Finite Group Databasebeta

← #13166023

Subgroups — #13166023

Order 1728.

394 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C211yesmaximal, characteristic
3C3×C3⋊#859C211yesmaximal, characteristic
4#10593623—31nomaximal
5#10593196—33nomaximal
6#10591547—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8#88806—41nomaximal
9C3×Q8×Dic3D311yescharacteristic
10C3×C4×SL(2,𝔽3)—31no—
11#63869—31no—
12Dic3×SL(2,𝔽3)D313yes—
13C3⋊#859—31no—
14C3⋊#859—33no—
15C3⋊#859—61no—
16C3⋊#859—63no—
17C3⋊#859D311yescharacteristic
18C3×#859—33no—
19#63662—31no—
20C3×#835—31no—
21#63575—31no—
22#63574—31no—
23Dic3⋊(C3⋊C8)—31no—
24C2×C3⋊(C3⋊Dic3)—41no—
25C2×C3×C3⋊Dic3—41no—
26C2×C32×Dic3—41no—
27C32×SL(2,𝔽3)Q811yescharacteristic
28#5389—93no—
29#5022—91no—
30#4796—91no—
31C2×C32×Q8C2×D311yescharacteristic
32C2×C3×SL(2,𝔽3)—21no—
33C2×C3×SL(2,𝔽3)—23no—
34C2×C3×SL(2,𝔽3)C2×D313yes—
35C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
36C4⋊(C3⋊Dic3)—31no—
37#3750—121no—
38C2×C3⋊(C3⋊C8)—31no—
39C3×C4⋊Dic3—31no—
40C3×C4⋊Dic3—31no—
41C3×C4×Dic3—31no—
42C2×C3×C3⋊C8—31no—
43#3724—61no—
44#3724—123no—
45Dic3⋊Dic3—61no—
46C22×C33—41no—
47C3⋊(C3⋊Dic3)—42no—
48C3×C3⋊Dic3—81no—
49C32×Dic3—81no—
50C3×C4×Q8—31no—
51Q8×Dic3C3⋊D311yescharacteristic
52#890—91no—
53#870—91no—
54C4×SL(2,𝔽3)—33no—
55#859—33no—
56#859—93no—
57#859—183no—
58C3×#103—91no—
59#835—31no—
60#835—91no—
61#835—91no—
62#835—181no—
63#818—91no—
64Dic3⋊C8—91no—
65#808—91no—
66#808—91no—
67#807—91no—
68C4⋊(C3⋊C8)—91no—
69C32×Q8—31no—
70C32×Q8Dic611yescharacteristic
71C2×C32×C4—31no—
72C2×C32×C4—121no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—61no—
75C2×C3⋊Dic3—121no—
76C2×C3⋊Dic3—123no—
77C2×C3⋊Dic3—241no—
78C2×C3⋊Dic3—243no—
79C2×C3×Dic3—31no—
80C2×C3×Dic3—43no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—123no—
83C2×C3×Dic3S411yescharacteristic
84C3×SL(2,𝔽3)—21no—
85C3×SL(2,𝔽3)—23no—
86C3×SL(2,𝔽3)Dic613yes—
87C3×SL(2,𝔽3)Dic611yescharacteristic
88C3⋊(C3⋊C8)—61no—
89C3×C3⋊C8—61no—
90#479—271noSylow
91C2×C33—41no—
92C2×C33—41no—
93C2×C33—41no—
94C2×C3×Q8—21no—
95C2×C3×Q8D3211yescharacteristic
96C2×C3×Q8C2×C3⋊D311yescharacteristic
97C2×SL(2,𝔽3)—23no—
98C2×SL(2,𝔽3)D3213yes—
99C2×C3×C8—91no—
100C3×C4⋊C4—91no—
101C3×C4⋊C4—91no—
102C3×C42—91no—
103C4⋊Dic3—31no—
104C4⋊Dic3—31no—
105C4⋊Dic3—91no—
106C4⋊Dic3—91no—
107C4⋊Dic3—181no—
108C4⋊Dic3—181no—
109Dic3⋊C4—181no—
110Dic3⋊C4—181no—
111Dic3⋊C4—181no—
112Dic3⋊C4—363no—
113C4×Dic3—31no—
114C2×C3⋊C8—31no—
115C2×C3⋊C8—91no—
116C2×C3⋊C8—91no—
117C2×C3⋊C8—181no—
118C22×C32—41no—
119C22×C32—43no—
120C22×C32—83no—
121C22×C32—81no—
122C22×C32C2×S411yescharacteristic
123C32×C4—31no—
124C32×C4—31no—
125C32×C4—241no—
126C3⋊Dic3—81no—
127C3⋊Dic3—122no—
128C3⋊Dic3—126no—
129C3⋊Dic3—121no—
130C3⋊Dic3—246no—
131C3⋊Dic3—242no—
132C3×Dic3—21no—
133C3×Dic3—61no—
134C3×Dic3—83no—
135C3×Dic3—121no—
136C3×Dic3—243no—
137#125—271no—
138C4×Q8—31no—
139#107—271no—
140#106—271no—
141C4⋊C8—271no—
142#103—91no—
143#103—271no—
144C33—41noSylow
145C3×Q8—21no—
146C3×Q8—31no—
147C3×Q8—31no—
148C3×Q8—61no—
149C3×Q8#62711yescharacteristic
150C3×Q8#63411yescharacteristic
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—61no—
155C2×C3×C4—91no—
156C2×C3×C4—123no—
157C2×C3×C4—181no—
158C2×Dic3—31no—
159C2×Dic3—61no—
160C2×Dic3—123no—
161C2×Dic3—181no—
162C2×Dic3—181no—
163C2×Dic3—363no—
164C2×Dic3—361no—
165C2×Dic3—723no—
166C2×Dic3C3⋊S411yescharacteristic
167SL(2,𝔽3)—23no—
168SL(2,𝔽3)#62713yes—
169C3×C8—181no—
170C3⋊C8—61no—
171C3⋊C8—181no—
172C3⋊C8—181no—
173C3⋊C8—361no—
174C2×C32—41no—
175C2×C32—41no—
176C2×C32—41no—
177C2×C32—43no—
178C2×C32—43no—
179C2×C32—43no—
180C2×C32—83no—
181C2×C32—83no—
182C2×C32—83no—
183C2×C32—81no—
184C2×C32—81no—
185C2×C32—81no—
186C2×C32A4⋊Q811yescharacteristic
187C2×C32#98311yescharacteristic
188C2×C32#98511yescharacteristic
189C2×Q8D3×C3⋊D311yescharacteristic
190C2×C8—91no—
191C2×C8—271no—
192C4⋊C4—91no—
193C4⋊C4—91no—
194C4⋊C4—271no—
195C4⋊C4—541no—
196C4⋊C4—541no—
197C42—91no—
198C22×C3—21no—
199C22×C3—43no—
200C22×C3—83no—
201C22×C3D3×S411yescharacteristic
202C22×C3C2×C3⋊S411yescharacteristic
203C3×C4—31no—
204C3×C4—31no—
205C3×C4—31no—
206C3×C4—31no—
207C3×C4—61no—
208C3×C4—61no—
209C3×C4—61no—
210C3×C4—181no—
211C3×C4—243no—
212C3×C4—361no—
213Dic3—21no—
214Dic3—61no—
215Dic3—121no—
216Dic3—243no—
217Dic3—366no—
218Dic3—361no—
219Dic3—361no—
220Dic3—721no—
221Dic3—726no—
222C32—41no—
223C32—43no—
224C32—83no—
225C32—81no—
226C32#538911yescharacteristic, complemented
227Q8—31no—
228Q8#629311yescharacteristic
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—91no—
232C2×C4—181no—
233C2×C4—541no—
234C8—181no—
235C8—541no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3—43no—
240C2×C3—43no—
241C2×C3—43no—
242C2×C3—83no—
243C2×C3—83no—
244C2×C3—83no—
245C2×C3#6419811yescharacteristic
246C2×C3#6420111yescharacteristic
247C2×C3#6420411yescharacteristic
248C2×C3#6425911yescharacteristic
249C2×C3#6426411yescharacteristic
250C2×C3#6426611yescharacteristic
251C22D3×C3⋊S411yescharacteristic
252C4—31no—
253C4—31no—
254C4—61no—
255C4—181no—
256C4—361no—
257C4—1081no—
258C3—21no—
259C3—43no—
260C3—83no—
261C3#1059319611yescharacteristic, complemented
262C3#1059362311yescharacteristic, complemented
263C2#1172419311yescharacteristic
264C2#1172419611yescharacteristic
265C2#1172419911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, prime radical, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)Q8
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
226. C32#5389semidirect
261. C3#10593196semidirect
262. C3#10593623semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
C3×C3⋊#8592
#105936233
#105931963
#105915473
#888064