Finite Group Databasebeta

← #13140131

Subgroups — #13140131

Order 1728.

759 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 759 conjugacy classes of subgroups — too many to draw. Only its 179 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
1C2×C4×#1084C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C212yesmaximal
3C2×#88449C214yesmaximal
4#10591349—32nomaximal
5C22×C4×H(3,𝔽3)C2211yescharacteristic
6C22×#1084C2212yes—
7C22×(C3⋊Dic3)⋊C3C2214yes—
8C4×#1084C414yescomplemented
9C4⋊((C3⋊Dic3)⋊C3)—24no—
10#88449—28no—
11C2×C4⋊(C3⋊Dic3)D312yes—
12C2×C3×C4⋊Dic3—32no—
13C2×C3×C4×Dic3—32no—
14C2×C3×C4×Dic3—61no—
15C2×Dic3⋊Dic3—38no—
16C23×H(3,𝔽3)C2311yescharacteristic
17C2×C4×H(3,𝔽3)C2×C412yes—
18C2×C4×H(3,𝔽3)D412yes—
19C2×C4×H(3,𝔽3)Q812yes—
20C2×#1084C2×C414yes—
21C2×#1084D414yes—
22C2×#1084Q814yes—
23C2×(C3⋊Dic3)⋊C3—28no—
24C2×(C3⋊Dic3)⋊C3—24no—
25#4975—92no—
26#4935—91no—
27C22×C32×C4—21no—
28C22×C32×C4C2×D312yes—
29C22×C3⋊Dic3C2×D314yes—
30C22×C3×Dic3—34no—
31C22×C3×Dic3—34no—
32C22×C3×Dic3—62no—
33C4⋊(C3⋊Dic3)—24no—
34C3×C4⋊Dic3—64no—
35C3×C4×Dic3—38no—
36C3×C4×Dic3—64no—
37Dic3⋊Dic3—616no—
38C22×H(3,𝔽3)C22×C411yescharacteristic
39C22×H(3,𝔽3)C2×D412yes—
40C22×H(3,𝔽3)C2×D411yescharacteristic
41C22×H(3,𝔽3)C2×Q811yescharacteristic
42C22×H(3,𝔽3)C2×Q812yes—
43C4×H(3,𝔽3)C4⋊C414yescomplemented
44#1084C4⋊C418yescomplemented
45(C3⋊Dic3)⋊C3—44no—
46C2×C3×C4⋊C4—92no—
47C2×C3×C42—91no—
48C2×C4⋊Dic3—32no—
49C2×C4⋊Dic3—92no—
50C2×C4⋊Dic3—94no—
51C2×Dic3⋊C4—98no—
52C2×C4×Dic3—32no—
53C2×C4×Dic3—61no—
54C23×C32—21no—
55C23×C32C22×D312yes—
56C2×C32×C4—22no—
57C2×C32×C4—22no—
58C2×C32×C4—22no—
59C2×C32×C4Dic614yes—
60C2×C32×C4C4×D314yes—
61C2×C32×C4#6714yes—
62C2×C3⋊Dic3—28no—
63C2×C3⋊Dic3—24no—
64C2×C3×Dic3—38no—
65C2×C3×Dic3—38no—
66C2×C3×Dic3—38no—
67C2×C3×Dic3—64no—
68C2×C3×Dic3—64no—
69C2×C3×Dic3—64no—
70C2×C3×Dic3—68no—
71C2×C3×Dic3—64no—
72#384—271noSylow
73C2×H(3,𝔽3)C2×C4⋊C412yes—
74C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
75C2×H(3,𝔽3)#12711yescharacteristic
76C2×H(3,𝔽3)C4⋊Q812yes—
77C2×H(3,𝔽3)C4⋊Q811yescharacteristic
78C22×C3×C4—32no—
79C22×C3×C4—61no—
80C22×C3×C4—92no—
81C22×C3×C4—94no—
82C22×C3×C4D3211yescharacteristic
83C22×Dic3—34no—
84C22×Dic3—34no—
85C22×Dic3—62no—
86C22×Dic3—94no—
87C3×C4⋊C4—184no—
88C3×C42—94no—
89C4⋊Dic3—64no—
90C4⋊Dic3—184no—
91C4⋊Dic3—188no—
92Dic3⋊C4—1816no—
93C4×Dic3—38no—
94C4×Dic3—64no—
95C22×C32—22no—
96C22×C32—22no—
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32C2×Dic612yes—
101C22×C32C2×C4×D312yes—
102C22×C32D3×D414yes—
103C22×C32D3×Q814yes—
104C22×C32C2×#6712yes—
105C32×C4—24no—
106C32×C4Dic3⋊C418yescomplemented
107C3⋊Dic3—44no—
108C3×Dic3—316no—
109C3×Dic3—68no—
110C3×Dic3—124no—
111C2×C4⋊C4—92no—
112C2×C4⋊C4—274no—
113C2×C42—91no—
114H(3,𝔽3)#38411yescharacteristic, complemented, Sylow
115C23×C3—32no—
116C23×C3—61no—
117C23×C3C2×D3211yescharacteristic
118C2×C3×C4—34no—
119C2×C3×C4—34no—
120C2×C3×C4—34no—
121C2×C3×C4—62no—
122C2×C3×C4—62no—
123C2×C3×C4—62no—
124C2×C3×C4—94no—
125C2×C3×C4—94no—
126C2×C3×C4—94no—
127C2×C3×C4—188no—
128C2×C3×C4—184no—
129C2×C3×C4#62412yes—
130C2×C3×C4#62512yes—
131C2×C3×C4#62712yes—
132C2×Dic3—38no—
133C2×Dic3—38no—
134C2×Dic3—38no—
135C2×Dic3—64no—
136C2×Dic3—64no—
137C2×Dic3—64no—
138C2×Dic3—68no—
139C2×Dic3—64no—
140C2×Dic3—188no—
141C2×Dic3—184no—
142C2×C32—21no—
143C2×C32—22no—
144C2×C32—21no—
145C2×C32—22no—
146C2×C32—21no—
147C2×C32#88814yes—
148C2×C32D3×C4⋊C414yes—
149C2×C32C2×Dic3⋊C412yes—
150C2×C32#94012yes—
151C2×C32#94412yes—
152C22×C4—92no—
153C22×C4—94no—
154C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
155C4⋊C4—184no—
156C4⋊C4—548no—
157C42—94no—
158C22×C3—34no—
159C22×C3—34no—
160C22×C3—32no—
161C22×C3—32no—
162C22×C3—32no—
163C22×C3—62no—
164C22×C3—62no—
165C22×C3—61no—
166C22×C3—61no—
167C22×C3—61no—
168C22×C3#379712yes—
169C22×C3#380212yes—
170C22×C3C2×#62411yescharacteristic
171C22×C3C2×#62511yescharacteristic
172C22×C3C2×#62711yescharacteristic
173C3×C4—38no—
174C3×C4—64no—
175C3×C4—98no—
176C3×C4—364no—
177C3×C4#372414yes—
178Dic3—316no—
179Dic3—68no—
180Dic3—124no—
181Dic3—364no—
182C32—21no—
183C32#497512yescomplemented
184C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
185C2×C4—94no—
186C2×C4—94no—
187C2×C4—94no—
188C2×C4—188no—
189C2×C4—184no—
190C2×C4#619612yes—
191C2×C4#619812yes—
192C2×C4#619912yes—
193C2×C3—32no—
194C2×C3—32no—
195C2×C3—34no—
196C2×C3—32no—
197C2×C3—34no—
198C2×C3—61no—
199C2×C3—61no—
200C2×C3—62no—
201C2×C3—61no—
202C2×C3—62no—
203C2×C3#6387311yescharacteristic
204C2×C3#6390012yes—
205C2×C3#6391411yescharacteristic
206C2×C3#6391912yes—
207C2×C3C2×#372411yescharacteristic
208C22#8865112yes—
209C22#8865512yes—
210C22C2×#619611yescharacteristic
211C22C2×#619811yescharacteristic
212C22C2×#619911yescharacteristic
213C4—98no—
214C4—364no—
215C4#8844914yescomplemented
216C3—32no—
217C3—61no—
218C3#1059135911yescharacteristic
219C2#1172109911yescharacteristic
220C2#1172111712yes—
221C2#1172114411yescharacteristic
222C2#1172115212yes—
223C2C2×#8844911yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#384
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. C4×#1084C4semidirect
43. C4×H(3,𝔽3)C4⋊C4semidirect
44. #1084C4⋊C4semidirect
106. C32×C4Dic3⋊C4semidirect
114. H(3,𝔽3)#384semidirect
183. C32#4975semidirect
215. C4#88449semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884492
#105913493