Finite Group Databasebeta

← #13150664

Subgroups — #13150664

Order 1728.

885 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 183 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
2#64135⋊C3C212yesmaximal
3#11721309C214yesmaximal
4#11721117C212yesmaximal, complemented
5C4×#6196C214yesmaximal, complemented
6#11721095C212yesmaximal, complemented
7#10595049—32nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C2×#6196C2214yes—
11C4×#1084C2211yescharacteristic
12C4×#1084C2212yes—
13C4×#1084C2211yescharacteristic
14#3757⋊C3C2214yes—
15C4⋊((C3⋊Dic3)⋊C3)C2214yes—
16C4×(C3⋊Dic3)⋊C3C2214yes—
17#88449C2214yes—
18#88448C2218yes—
19#64135D312yes—
20C3×#926—32no—
21C2×C3×C4×Dic3—32no—
22C2×C3×C4×Dic3—61no—
23#63964—38no—
24#63918—34no—
25C4×#627—38no—
26#63869—34no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)Q814yescomplemented
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—24no—
32C2×#1084C2312yes—
33C2×#1084C2312yes—
34C2×(C3⋊Dic3)⋊C3C2318yes—
35#6196—28no—
36#5517—92no—
37#5470—91no—
38C22×C32×C4—21no—
39C22×C32×C4C2×D312yes—
40C22×C3×Dic3—34no—
41C22×C3×Dic3—62no—
42C2×#627—38no—
43#3757C2×D314yes—
44C4⋊(C3⋊Dic3)C2×D314yes—
45C4×C3⋊Dic3C2×D314yes—
46C3×#217—34no—
47C3×C4⋊Dic3—34no—
48C3×C4×Dic3—32no—
49C3×C4×Dic3—34no—
50C3×C4×Dic3—32no—
51C3×C4×Dic3—34no—
52C3×C4×Dic3—62no—
53C3×C4×Dic3—61no—
54C3×C4×Dic3—61no—
55#3724—38no—
56Dic3⋊Dic3—38no—
57Dic3⋊Dic3—38no—
58C22×H(3,𝔽3)—22no—
59C22×H(3,𝔽3)C2×Q812yes—
60C22×H(3,𝔽3)C2411yescharacteristic
61C4×H(3,𝔽3)C2×Q812yes—
62C4×H(3,𝔽3)C2×Q812yes—
63#1084—22no—
64#1084#4114yes—
65(C3⋊Dic3)⋊C3—28no—
66C3×#117—92no—
67C2×C3×C42—91no—
68#926—32no—
69#926—92no—
70#924—94no—
71C2×C4×Dic3—32no—
72C2×C4×Dic3—61no—
73#890—94no—
74#888—94no—
75#878—98no—
76#870—92no—
77#869—92no—
78C4×Dic6—94no—
79C4×Dic6—98no—
80C23×C32—21no—
81C23×C32C22×D312yes—
82C2×C32×C4—24no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4Dic618yescomplemented
86C2×C32×C4C22×D312yes—
87C2×C32×C4C22×D312yes—
88C2×C3⋊Dic3C22×D318yes—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—38no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—68no—
95C2×C3×Dic3—68no—
96#627—616no—
97#533—271noSylow
98C2×H(3,𝔽3)—22no—
99C2×H(3,𝔽3)C22×Q811yescharacteristic
100C2×H(3,𝔽3)C2×#4112yes—
101C22×C3×C4—32no—
102C22×C3×C4—61no—
103C22×C3×C4—92no—
104C22×C3×C4D3211yescharacteristic
105C22×Dic3—34no—
106C22×Dic3—62no—
107C2×Dic6—94no—
108C2×Dic6—98no—
109C3×C4⋊C4—94no—
110C3×C22⋊C4—94no—
111C3×C42—94no—
112C3×C42—92no—
113C3×C42—91no—
114C3×C42—91no—
115#217—34no—
116#217—94no—
117C4⋊Dic3—34no—
118C4⋊Dic3—94no—
119C4⋊Dic3—94no—
120C4⋊Dic3—98no—
121Dic3⋊C4—98no—
122Dic3⋊C4—98no—
123Dic3⋊C4—98no—
124C4×Dic3—34no—
125C4×Dic3—32no—
126C4×Dic3—32no—
127C4×Dic3—34no—
128C4×Dic3—61no—
129C4×Dic3—61no—
130C4×Dic3—62no—
131C4×Dic3—94no—
132C22×C32—24no—
133C22×C32—24no—
134C22×C32—21no—
135C22×C32—22no—
136C22×C32C2×Dic614yes—
137C22×C32C23×D312yes—
138C32×C4—22no—
139C32×C4—22no—
140C32×C4C2×Dic614yes—
141C32×C4C2×Dic614yes—
142C3⋊Dic3—28no—
143C3×Dic3—38no—
144C3×Dic3—64no—
145C3×Dic3—64no—
146C3×Dic3—68no—
147C3×Dic3—64no—
148C4⋊Q8—272no—
149#125—272no—
150C22⋊Q8—274no—
151C4×Q8—274no—
152#117—92no—
153C2×C42—91no—
154H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
155C23×C3—32no—
156C23×C3—61no—
157C23×C3C2×D3211yescharacteristic
158C2×C3×C4—38no—
159C2×C3×C4—32no—
160C2×C3×C4—32no—
161C2×C3×C4—64no—
162C2×C3×C4—61no—
163C2×C3×C4—61no—
164C2×C3×C4—92no—
165C2×C3×C4—98no—
166C2×C3×C4—92no—
167C2×C3×C4—184no—
168C2×C3×C4#62714yes—
169C2×C3×C4C2×D3211yescharacteristic
170C2×C3×C4C2×D3211yescharacteristic
171C2×Dic3—34no—
172C2×Dic3—34no—
173C2×Dic3—38no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—68no—
177C2×Dic3—68no—
178C2×Dic3—98no—
179Dic6—1816no—
180Dic6—188no—
181C2×C32—24no—
182C2×C32—24no—
183C2×C32—22no—
184C2×C32—21no—
185C2×C32C22×Dic612yes—
186C2×C32D3×#4114yes—
187C2×Q8—274no—
188C22×C4—92no—
189C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
190C4⋊C4—94no—
191C4⋊C4—278no—
192C4⋊C4—274no—
193C22⋊C4—94no—
194C42—91no—
195C42—91no—
196C42—92no—
197C42—94no—
198C22×C3—22no—
199C22×C3—32no—
200C22×C3—34no—
201C22×C3—64no—
202C22×C3—64no—
203C22×C3—61no—
204C22×C3—62no—
205C22×C3C2×#62712yes—
206C22×C3C22×D3211yescharacteristic
207C3×C4—34no—
208C3×C4—34no—
209C3×C4—62no—
210C3×C4—62no—
211C3×C4—94no—
212C3×C4—188no—
213C3×C4—182no—
214C3×C4C2×#62712yes—
215C3×C4C2×#62712yes—
216Dic3—38no—
217Dic3—64no—
218Dic3—64no—
219Dic3—64no—
220Dic3—68no—
221Dic3—188no—
222C32—21no—
223C32#551712yescomplemented
224C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
225Q8—548no—
226C2×C4—92no—
227C2×C4—92no—
228C2×C4—98no—
229C2×C4—184no—
230C2×C4#619614yescomplemented
231C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
232C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C2×C3—22no—
234C2×C3—32no—
235C2×C3—34no—
236C2×C3—64no—
237C2×C3—64no—
238C2×C3—61no—
239C2×C3—62no—
240C2×C3#6430612yes—
241C2×C3C22×#62711yescharacteristic
242C22—22no—
243C22C2×#619612yes—
244C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
245C4—94no—
246C4—182no—
247C4—188no—
248C4C2×#619612yes—
249C4C2×#619612yes—
250C3—32no—
251C3—61no—
252C3#1059505011yescharacteristic
253C2—22no—
254C2#1172249112yes—
255C2C22×#619611yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11721117C2semidirect
5. C4×#6196C2semidirect
6. #11721095C2semidirect
28. C2×C4×H(3,𝔽3)Q8semidirect
85. C2×C32×C4Dic6semidirect
154. H(3,𝔽3)#533semidirect
223. C32#5517semidirect
230. C2×C4#6196semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
#64135⋊C32
#117213092
#117211172
C4×#61962
#117210952
#105950493