Finite Group Databasebeta

← #13158836

Subgroups — #13158836

Order 1728.

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

normal subgroups only

This group has 800 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
1#11723911C211yesmaximal, characteristic, complemented
2C32×#875C211yesmaximal, characteristic
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722038C212yesmaximal, complemented
5#11722032C212yesmaximal
6#10592755—31nomaximal
7#10590274—34nomaximal
8C2×C32×D12C2211yescharacteristic
9C2×C32×Dic6C2211yescharacteristic
10C33×C42C2211yescharacteristic
11C4⋊(C3⋊(C3⋊Dic3))C2212yes—
12#88857—22no—
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×(C2×D3)⋊C4—22no—
15C32×#124—31no—
16#64079—31no—
17#64079—34no—
18#64079—64no—
19C3×#875D314yes—
20#63662—32no—
21#63659—32no—
22C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
23C3×C4×C3⋊C8—34no—
24#63566—38no—
25#63563—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#5056—94no—
35#4697—91no—
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×D311yescharacteristic
43C32×C42C2×D314yes—
44C4⋊(C3⋊Dic3)—32no—
45C4⋊(C3⋊Dic3)—38no—
46C4⋊(C3⋊Dic3)—68no—
47#3750—616no—
48#3750—62no—
49#3750—68no—
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×(C2×D3)⋊C4—28no—
52C2×C3×C3⋊C8—38no—
53C22×C33C2×D411yescharacteristic
54C2×C32×D3—41no—
55C2×C32×D3—42no—
56C33×C4—22no—
57C33×C4#4112yes—
58C3⋊(C3⋊Dic3)—42no—
59C32×Dic3—41no—
60C3×#124—34no—
61#875C3⋊D311yescharacteristic
62#870—94no—
63#870—91no—
64#870—184no—
65C3×C4×C8—91no—
66#835—98no—
67#832—98no—
68#821—92no—
69#816—92no—
70C4×C3⋊C8—34no—
71C23×C32—61no—
72C22×C3×D3—24no—
73C32×Q8—62no—
74C32×D4—62no—
75C2×C32×C4—28no—
76C2×C32×C4—24no—
77C2×C32×C4—61no—
78C2×C32×C4D1212yes—
79C2×C32×C4#6718yes—
80C2×C32×C4C22×D314yes—
81C2×C32×C4C22×D311yescharacteristic
82C2×C3⋊Dic3—616no—
83C2×C3⋊Dic3—62no—
84C2×C3⋊Dic3—68no—
85C2×C3×Dic3—24no—
86C3×D12—28no—
87C3×Dic6—28no—
88C3⋊(C3⋊C8)—22no—
89C3×C3⋊C8—68no—
90#488—271noSylow
91C2×C33#12411yescharacteristic
92C2×C33#13512yes—
93C32×D3—81no—
94C2×C3×Q8—34no—
95C2×C3×D4—34no—
96C2×D12C2×C3⋊D311yescharacteristic
97C2×Dic6C2×C3⋊D311yescharacteristic
98C2×C3×C8—92no—
99C3×C22⋊C4—68no—
100C3×C42—24no—
101C3×C42D3214yes—
102C3×C42C2×C3⋊D311yescharacteristic
103(C2×D3)⋊C4—22no—
104C4⋊Dic3—98no—
105C4⋊Dic3—92no—
106C4⋊Dic3—188no—
107Dic3⋊C4—1816no—
108Dic3⋊C4—188no—
109Dic3⋊C4—182no—
110C2×C3⋊C8—38no—
111C22×C32—24no—
112C22×C32—121no—
113C22×C32—122no—
114C22×C32C2×D1211yescharacteristic
115C22×C32C2×#6714yes—
116C2×C3×D3—44no—
117C2×C3×D3—48no—
118C32×C4—22no—
119C32×C4—28no—
120C32×C4—28no—
121C32×C4—48no—
122C32×C4—121no—
123C32×C4#23512yes—
124C32×C4#23718yes—
125C3⋊Dic3—1216no—
126C3⋊Dic3—122no—
127C3⋊Dic3—128no—
128C3×Dic3—44no—
129#125—271no—
130#124—31no—
131#103—272no—
132#102—272no—
133C4×C8—91no—
134C33#48811yescharacteristic, complemented, Sylow
135C23×C3—64no—
136C22×D3—21no—
137C3×Q8—68no—
138C3×D4—68no—
139C2×C3×C4—28no—
140C2×C3×C4—24no—
141C2×C3×C4—64no—
142C2×C3×C4#62618yes—
143C2×C3×C4#63812yes—
144C2×C3×C4C2×D3214yes—
145C2×C3×C4C22×C3⋊D311yescharacteristic
146C2×Dic3—21no—
147C2×Dic3—1816no—
148C2×Dic3—188no—
149C2×Dic3—182no—
150D12—22no—
151Dic6—22no—
152C3×C8—182no—
153C3⋊C8—68no—
154C2×C32—28no—
155C2×C32—24no—
156C2×C32—241no—
157C2×C32#87511yescharacteristic
158C2×C32#90412yes—
159C2×C32#93614yes—
160C2×C32#95018yes—
161C3×D3—84no—
162C2×Q8—31no—
163C2×D4—31no—
164C2×C8—92no—
165C4⋊C4—272no—
166C4⋊C4—542no—
167C22⋊C4—62no—
168C42D3×C3⋊D311yescharacteristic
169C22×C3—24no—
170C22×C3—128no—
171C22×C3—124no—
172C22×C3C2×#62614yes—
173C22×C3C2×#63811yescharacteristic
174C2×D3—41no—
175C2×D3—42no—
176C3×C4—28no—
177C3×C4—22no—
178C3×C4—28no—
179C3×C4—48no—
180C3×C4—124no—
181C3×C4#379518yes—
182C3×C4#383012yes—
183Dic3—41no—
184Dic3—3616no—
185Dic3—368no—
186Dic3—362no—
187C32—24no—
188C32#469711yescharacteristic, complemented
189C32#505614yescomplemented
190C23—61no—
191Q8—62no—
192D4—62no—
193C2×C4—61no—
194C2×C4—542no—
195C2×C4#629112yes—
196C2×C4C2×D3×C3⋊D311yescharacteristic
197C8—182no—
198C2×C3—28no—
199C2×C3—24no—
200C2×C3—244no—
201C2×C3#6382918yes—
202C2×C3#6386014yes—
203C2×C3#6414511yescharacteristic
204C2×C3#6415912yes—
205D3—81no—
206C22—122no—
207C22—121no—
208C22C2×#629111yescharacteristic
209C4—22no—
210C4—121no—
211C4—1082no—
212C4#8901412yes—
213C3—24no—
214C3#1059027414yescomplemented
215C3#1059275511yescharacteristic, complemented
216C2—241no—
217C2#1172337712yes—
218C2#1172342911yescharacteristic
subgroupG/H
centerC22C2×#6291
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. #11723911C2semidirect
3. C3×C4×C3⋊(C3⋊C8)C2semidirect
4. #11722038C2semidirect
22. C4×C3⋊(C3⋊C8)D3semidirect
134. C33#488semidirect
188. C32#4697semidirect
189. C32#5056semidirect
214. C3#10590274semidirect
215. C3#10592755semidirect

Maximal subgroups

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

subgroupindex
#117239112
C32×#8752
C3×C4×C3⋊(C3⋊C8)2
#117220382
#117220322
#105927553
#105902743