Finite Group Databasebeta

← #13166098

Subgroups — #13166098

Order 1728.

507 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 58 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×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×C3⋊#859C212yesmaximal, complemented
3D3×#859—32nomaximal
4#10591552—31nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C2×#6294—41nomaximal
7C3⋊D3×SL(2,𝔽3)C412yescomplemented
8C2×Q8×C3⋊D3D311yescharacteristic
9C2×D3×SL(2,𝔽3)—32no—
10C2×D3×SL(2,𝔽3)—61no—
11#63900—31no—
12C2×#3710—31no—
13C3⋊#859D312yescomplemented
14C3×#859—32no—
15C3×#859—62no—
16C3×#835—32no—
17#63567—32no—
18C22×C3×C3⋊D3—41no—
19C2×C3×C3⋊Dic3—42no—
20C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
21#6294—44no—
22C2×#859—91no—
23D3×#103—92no—
24C2×C32×Q8C2×D311yescharacteristic
25Q8×C3⋊D3—32no—
26Q8×C3⋊D3Dic312yescomplemented
27C2×C4×C3⋊D3—31no—
28C2×#634—31no—
29C2×C3×SL(2,𝔽3)—22no—
30C2×C3×SL(2,𝔽3)—21no—
31C2×C3×SL(2,𝔽3)—41no—
32C2×C3×SL(2,𝔽3)C2×D312yes—
33C2×#624—61no—
34C2×D3×Dic3—122no—
35D3×SL(2,𝔽3)—34no—
36D3×SL(2,𝔽3)—62no—
37C3×C4⋊Dic3—32no—
38C2×C3×C3⋊C8—32no—
39#3724—61no—
40#3710—62no—
41C22×C33—41no—
42C2×C3×C3⋊D3—42no—
43C2×C3×C3⋊D3—42no—
44C2×C3×C3⋊D3—42no—
45C3×C3⋊Dic3—44no—
46C2×D3×Q8—32no—
47C2×D3×Q8—61no—
48C22×SL(2,𝔽3)—91no—
49C2×D3×C8—92no—
50D3×C4⋊C4—92no—
51#859—32no—
52#859—62no—
53C3×#103—92no—
54#835—32no—
55#816—92no—
56#810—92no—
57C22×C3⋊D3S411yescharacteristic
58C22×C3×D3—122no—
59C22×C3×D3—241no—
60C32×Q8—31no—
61C32×Q8C2×Dic311yescharacteristic, complemented
62C2×C32×C4—31no—
63C2×C3⋊Dic3—31no—
64C2×C3⋊Dic3—42no—
65C4×C3⋊D3—32no—
66C4×C3⋊D3—32no—
67#634—32no—
68#634—61no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—122no—
71C2×C3×Dic3—242no—
72C3×SL(2,𝔽3)—22no—
73C3×SL(2,𝔽3)—21no—
74C3×SL(2,𝔽3)—41no—
75C3×SL(2,𝔽3)C4×D312yescomplemented
76#624—122no—
77D3×Dic3—128no—
78C3×C3⋊C8—62no—
79C2×#103—271noSylow
80C2×C33—41no—
81C2×C33—41no—
82C2×C33—41no—
83C3×C3⋊D3—44no—
84C2×C3×Q8—21no—
85C2×C3×Q8D3212yes—
86C22×Dic3—361no—
87D3×Q8—34no—
88D3×Q8—62no—
89D3×Q8—94no—
90D3×Q8—182no—
91C2×C4×D3—92no—
92C2×C4×D3—181no—
93C2×C4×D3—182no—
94C2×Dic6—92no—
95C2×Dic6—181no—
96C2×SL(2,𝔽3)—22no—
97C2×SL(2,𝔽3)—41no—
98C2×SL(2,𝔽3)—92no—
99C2×SL(2,𝔽3)D3211yescharacteristic
100C2×C3×C8—92no—
101C3×C4⋊C4—92no—
102C4⋊Dic3—32no—
103Dic3⋊C4—182no—
104C2×C3⋊C8—32no—
105D3×C8—184no—
106C22×C32—42no—
107C22×C32—81no—
108C22×C32—82no—
109C22×C32—161no—
110C22×C32C2×S411yescharacteristic
111C2×C3⋊D3#22612yes—
112C2×C3⋊D3GL(2,𝔽3)12yes—
113C2×C3⋊D3SL(2,ℤ/4)12yes—
114C2×C3×D3—124no—
115C2×C3×D3—124no—
116C2×C3×D3—124no—
117C2×C3×D3—242no—
118C2×C3×D3—242no—
119C2×C3×D3—242no—
120C32×C4—31no—
121C32×C4—31no—
122C3⋊Dic3—32no—
123C3⋊Dic3—44no—
124C3×Dic3—124no—
125C3×Dic3—122no—
126C3×Dic3—244no—
127C22×Q8—91no—
128C22×C8—271no—
129C2×C4⋊C4—271no—
130#103—92no—
131#103—272no—
132C33—41noSylow
133C23×C3—361no—
134C22×D3—32no—
135C22×D3—61no—
136C3×Q8—21no—
137C3×Q8—32no—
138C3×Q8—61no—
139C3×Q8D3×Dic312yescomplemented
140C2×C3×C4—32no—
141C2×C3×C4—61no—
142C2×C3×C4—182no—
143C2×Dic3—62no—
144C2×Dic3—92no—
145C2×Dic3—122no—
146C2×Dic3—181no—
147C2×Dic3—242no—
148C2×Dic3—364no—
149C4×D3—94no—
150C4×D3—94no—
151C4×D3—182no—
152C4×D3—182no—
153C4×D3—364no—
154Dic6—94no—
155Dic6—182no—
156Dic6—182no—
157Dic6—182no—
158SL(2,𝔽3)—22no—
159SL(2,𝔽3)—41no—
160SL(2,𝔽3)#62411yescharacteristic, complemented
161C3×C8—182no—
162C3⋊C8—62no—
163C2×C32—42no—
164C2×C32—42no—
165C2×C32—42no—
166C2×C32—81no—
167C2×C32—81no—
168C2×C32—81no—
169C2×C32—82no—
170C2×C32—82no—
171C2×C32—82no—
172C2×C32—161no—
173C2×C32—161no—
174C2×C32—161no—
175C2×C32C2×#22611yescharacteristic
176C2×C32C2×GL(2,𝔽3)11yescharacteristic
177C2×C32C2×SL(2,ℤ/4)11yescharacteristic
178C3⋊D3#85914yescomplemented
179C3×D3—128no—
180C3×D3—244no—
181C2×Q8—92no—
182C2×Q8—271no—
183C2×Q8—272no—
184C2×Q8#111311yescharacteristic
185C22×C4—271no—
186C22×C4—541no—
187C2×C8—92no—
188C2×C8—542no—
189C4⋊C4—92no—
190C4⋊C4—541no—
191C22×C3—21no—
192C22×C3—41no—
193C22×C3—82no—
194C22×C3—161no—
195C22×C3—362no—
196C22×C3—362no—
197C22×C3—362no—
198C22×C3D3×S412yes—
199C2×D3—34no—
200C2×D3—34no—
201C2×D3—34no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—62no—
205C3×C4—32no—
206C3×C4—32no—
207C3×C4—61no—
208C3×C4—61no—
209C3×C4—362no—
210Dic3—94no—
211Dic3—124no—
212Dic3—122no—
213Dic3—182no—
214Dic3—244no—
215C32—42no—
216C32—81no—
217C32—82no—
218C32—161no—
219C32C2×#85911yescharacteristic, complemented
220C23—91no—
221Q8—31no—
222Q8—272no—
223Q8—541no—
224Q8#629411yescharacteristic, complemented
225C2×C4—31no—
226C2×C4—182no—
227C2×C4—272no—
228C2×C4—272no—
229C2×C4—271no—
230C2×C4—1082no—
231C8—182no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—82no—
239C2×C3—82no—
240C2×C3—82no—
241C2×C3—161no—
242C2×C3—161no—
243C2×C3—161no—
244C2×C3—364no—
245C2×C3D3×#22612yes—
246C2×C3D3×GL(2,𝔽3)12yes—
247C2×C3D3×SL(2,ℤ/4)12yes—
248D3—38no—
249D3—64no—
250C22—92no—
251C22—92no—
252C22—92no—
253C22#8910111yescharacteristic
254C4—31no—
255C4—31no—
256C4—272no—
257C4—362no—
258C3—21no—
259C3—41no—
260C3—82no—
261C3—161no—
262C3D3×#85912yescomplemented
263C2—94no—
264C2#1172420911yescharacteristic
265C2#1172421111yescharacteristic
266C2#1172421511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#859C2semidirect
7. C3⋊D3×SL(2,𝔽3)C4semidirect
13. C3⋊#859D3semidirect
20. C32×SL(2,𝔽3)C2×C4semidirect
26. Q8×C3⋊D3Dic3semidirect
61. C32×Q8C2×Dic3semidirect
75. C3×SL(2,𝔽3)C4×D3semidirect
139. C3×Q8D3×Dic3semidirect
160. SL(2,𝔽3)#624semidirect
178. C3⋊D3#859semidirect
219. C32C2×#859semidirect
224. Q8#6294semidirect
262. C3D3×#859semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C3×C3⋊#8592
D3×#8593
#105915523
C2×#62944