Finite Group Databasebeta

← #13158831

Subgroups — #13158831

Order 1728.

752 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 164 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×#64079C211yesmaximal, characteristic, complemented
2C32×#875C211yesmaximal, characteristic
3C4×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4#11722037C212yesmaximal, complemented
5#11722031C212yesmaximal
6#10592755—31nomaximal
7#10590278—34nomaximal
8C2×C32×D12C2211yescharacteristic
9C2×C32×Dic6C2211yescharacteristic
10C33×C42C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C3×#3750—22no—
14C32×(C2×D3)⋊C4—22no—
15C32×#124—31no—
16#64079D311yescharacteristic, complemented
17C3×#875D314yes—
18C3×#870—34no—
19#63662—32no—
20#63659—32no—
21C4×C3⋊(C3⋊C8)—31no—
22C4×C3⋊(C3⋊C8)—34no—
23C4×C3⋊(C3⋊C8)—64no—
24#63565—38no—
25#63562—38no—
26C22×C32×D3—21no—
27C2×C33×C4D412yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3—22no—
30C2×C32×Dic3—21no—
31C32×D12—22no—
32C32×Dic6—22no—
33C3⋊(C3⋊(C3⋊C8))—22no—
34#5068—91no—
35#5056—94no—
36C2×C32×Q8—31no—
37C2×C32×D4—31no—
38C2×C3×D12C2×D314yes—
39C2×C3×Dic6C2×D314yes—
40C32×C22⋊C4—62no—
41C32×C42—24no—
42C32×C42C2×D314yes—
43C32×C42C2×D311yescharacteristic
44C4⋊(C3⋊Dic3)C2×D312yes—
45#3750—22no—
46C2×C3⋊(C3⋊C8)—32no—
47C2×C3⋊(C3⋊C8)—38no—
48C2×C3⋊(C3⋊C8)—68no—
49C3×(C2×D3)⋊C4—28no—
50C3×C4⋊Dic3—38no—
51C3×Dic3⋊C4—68no—
52C22×C33C2×D411yescharacteristic
53C2×C32×D3—41no—
54C2×C32×D3—42no—
55C33×C4—22no—
56C33×C4#4112yes—
57C3×C3⋊Dic3—42no—
58C32×Dic3—41no—
59C3×#125—91no—
60C3×#124—34no—
61#875C3⋊D311yescharacteristic
62#870—34no—
63#835—98no—
64#832—98no—
65#810—92no—
66#809—92no—
67C4×C3⋊C8—94no—
68C4×C3⋊C8—91no—
69C4×C3⋊C8—184no—
70C23×C32—61no—
71C22×C3×D3—24no—
72C32×Q8—62no—
73C32×D4—62no—
74C2×C32×C4—28no—
75C2×C32×C4—24no—
76C2×C32×C4—61no—
77C2×C32×C4#6718yes—
78C2×C32×C4#6712yes—
79C2×C32×C4C22×D314yes—
80C2×C32×C4C22×D311yescharacteristic
81C2×C3⋊Dic3—22no—
82C2×C3×Dic3—24no—
83C2×C3×Dic3—68no—
84C3×D12—28no—
85C3×Dic6—28no—
86C3⋊(C3⋊C8)—616no—
87C3⋊(C3⋊C8)—68no—
88C3⋊(C3⋊C8)—62no—
89#488—271noSylow
90C2×C33#12411yescharacteristic
91C2×C33#13512yes—
92C32×D3—81no—
93C2×C3×Q8—34no—
94C2×C3×D4—34no—
95C2×D12C2×C3⋊D311yescharacteristic
96C2×Dic6C2×C3⋊D311yescharacteristic
97C3×C4⋊C4—92no—
98C3×C4⋊C4—182no—
99C3×C22⋊C4—68no—
100C3×C42—24no—
101C3×C42D3214yes—
102C3×C42C2×C3⋊D311yescharacteristic
103(C2×D3)⋊C4—22no—
104C4⋊Dic3—38no—
105Dic3⋊C4—68no—
106C2×C3⋊C8—98no—
107C2×C3⋊C8—92no—
108C2×C3⋊C8—188no—
109C22×C32—24no—
110C22×C32—121no—
111C22×C32—122no—
112C22×C32C2×#6714yes—
113C22×C32C2×#6711yescharacteristic
114C2×C3×D3—44no—
115C2×C3×D3—48no—
116C32×C4—216no—
117C32×C4—28no—
118C32×C4—22no—
119C32×C4—28no—
120C32×C4—121no—
121C32×C4#23718yes—
122C32×C4#23912yes—
123C3⋊Dic3—42no—
124C3×Dic3—44no—
125C3×Dic3—128no—
126#125—91no—
127#124—31no—
128#103—272no—
129#102—272no—
130C4×C8—271no—
131C33#48811yescharacteristic, complemented, Sylow
132C23×C3—64no—
133C22×D3—21no—
134C3×Q8—68no—
135C3×D4—68no—
136C2×C3×C4—28no—
137C2×C3×C4—24no—
138C2×C3×C4—64no—
139C2×C3×C4—182no—
140C2×C3×C4#62518yes—
141C2×C3×C4#63812yes—
142C2×C3×C4C2×D3214yes—
143C2×C3×C4C22×C3⋊D311yescharacteristic
144C2×Dic3—21no—
145C2×Dic3—68no—
146D12—22no—
147Dic6—22no—
148C3⋊C8—1816no—
149C3⋊C8—188no—
150C3⋊C8—182no—
151C2×C32—28no—
152C2×C32—24no—
153C2×C32—241no—
154C2×C32#93614yes—
155C2×C32#94711yescharacteristic
156C2×C32#95018yes—
157C2×C32#95012yes—
158C3×D3—84no—
159C2×Q8—31no—
160C2×D4—31no—
161C2×C8—272no—
162C4⋊C4—92no—
163C4⋊C4—182no—
164C22⋊C4—62no—
165C42D3×C3⋊D311yescharacteristic
166C22×C3—24no—
167C22×C3—128no—
168C22×C3—124no—
169C22×C3C2×#62514yes—
170C22×C3C2×#63811yescharacteristic
171C2×D3—41no—
172C2×D3—42no—
173C3×C4—216no—
174C3×C4—28no—
175C3×C4—22no—
176C3×C4—28no—
177C3×C4—124no—
178C3×C4—362no—
179C3×C4#379618yes—
180C3×C4#383012yes—
181Dic3—41no—
182Dic3—128no—
183C32—24no—
184C32#505614yescomplemented
185C32#506811yescharacteristic, complemented
186C23—61no—
187Q8—62no—
188D4—62no—
189C2×C4—61no—
190C2×C4—182no—
191C2×C4#629012yes—
192C2×C4C2×D3×C3⋊D311yescharacteristic
193C8—542no—
194C2×C3—28no—
195C2×C3—24no—
196C2×C3—244no—
197C2×C3#6382218yes—
198C2×C3#6386314yes—
199C2×C3#6414511yescharacteristic
200C2×C3#6415912yes—
201D3—81no—
202C22—122no—
203C22—121no—
204C22C2×#629011yescharacteristic
205C4—22no—
206C4—121no—
207C4—362no—
208C4#8901512yes—
209C3—24no—
210C3#1059027814yescomplemented
211C3#1059275511yescharacteristic, complemented
212C2—241no—
213C2#1172336312yes—
214C2#1172343411yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC42D3×C3⋊D3
hypocenterC33#488
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×#64079C2semidirect
3. C4×C3⋊(C3⋊(C3⋊C8))C2semidirect
4. #11722037C2semidirect
16. #64079D3semidirect
131. C33#488semidirect
184. C32#5056semidirect
185. C32#5068semidirect
210. C3#10590278semidirect
211. C3#10592755semidirect

Maximal subgroups

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

subgroupindex
C3×#640792
C32×#8752
C4×C3⋊(C3⋊(C3⋊C8))2
#117220372
#117220312
#105927553
#105902783