
127C32×C4D3×Q811yescharacteristic
128C3⋊Dic3—42no—
129C3×Dic3—124no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C3×Dic3—122no—
133#125—91no—
134#125—91no—
135#107—272no—
136#106—272no—
137C4×C8—91no—
138H(3,𝔽3)#49911yescharacteristic, complemented, Sylow
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—62no—
144C2×C3×C4—61no—
145C2×C3×C4—182no—
146C2×C3×C4—182no—
147C2×C3×C4#62612yes—
148C2×C3×C4C2×D3211yescharacteristic
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—64no—
152C2×Dic3—62no—
153C2×Dic3—182no—
154C3×C8—92no—
155C3×C8—92no—
156C3⋊C8—32no—
157C3⋊C8—32no—
158C3⋊C8—92no—
159C3⋊C8—92no—
160C2×C32—22no—
161C2×C32—21no—
162C2×C32#86911yescharacteristic
163C2×C32#90412yes—
164C2×C32#94411yescharacteristic
165C2×C32#95012yes—
166C2×C8—91no—
167C2×C8—91no—
168C4⋊C4—92no—
169C4⋊C4—92no—
170C4⋊C4—182no—
171C4⋊C4—182no—
172C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—61no—
176C22×C3C2×#62611yescharacteristic
177C3×C4—22no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—62no—
183C3×C4—62no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—122no—
187C3×C4—362no—
188C3×C4—362no—
189C3×C4D3×Dic611yescharacteristic
190C3×C4D3×Dic611yescharacteristic
191Dic3—122no—
192Dic3—122no—
193Dic3—124no—
194Dic3—122no—
195Dic3—362no—
196C32—21no—
197C32#468311yescharacteristic, complemented
198C32#506511yescharacteristic, complemented
199C2×C4—182no—
200C2×C4—182no—
201C2×C4#620012yes—
202C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
203C8—92no—
204C8—92no—
205C2×C3—32no—
206C2×C3—32no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—62no—
210C2×C3—61no—
211C2×C3#6382912yes—
212C2×C3#6386611yescharacteristic
213C22C2×#620011yescharacteristic
214C4—22no—
215C4—362no—
216C4—362no—
217C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
218C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
219C3—31no—
220C3—31no—
221C3—61no—
222C3#1059030411yescharacteristic
223C2#1172107312yes—
224C2#1172109011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#499
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
138. H(3,𝔽3)#499semidirect
197. C32#4683semidirect
198. C32#5065semidirect

Maximal subgroups

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

subgroupindex
#117217162
#64079⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#117203072
#117203062
#105903143
#105903043