Finite Group Databasebeta

← #13140061

Subgroups — #13140061

Order 1728.

731 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 155 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×#88447C212yesmaximal
3#63658⋊C3C212yesmaximal
4#11720334C212yesmaximal
5#10591239—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C2214yes—
10C4×#1084C414yescomplemented
11#88447C418yescomplemented
12C2×C3×C4×Dic3—32no—
13C2×C3×C4×Dic3—61no—
14C2×Dic32—34no—
15#63658D312yes—
16C3×C3⋊#95—32no—
17#63579—34no—
18C23×H(3,𝔽3)C2311yescharacteristic
19C2×C4×H(3,𝔽3)—22no—
20C2×C4×H(3,𝔽3)C2×C412yes—
21C2×#1084—22no—
22C2×#1084C2×C412yes—
23C2×#1084C2×C414yes—
24C2×#1084C2×C412yes—
25C2×(C3⋊Dic3)⋊C3—24no—
26C2×(C3⋊Dic3)⋊C3—24no—
27C2×(C3⋊Dic3)⋊C3C2×C418yes—
28#4932—91no—
29#4648—92no—
30C22×C32×C4—21no—
31C22×C32×C4C2×D312yes—
32C22×C3⋊Dic3C2×D314yes—
33C22×C3×Dic3—32no—
34C22×C3×Dic3—32no—
35C22×C3×Dic3—34no—
36C22×C3×Dic3—61no—
37C22×C3×Dic3—61no—
38C3×C4×Dic3—38no—
39C3×C4×Dic3—64no—
40Dic32—316no—
41C22×H(3,𝔽3)C22×C412yes—
42C22×H(3,𝔽3)C22×C411yescharacteristic
43C22×H(3,𝔽3)#4112yes—
44C22×H(3,𝔽3)#4112yes—
45C4×H(3,𝔽3)—22no—
46#1084—22no—
47#1084C4214yes—
48(C3⋊Dic3)⋊C3—28no—
49C2×C3×C42—91no—
50C2×C4×Dic3—32no—
51C2×C4×Dic3—61no—
52C2×C4×Dic3—92no—
53C2×C4×Dic3—94no—
54C3×#95—92no—
55C3⋊#95—32no—
56C3⋊#95—92no—
57C3⋊#95—92no—
58C3⋊#95—94no—
59C23×C32—21no—
60C23×C32C22×D312yes—
61C2×C32×C4—24no—
62C2×C32×C4—24no—
63C2×C32×C4—22no—
64C2×C32×C4C4×D314yes—
65C2×C3⋊Dic3—24no—
66C2×C3⋊Dic3—24no—
67C2×C3⋊Dic3C4×D318yes—
68C2×C3×Dic3—38no—
69C2×C3×Dic3—34no—
70C2×C3×Dic3—38no—
71C2×C3×Dic3—34no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—64no—
75C2×C3×Dic3—64no—
76C2×C3×Dic3—64no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—64no—
79C2×C3×Dic3—62no—
80#376—271noSylow
81C2×H(3,𝔽3)C2×C4211yescharacteristic
82C2×H(3,𝔽3)#11711yescharacteristic
83C2×H(3,𝔽3)#11714yes—
84C2×H(3,𝔽3)#11711yescharacteristic
85C22×C3×C4—32no—
86C22×C3×C4—61no—
87C22×C3×C4—91no—
88C22×C3×C4—94no—
89C22×C3×C4—91no—
90C22×C3×C4D3211yescharacteristic
91C22×Dic3—32no—
92C22×Dic3—32no—
93C22×Dic3—34no—
94C22×Dic3—61no—
95C22×Dic3—61no—
96C22×Dic3—94no—
97C3×C42—94no—
98C4×Dic3—38no—
99C4×Dic3—64no—
100C4×Dic3—98no—
101C4×Dic3—916no—
102C22×C32—22no—
103C22×C32—22no—
104C22×C32—22no—
105C22×C32—21no—
106C22×C32C2×C4×D314yes—
107C22×C32C2×C4×D312yes—
108C22×C32#23712yes—
109C22×C32#23714yes—
110C22×C32#23912yes—
111C32×C4—24no—
112C32×C4—24no—
113C3⋊Dic3—28no—
114C3×Dic3—38no—
115C3×Dic3—64no—
116C3×Dic3—64no—
117C3×Dic3—68no—
118C3×Dic3—64no—
119C2×C42—91no—
120C2×C42—272no—
121#95—92no—
122#95—272no—
123H(3,𝔽3)#37611yescharacteristic, complemented, Sylow
124C23×C3—32no—
125C23×C3—61no—
126C23×C3C2×D3211yescharacteristic
127C2×C3×C4—22no—
128C2×C3×C4—34no—
129C2×C3×C4—64no—
130C2×C3×C4—64no—
131C2×C3×C4—62no—
132C2×C3×C4—94no—
133C2×C3×C4—92no—
134C2×C3×C4—98no—
135C2×C3×C4—92no—
136C2×C3×C4—184no—
137C2×C3×C4—184no—
138C2×C3×C4—182no—
139C2×C3×C4#62412yes—
140C2×Dic3—34no—
141C2×Dic3—38no—
142C2×Dic3—34no—
143C2×Dic3—38no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—64no—
147C2×Dic3—62no—
148C2×Dic3—62no—
149C2×Dic3—64no—
150C2×Dic3—64no—
151C2×Dic3—64no—
152C2×Dic3—98no—
153C2×Dic3—184no—
154C2×Dic3—184no—
155C2×C32—21no—
156C2×C32—24no—
157C2×C32—21no—
158C2×C32—21no—
159C2×C32C42×D312yes—
160C2×C32#87714yes—
161C2×C32#87712yes—
162C2×C32#89212yes—
163C2×C32#89214yes—
164C22×C4—91no—
165C22×C4—94no—
166C22×C4—91no—
167C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
168C42—94no—
169C42—278no—
170C22×C3—32no—
171C22×C3—32no—
172C22×C3—34no—
173C22×C3—34no—
174C22×C3—32no—
175C22×C3—62no—
176C22×C3—62no—
177C22×C3—62no—
178C22×C3—61no—
179C22×C3#379612yes—
180C22×C3C4×D3212yes—
181C22×C3#380512yes—
182C22×C3C2×#62411yescharacteristic
183C3×C4—22no—
184C3×C4—64no—
185C3×C4—64no—
186C3×C4—94no—
187C3×C4—188no—
188C3×C4—182no—
189Dic3—38no—
190Dic3—64no—
191Dic3—64no—
192Dic3—64no—
193Dic3—68no—
194Dic3—188no—
195C32—21no—
196C32#464812yescomplemented
197C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
198C2×C4—22no—
199C2×C4—94no—
200C2×C4—92no—
201C2×C4—98no—
202C2×C4—92no—
203C2×C4—182no—
204C2×C4—184no—
205C2×C4—184no—
206C2×C4#619912yes—
207C2×C3—32no—
208C2×C3—32no—
209C2×C3—34no—
210C2×C3—34no—
211C2×C3—32no—
212C2×C3—61no—
213C2×C3—61no—
214C2×C3—64no—
215C2×C3—61no—
216C2×C3#6384911yescharacteristic
217C2×C3#6387814yes—
218C2×C3C4×#62411yescharacteristic
219C2×C3#6395711yescharacteristic
220C22#8864912yes—
221C22C4×((C3⋊D3)⋊C3)⋊C212yes—
222C22#8867212yes—
223C22C2×#619911yescharacteristic
224C4—22no—
225C4—94no—
226C4—182no—
227C4—188no—
228C3—32no—
229C3—61no—
230C3#1059125211yescharacteristic
231C2#1172108911yescharacteristic
232C2#1172112114yes—
233C2C4×#619911yescharacteristic
234C2#1172129811yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
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)#376
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
10. C4×#1084C4semidirect
11. #88447C4semidirect
123. H(3,𝔽3)#376semidirect
196. C32#4648semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×#884472
#63658⋊C32
#117203342
#105912393