Finite Group Databasebeta

← #13147754

Subgroups — #13147754

Order 1728.

885 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C42×#272C211yesmaximal, characteristic
2#64078⋊C3C212yesmaximal, complemented
3#11721095C212yesmaximal, complemented
4#11721094C214yesmaximal
5#11721089C212yesmaximal
6#11721084C214yesmaximal, complemented
7#10593859—32nomaximal
8C2×C4×#272C2212yes—
9C2×C4×#272C2211yescharacteristic
10C2×#634⋊C3C2214yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2212yes—
13C4×#1084C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2214yes—
15C4⋊((C3⋊Dic3)⋊C3)C2214yes—
16#88451C2214yes—
17#88448C2218yes—
18#88447C2214yes—
19#64078D312yescomplemented
20C3×C42×D3—32no—
21C3×C42×D3—61no—
22C3×#869—32no—
23#63869—34no—
24#63857—34no—
25#63856—34no—
26#63844—34no—
27Dic3×Dic6—34no—
28#63841—34no—
29C22×#272C2311yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×C4×H(3,𝔽3)C2312yes—
32C2×#1084C2312yes—
33C2×#1084C2311yescharacteristic
34C4×#272—24no—
35C4×#272Q814yescomplemented
36C2×(C3⋊Dic3)⋊C3C2318yes—
37#634⋊C3—28no—
38#5728—92no—
39#5470—91no—
40C2×#634C2×D314yes—
41C2×C3×C4×D3—32no—
42C2×C3×C4×D3—32no—
43C2×C3×C4×D3—32no—
44C2×C3×C4×D3—61no—
45C2×C3×C4×D3—62no—
46C2×C3×Dic6—34no—
47C32×C42—21no—
48C32×C42C2×D312yes—
49C4⋊(C3⋊Dic3)C2×D314yes—
50C4⋊(C3⋊Dic3)C2×D314yes—
51C3×C4⋊Dic3—34no—
52C3×C4⋊Dic3—34no—
53C3×C4×Dic3—32no—
54C3×C4×Dic3—32no—
55C3×C4×Dic3—32no—
56C3×C4×Dic3—62no—
57C3×C4×Dic3—61no—
58#3724—38no—
59Dic3⋊Dic3—38no—
60(C2×D3)⋊Dic3—38no—
61Dic32—38no—
62C22×H(3,𝔽3)C2411yescharacteristic
63C2×#272—22no—
64C2×#272C2×Q812yes—
65C4×H(3,𝔽3)C2×Q812yes—
66C4×H(3,𝔽3)#4114yes—
67#1084—22no—
68#1084C2×Q812yes—
69(C3⋊Dic3)⋊C3—28no—
70C3×C4⋊Q8—92no—
71C2×C3×C42—91no—
72#946—94no—
73Q8×Dic3—94no—
74#926—92no—
75#924—94no—
76#897—94no—
77#892—94no—
78#890—94no—
79#887—94no—
80C42×D3—32no—
81C42×D3—61no—
82#870—92no—
83#869—32no—
84#869—92no—
85C4×Dic6—94no—
86C22×C3×D3—32no—
87C22×C3×D3—61no—
88C2×C32×C4—22no—
89C2×C32×C4—21no—
90C2×C32×C4C22×D312yes—
91C2×C32×C4C22×D312yes—
92C2×C32×C4C22×D312yes—
93C2×C3⋊Dic3C22×D318yes—
94#634—28no—
95C2×C3×Dic3—32no—
96C2×C3×Dic3—38no—
97C2×C3×Dic3—32no—
98C2×C3×Dic3—32no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—62no—
101C3×C4×D3—38no—
102C3×C4×D3—64no—
103C3×C4×D3—68no—
104C3×C4×D3—64no—
105C3×C4×D3—64no—
106C3×Dic6—68no—
107#533—271noSylow
108C2×H(3,𝔽3)C22×Q811yescharacteristic
109C2×H(3,𝔽3)C2×#4112yes—
110#272—22no—
111C2×C3×Q8—94no—
112C22×C3×C4—91no—
113C22×C3×C4—92no—
114C2×C4×D3—32no—
115C2×C4×D3—32no—
116C2×C4×D3—32no—
117C2×C4×D3—61no—
118C2×C4×D3—62no—
119C2×Dic6—34no—
120C2×Dic6—94no—
121C3×C4⋊C4—94no—
122C3×C4⋊C4—94no—
123C3×C42—32no—
124C3×C42—61no—
125C3×C42—92no—
126C3×C42—91no—
127C3×C42D3211yescharacteristic
128#217—94no—
129(C2×D3)⋊C4—98no—
130C4⋊Dic3—34no—
131C4⋊Dic3—34no—
132C4⋊Dic3—94no—
133C4⋊Dic3—94no—
134C4⋊Dic3—98no—
135Dic3⋊C4—98no—
136Dic3⋊C4—98no—
137C4×Dic3—32no—
138C4×Dic3—32no—
139C4×Dic3—32no—
140C4×Dic3—62no—
141C4×Dic3—61no—
142C4×Dic3—94no—
143C4×Dic3—98no—
144C22×C32—21no—
145C22×C32C23×D312yes—
146C2×C3×D3—34no—
147C2×C3×D3—62no—
148C2×C3×D3—64no—
149C2×C3×D3—62no—
150C2×C3×D3—62no—
151C32×C4—24no—
152C32×C4—22no—
153C32×C4#23714yes—
154C32×C4D3×Q814yes—
155C32×C4#23914yes—
156C3⋊Dic3—28no—
157C3×Dic3—34no—
158C3×Dic3—62no—
159C3×Dic3—64no—
160C3×Dic3—62no—
161C3×Dic3—68no—
162C3×Dic3—62no—
163C4⋊Q8—92no—
164#125—272no—
165C22⋊Q8—274no—
166C4×Q8—274no—
167#117—272no—
168C2×C42—91no—
169H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
170C23×C3—91no—
171C22×D3—32no—
172C22×D3—61no—
173C3×Q8—188no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—62no—
178C2×C3×C4—61no—
179C2×C3×C4—91no—
180C2×C3×C4—94no—
181C2×C3×C4—98no—
182C2×C3×C4—92no—
183C2×C3×C4—184no—
184C2×C3×C4C2×D3212yes—
185C2×C3×C4C2×D3211yescharacteristic
186C2×Dic3—32no—
187C2×Dic3—32no—
188C2×Dic3—32no—
189C2×Dic3—38no—
190C2×Dic3—61no—
191C2×Dic3—62no—
192C2×Dic3—98no—
193C4×D3—38no—
194C4×D3—64no—
195C4×D3—64no—
196C4×D3—64no—
197C4×D3—68no—
198Dic6—68no—
199Dic6—188no—
200C2×C32—22no—
201C2×C32—21no—
202C2×C32C2×#23712yes—
203C2×C32C2×D3×Q812yes—
204C2×C32C2×#23912yes—
205C3×D3—64no—
206C3×D3—64no—
207C2×Q8—94no—
208C22×C4—91no—
209C22×C4—92no—
210C4⋊C4—94no—
211C4⋊C4—94no—
212C4⋊C4—278no—
213C22⋊C4—274no—
214C42—92no—
215C42—91no—
216C42—274no—
217C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C22×C3—32no—
219C22×C3—61no—
220C22×C3—92no—
221C22×C3—182no—
222C22×C3C22×D3211yescharacteristic
223C2×D3—34no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—64no—
228C3×C4—34no—
229C3×C4—34no—
230C3×C4—34no—
231C3×C4—64no—
232C3×C4—62no—
233C3×C4—92no—
234C3×C4—188no—
235C3×C4—182no—
236C3×C4#379614yes—
237C3×C4#379712yes—
238Dic3—34no—
239Dic3—62no—
240Dic3—62no—
241Dic3—62no—
242Dic3—64no—
243Dic3—68no—
244Dic3—188no—
245C32—21no—
246C32#572812yescomplemented
247C23—91no—
248Q8—188no—
249C2×C4—91no—
250C2×C4—94no—
251C2×C4—92no—
252C2×C4—98no—
253C2×C4—184no—
254C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
255C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
256C2×C3—32no—
257C2×C3—32no—
258C2×C3—32no—
259C2×C3—62no—
260C2×C3—61no—
261C2×C3—182no—
262C2×C3C2×#379612yes—
263C2×C3C2×#379711yescharacteristic
264D3—64no—
265D3—64no—
266C22—92no—
267C22—182no—
268C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C4—92no—
270C4—182no—
271C4—188no—
272C4#8864914yes—
273C4#8865112yes—
274C3—32no—
275C3—61no—
276C3#1059386611yescharacteristic
277C2—182no—
278C2C2×#8864912yes—
279C2C2×#8865111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. #11721095C2semidirect
6. #11721084C2semidirect
19. #64078D3semidirect
35. C4×#272Q8semidirect
169. H(3,𝔽3)#533semidirect
246. C32#5728semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
#64078⋊C32
#117210952
#117210942
#117210892
#117210842
#105938593