Finite Group Databasebeta

← #13150613

Subgroups — #13150613

Order 1728.

1087 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1087 conjugacy classes of subgroups — too many to draw. Only its 219 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
1C22×#6229C211yesmaximal, characteristic
2#64135⋊C3C212yesmaximal
3#11721298C214yesmaximal
4#11721083C218yesmaximal, complemented
5#10594818—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C2212yes—
8C2×#6229C418yescomplemented
9C2×#6229C2214yes—
10#3757⋊C3C2214yes—
11C4⋊((C3⋊Dic3)⋊C3)C2214yes—
12C4×(C3⋊Dic3)⋊C3C2214yes—
13#88449C2218yes—
14#88447C2218yes—
15C22×C3×Dic6—32no—
16C22×C3×Dic6—61no—
17#64135D312yes—
18C3×#926—32no—
19#63955—38no—
20#63843—38no—
21Dic3×Dic6—38no—
22C23×H(3,𝔽3)C2311yescharacteristic
23C2×C4×H(3,𝔽3)C2×C414yes—
24C2×C4×H(3,𝔽3)C2312yes—
25C2×#1084C2×C418yes—
26C2×#1084C2314yes—
27#6229C2×C4116yes—
28C2×(C3⋊Dic3)⋊C3C2318yes—
29#5812—91no—
30#5516—92no—
31C22×C32×C4—21no—
32C22×C32×C4C2×D312yes—
33C22×C3×Dic3—34no—
34C22×C3×Dic3—62no—
35C2×C3×Dic6—316no—
36C2×C3×Dic6—34no—
37C2×C3×Dic6—34no—
38C2×C3×Dic6—68no—
39C2×C3×Dic6—64no—
40#3757C2×D314yes—
41C4⋊(C3⋊Dic3)C2×D314yes—
42C4×C3⋊Dic3C2×D314yes—
43C3×#217—34no—
44C3×C4⋊Dic3—34no—
45C3×C4×Dic3—34no—
46Dic3⋊Dic3—316no—
47Dic32—316no—
48C22×H(3,𝔽3)—22no—
49C22×H(3,𝔽3)C22×C412yes—
50C22×H(3,𝔽3)C2411yescharacteristic
51C4×H(3,𝔽3)C22×C414yes—
52#1084C22×C418yes—
53(C3⋊Dic3)⋊C3—28no—
54C22×C3×Q8—91no—
55C22×Dic6—32no—
56C22×Dic6—61no—
57C3×#117—92no—
58Q8×Dic3—98no—
59#926—32no—
60#926—92no—
61#926—94no—
62#887—98no—
63#877—98no—
64C4×Dic6—98no—
65C23×C32—21no—
66C23×C32C22×D312yes—
67C2×C32×C4—24no—
68C2×C32×C4—22no—
69C2×C32×C4C4×D318yes—
70C2×C32×C4C22×D312yes—
71C2×C32×C4C22×D312yes—
72C2×C3⋊Dic3C22×D318yes—
73C2×C3×Dic3—316no—
74C2×C3×Dic3—38no—
75C2×C3×Dic3—38no—
76C2×C3×Dic3—68no—
77C2×C3×Dic3—64no—
78C3×Dic6—316no—
79C3×Dic6—316no—
80C3×Dic6—616no—
81#519—271noSylow
82C2×H(3,𝔽3)—22no—
83C2×H(3,𝔽3)C23×C411yescharacteristic
84C2×H(3,𝔽3)2−1+412yes—
85C2×C3×Q8—98no—
86C2×C3×Q8—94no—
87C22×C3×C4—32no—
88C22×C3×C4—61no—
89C22×C3×C4—92no—
90C22×C3×C4D3211yescharacteristic
91C22×Dic3—34no—
92C22×Dic3—62no—
93C2×Dic6—34no—
94C2×Dic6—34no—
95C2×Dic6—316no—
96C2×Dic6—68no—
97C2×Dic6—64no—
98C3×C4⋊C4—94no—
99C3×C22⋊C4—94no—
100C3×C42—94no—
101#217—34no—
102#217—94no—
103C4⋊Dic3—34no—
104C4⋊Dic3—94no—
105C4⋊Dic3—98no—
106Dic3⋊C4—916no—
107C4×Dic3—34no—
108C4×Dic3—94no—
109C4×Dic3—98no—
110C4×Dic3—916no—
111C22×C32—24no—
112C22×C32—24no—
113C22×C32—21no—
114C22×C32—22no—
115C22×C32C2×C4×D314yes—
116C22×C32C23×D312yes—
117C32×C4—24no—
118C32×C4C2×C4×D314yes—
119C32×C4C2×C4×D314yes—
120C3⋊Dic3—28no—
121C3×Dic3—316no—
122C3×Dic3—68no—
123C3×Dic3—68no—
124C22×Q8—91no—
125C4×Q8—278no—
126#117—92no—
127#117—274no—
128H(3,𝔽3)#51911yescharacteristic, complemented, Sylow
129C23×C3—32no—
130C23×C3—61no—
131C23×C3C2×D3211yescharacteristic
132C3×Q8—916no—
133C2×C3×C4—38no—
134C2×C3×C4—32no—
135C2×C3×C4—32no—
136C2×C3×C4—64no—
137C2×C3×C4—62no—
138C2×C3×C4—94no—
139C2×C3×C4—98no—
140C2×C3×C4—98no—
141C2×C3×C4#62414yes—
142C2×C3×C4C2×D3212yes—
143C2×Dic3—38no—
144C2×Dic3—316no—
145C2×Dic3—38no—
146C2×Dic3—64no—
147C2×Dic3—68no—
148C2×Dic3—98no—
149Dic6—316no—
150Dic6—316no—
151Dic6—616no—
152C2×C32—24no—
153C2×C32—24no—
154C2×C32—22no—
155C2×C32—21no—
156C2×C32C22×C4×D312yes—
157C2×C32#101014yes—
158C2×Q8—98no—
159C2×Q8—94no—
160C22×C4—92no—
161C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
162C4⋊C4—94no—
163C4⋊C4—278no—
164C22⋊C4—94no—
165C42—94no—
166C42—278no—
167C22×C3—22no—
168C22×C3—32no—
169C22×C3—34no—
170C22×C3—64no—
171C22×C3—64no—
172C22×C3—61no—
173C22×C3—62no—
174C22×C3C2×#62412yes—
175C22×C3C22×D3211yescharacteristic
176C3×C4—34no—
177C3×C4—34no—
178C3×C4—64no—
179C3×C4—98no—
180C3×C4—188no—
181C3×C4C2×#62414yes—
182Dic3—316no—
183Dic3—68no—
184Dic3—68no—
185Dic3—188no—
186C32—21no—
187C32#551612yescomplemented
188C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
189Q8—916no—
190C2×C4—94no—
191C2×C4—98no—
192C2×C4—98no—
193C2×C4#619914yes—
194C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
195C2×C3—22no—
196C2×C3—32no—
197C2×C3—34no—
198C2×C3—64no—
199C2×C3—64no—
200C2×C3—61no—
201C2×C3—62no—
202C2×C3#6429812yes—
203C2×C3C22×#62411yescharacteristic
204C22—22no—
205C22C2×#619912yes—
206C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
207C4—98no—
208C4—188no—
209C4C2×#619914yes—
210C3—32no—
211C3—61no—
212C3#1059482811yescharacteristic
213C2—22no—
214C2#1172248312yes—
215C2C22×#619911yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#519
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11721083C2semidirect
8. C2×#6229C4semidirect
128. H(3,𝔽3)#519semidirect
187. C32#5516semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
#64135⋊C32
#117212982
#117210832
#105948183