Finite Group Databasebeta

← #13140122

Subgroups — #13140122

Order 1728.

787 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 163 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×#88449C212yesmaximal
3C2×#88447C212yesmaximal
4#63658⋊C3C212yesmaximal
5#10591242—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C2214yes—
10C4×#1084C414yescomplemented
11#88449C418yescomplemented
12#88447—24no—
13C2×C3×C4×Dic3—32no—
14C2×C3×C4×Dic3—61no—
15C2×Dic3⋊Dic3—34no—
16C2×Dic32—34no—
17#63658D312yes—
18C3×C3⋊#95—32no—
19C23×H(3,𝔽3)C2311yescharacteristic
20C2×C4×H(3,𝔽3)—22no—
21C2×C4×H(3,𝔽3)C2×C412yes—
22C2×#1084C2×C414yes—
23C2×#1084C2×C412yes—
24C2×#1084C2×C412yes—
25C2×#1084D412yes—
26C2×#1084Q812yes—
27C2×(C3⋊Dic3)⋊C3—24no—
28C2×(C3⋊Dic3)⋊C3—24no—
29C2×(C3⋊Dic3)⋊C3C2×C418yes—
30C4×C4⋊Dic3—91no—
31#4649—92no—
32C22×C32×C4—21no—
33C22×C32×C4C2×D312yes—
34C22×C3⋊Dic3C2×D314yes—
35C22×C3×Dic3—32no—
36C22×C3×Dic3—32no—
37C22×C3×Dic3—34no—
38C22×C3×Dic3—61no—
39C22×C3×Dic3—61no—
40C3×C4×Dic3—38no—
41C3×C4×Dic3—64no—
42Dic3⋊Dic3—316no—
43Dic32—68no—
44C22×H(3,𝔽3)C22×C411yescharacteristic
45C22×H(3,𝔽3)C22×C412yes—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C22×H(3,𝔽3)C2×Q811yescharacteristic
48C22×H(3,𝔽3)#4112yes—
49C4×H(3,𝔽3)—22no—
50#1084C4214yescomplemented
51#1084C4⋊C414yescomplemented
52(C3⋊Dic3)⋊C3—28no—
53C2×C3×C42—91no—
54C2×C4⋊Dic3—92no—
55C2×Dic3⋊C4—94no—
56C2×C4×Dic3—32no—
57C2×C4×Dic3—61no—
58C2×C4×Dic3—92no—
59C2×C4×Dic3—94no—
60C3×#95—92no—
61C3⋊#95—32no—
62C3⋊#95—92no—
63C23×C32—21no—
64C23×C32C22×D312yes—
65C2×C32×C4—24no—
66C2×C32×C4—24no—
67C2×C32×C4—22no—
68C2×C32×C4C4×D314yes—
69C2×C3⋊Dic3—24no—
70C2×C3⋊Dic3—24no—
71C2×C3⋊Dic3C4×D318yes—
72C2×C3×Dic3—38no—
73C2×C3×Dic3—34no—
74C2×C3×Dic3—34no—
75C2×C3×Dic3—34no—
76C2×C3×Dic3—34no—
77C2×C3×Dic3—38no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—64no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—64no—
84C2×C3×Dic3—64no—
85C4×C4⋊C4—271noSylow
86C2×H(3,𝔽3)C2×C4211yescharacteristic
87C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
88C2×H(3,𝔽3)#11711yescharacteristic
89C2×H(3,𝔽3)C4×D412yes—
90C2×H(3,𝔽3)C4×Q812yes—
91C22×C3×C4—32no—
92C22×C3×C4—61no—
93C22×C3×C4—91no—
94C22×C3×C4—91no—
95C22×C3×C4—94no—
96C22×C3×C4D3211yescharacteristic
97C22×Dic3—32no—
98C22×Dic3—32no—
99C22×Dic3—34no—
100C22×Dic3—61no—
101C22×Dic3—61no—
102C22×Dic3—94no—
103C3×C42—94no—
104C4⋊Dic3—98no—
105Dic3⋊C4—916no—
106C4×Dic3—38no—
107C4×Dic3—64no—
108C4×Dic3—188no—
109C4×Dic3—184no—
110C22×C32—21no—
111C22×C32—22no—
112C22×C32—21no—
113C22×C32—22no—
114C22×C32—21no—
115C22×C32C2×C4×D314yes—
116C22×C32C2×C4×D312yes—
117C22×C32D3×D412yes—
118C22×C32#23714yes—
119C22×C32D3×Q812yes—
120C32×C4—24no—
121C32×C4—24no—
122C3⋊Dic3—28no—
123C3×Dic3—38no—
124C3×Dic3—38no—
125C3×Dic3—64no—
126C3×Dic3—64no—
127C3×Dic3—68no—
128C2×C4⋊C4—272no—
129C2×C42—91no—
130C2×C42—272no—
131#95—92no—
132H(3,𝔽3)C4×C4⋊C411yescharacteristic, complemented, Sylow
133C23×C3—32no—
134C23×C3—61no—
135C23×C3C2×D3211yescharacteristic
136C2×C3×C4—22no—
137C2×C3×C4—34no—
138C2×C3×C4—64no—
139C2×C3×C4—64no—
140C2×C3×C4—62no—
141C2×C3×C4—92no—
142C2×C3×C4—94no—
143C2×C3×C4—92no—
144C2×C3×C4—92no—
145C2×C3×C4—92no—
146C2×C3×C4—98no—
147C2×C3×C4—184no—
148C2×C3×C4—184no—
149C2×C3×C4#62412yes—
150C2×Dic3—34no—
151C2×Dic3—34no—
152C2×Dic3—34no—
153C2×Dic3—38no—
154C2×Dic3—34no—
155C2×Dic3—38no—
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—64no—
160C2×Dic3—62no—
161C2×Dic3—64no—
162C2×Dic3—64no—
163C2×Dic3—98no—
164C2×Dic3—184no—
165C2×Dic3—184no—
166C2×C32—22no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32—22no—
170C2×C32—21no—
171C2×C32C42×D312yes—
172C2×C32#87712yes—
173C2×C32#67⋊C414yes—
174C2×C32#88714yes—
175C2×C32D3×C4⋊C412yes—
176C22×C4—91no—
177C22×C4—91no—
178C22×C4—94no—
179C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
180C4⋊C4—278no—
181C42—94no—
182C42—544no—
183C22×C3—32no—
184C22×C3—34no—
185C22×C3—32no—
186C22×C3—34no—
187C22×C3—32no—
188C22×C3—61no—
189C22×C3—62no—
190C22×C3—61no—
191C22×C3—62no—
192C22×C3—61no—
193C22×C3#379711yescharacteristic
194C22×C3C4×D3212yes—
195C22×C3#380211yescharacteristic
196C22×C3#380512yes—
197C22×C3C2×#62411yescharacteristic
198C3×C4—22no—
199C3×C4—64no—
200C3×C4—64no—
201C3×C4—94no—
202C3×C4—94no—
203C3×C4—188no—
204Dic3—38no—
205Dic3—38no—
206Dic3—64no—
207Dic3—64no—
208Dic3—68no—
209Dic3—188no—
210C32—21no—
211C32#464912yescomplemented
212C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
213C2×C4—22no—
214C2×C4—92no—
215C2×C4—94no—
216C2×C4—92no—
217C2×C4—92no—
218C2×C4—92no—
219C2×C4—98no—
220C2×C4—184no—
221C2×C4—184no—
222C2×C4#619912yes—
223C2×C3—32no—
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—32no—
227C2×C3—32no—
228C2×C3—61no—
229C2×C3—62no—
230C2×C3—62no—
231C2×C3—61no—
232C2×C3—61no—
233C2×C3#6383812yes—
234C2×C3C4×#62411yescharacteristic
235C2×C3#6390011yescharacteristic
236C2×C3#6390212yes—
237C2×C3#6395711yescharacteristic
238C22#8865111yescharacteristic
239C22C4×((C3⋊D3)⋊C3)⋊C212yes—
240C22#8865511yescharacteristic
241C22#8867212yes—
242C22C2×#619911yescharacteristic
243C4—22no—
244C4—94no—
245C4—94no—
246C4—188no—
247C3—32no—
248C3—61no—
249C3#1059125711yescharacteristic
250C2#1172107812yes—
251C2#1172113812yes—
252C2C4×#619911yescharacteristic
253C2#1172115211yescharacteristic
254C2#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)C4×C4⋊C4
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. #88449C4semidirect
50. #1084C42semidirect
51. #1084C4⋊C4semidirect
132. H(3,𝔽3)C4×C4⋊C4semidirect
211. C32#4649semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×#884492
C2×#884472
#63658⋊C32
#105912423