Finite Group Databasebeta

← #13147110

Subgroups — #13147110

Order 1728.

292 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 47 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×D9×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88615C211yesmaximal, characteristic, complemented
3C4×C9×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5576⋊C3—31nomaximal
5#10589678C311yesmaximal, characteristic, complemented
6C2×C9×SL(2,𝔽3)C2211yescharacteristic
7D9×SL(2,𝔽3)—21no—
8#88615—21no—
9C3×(C2×D9)⋊C4—41nomaximal
10C2×D3×SL(2,𝔽3)—31no—
11C2×C3⋊(#41⋊C3)—31no—
12C3×C4×SL(2,𝔽3)D311yescharacteristic
13C2×#3701C2×C311yescharacteristic, complemented
14C2×Q8×D9C2×C311yescharacteristic
15C4×Q8×C9C2×C311yescharacteristic, complemented
16#63459—31no—
17#63457—31no—
18#63437—31no—
19C4×D36—31no—
20C4×Q8⋊C9—21no—
21C22×C3×D9—41no—
22C2×C3×C4×C9—41no—
23C2×C3×Dic9—41no—
24C9×SL(2,𝔽3)D411yescharacteristic, complemented
25#5576—31no—
26#548⋊C3—91no—
27C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
28D3×SL(2,𝔽3)—61no—
29C3⋊(#41⋊C3)—61no—
30C3×(C2×D3)⋊C4—121no—
31C2×Q8×C9C22×C311yescharacteristic
32#3701—21no—
33#3701—61no—
34Q8×D9—21no—
35Q8×D9—61no—
36C2×D36—31no—
37C2×D36—31no—
38C2×C4×D9—31no—
39C2×C4×D9—31no—
40C2×Dic18—31no—
41C2×Q8⋊C9—21no—
42C9×C4⋊C4—31no—
43C42×C9—31no—
44(C2×D9)⋊C4—32no—
45(C2×D9)⋊C4—31no—
46(C2×D9)⋊C4A411yescharacteristic
47C4⋊Dic9—31no—
48C22×C3×C9—41no—
49C2×C3×D9—42no—
50C2×C3×D9—81no—
51C2×C3×D9—81no—
52C3×C4×C9—81no—
53C3×Dic9—81no—
54C2×#239—31no—
55C2×D3×Q8—31no—
56C2×#41⋊C3—91no—
57C22×SL(2,𝔽3)—91no—
58C3×C4×Q8C3×D311yescharacteristic
59#897—91no—
60#895—91no—
61#875—91no—
62C4×D12—91no—
63C4×SL(2,𝔽3)—21no—
64C4×SL(2,𝔽3)D911yescharacteristic, complemented
65C22×C3×D3—121no—
66C2×C32×C4—41no—
67C2×C3×Dic3—121no—
68C3×SL(2,𝔽3)D1211yescharacteristic
69C22×D9—31no—
70C22×D9C2×A411yescharacteristic
71Q8×C9—31no—
72Q8×C9C3×D411yescharacteristic
73C2×C4×C9—31no—
74C2×C4×C9—31no—
75C2×C4×C9—81no—
76C2×C4×C9C2×A411yescharacteristic
77C2×Dic9—31no—
78C2×Dic9C2×A411yescharacteristic
79D36—61no—
80D36—61no—
81D36—62no—
82C4×D9—61no—
83C4×D9—61no—
84C4×D9—61no—
85C4×D9—61no—
86Dic18—61no—
87Dic18—61no—
88Q8⋊C9—21no—
89#548—91noSylow
90C2×C3×C9—41no—
91C2×C3×C9—41no—
92C2×C3×C9—41no—
93C3×D9—82no—
94C2×C3×Q8C2×C3×D311yescharacteristic
95#239—61no—
96#239—181no—
97D3×Q8—61no—
98D3×Q8—181no—
99C2×D12—91no—
100C2×D12—91no—
101C2×C4×D3—91no—
102C2×C4×D3—91no—
103C2×Dic6—91no—
104#41⋊C3—181no—
105C2×SL(2,𝔽3)—21no—
106C2×SL(2,𝔽3)—181no—
107C2×SL(2,𝔽3)C2×D911yescharacteristic
108C3×C4⋊C4—31no—
109C3×C22⋊C4—361no—
110C3×C42—31no—
111(C2×D3)⋊C4—31no—
112(C2×D3)⋊C4—92no—
113(C2×D3)⋊C4—91no—
114C4⋊Dic3—91no—
115C22×C32—41no—
116C2×C3×D3—122no—
117C2×C3×D3—241no—
118C2×C3×D3—241no—
119C32×C4—81no—
120C3×Dic3—241no—
121C22×C9—81no—
122C22×C9C22×A411yescharacteristic
123C2×D9—21no—
124C2×D9—21no—
125C2×D9—61no—
126C2×D9—61no—
127C2×D9—61no—
128C2×D9#41⋊C312yescomplemented
129C4×C9—21no—
130C4×C9—31no—
131C4×C9—31no—
132C4×C9—61no—
133C4×C9—82no—
134Dic9—21no—
135Dic9—61no—
136C2×#41—91no—
137C22×Q8—91no—
138#124—271no—
139C22⋊Q8—271no—
140#121—271no—
141C4×Q8C3×D911yescharacteristic, complemented
142C4×D4—271no—
143C3×C9—41noSylow
144C23×C3—361no—
145C22×D3—31no—
146C22×D3—91no—
147C3×Q8—31no—
148C3×Q8C3×D1211yescharacteristic
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—41no—
152C2×C3×C4—81no—
153C2×C3×C4—361no—
154C2×C3×C4D3×A411yescharacteristic
155C2×Dic3—31no—
156C2×Dic3—91no—
157D12—181no—
158D12—181no—
159D12—182no—
160C4×D3—181no—
161C4×D3—181no—
162C4×D3—181no—
163C4×D3—181no—
164Dic6—181no—
165Dic6—181no—
166SL(2,𝔽3)—21no—
167SL(2,𝔽3)D3611yescharacteristic, complemented
168C2×C32—41no—
169C2×C32—41no—
170C2×C32—41no—
171C3×D3—242no—
172C2×C9—81no—
173C2×C9—81no—
174C2×C9—81no—
175C2×C9D4×A411yescharacteristic
176C2×C9C2×#41⋊C311yescharacteristic
177C2×C92−1+4⋊C311yescharacteristic
178D9—22no—
179D9—121no—
180#41—181no—
181#41—541no—
182C2×Q8—181no—
183C2×Q8—271no—
184C2×Q8—541no—
185C2×Q8C2×C3×D911yescharacteristic
186C2×D4—271no—
187C2×D4—271no—
188C22×C4—271no—
189C22×C4—271no—
190C4⋊C4—31no—
191C4⋊C4—271no—
192C22⋊C4—91no—
193C22⋊C4—272no—
194C22⋊C4—271no—
195C42—31no—
196C22×C3—41no—
197C22×C3—81no—
198C22×C3—362no—
199C22×C3—721no—
200C22×C3—721no—
201C22×C3C2×D3×A411yescharacteristic
202C2×D3—32no—
203C2×D3—61no—
204C2×D3—61no—
205C2×D3—181no—
206C2×D3—181no—
207C2×D3—181no—
208C3×C4—21no—
209C3×C4—31no—
210C3×C4—31no—
211C3×C4—61no—
212C3×C4—81no—
213C3×C4—82no—
214C3×C4—721no—
215Dic3—61no—
216Dic3—181no—
217C32—41no—
218C9—81no—
219C9#548⋊C311yescharacteristic, complemented
220C23—91no—
221C23—271no—
222Q8—31no—
223Q8—541no—
224Q8—541no—
225Q8C3×D3611yescharacteristic
226D4—541no—
227D4—541no—
228D4—542no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—91no—
232C2×C4—271no—
233C2×C4—541no—
234C2×C4—541no—
235C2×C4—541no—
236C2×C4—541no—
237C2×C4A4×D911yescharacteristic
238C2×C3—41no—
239C2×C3—41no—
240C2×C3—41no—
241C2×C3—81no—
242C2×C3—81no—
243C2×C3—81no—
244C2×C3—722no—
245C2×C3A4×D1211yescharacteristic
246C2×C3#1007⋊C311yescharacteristic
247C2×C3D3×#41⋊C311yescharacteristic
248D3—62no—
249D3—361no—
250C22—92no—
251C22—181no—
252C22—181no—
253C22—541no—
254C22—541no—
255C22—541no—
256C22C2×A4×D911yescharacteristic
257C4—21no—
258C4—31no—
259C4—31no—
260C4—61no—
261C4—181no—
262C4—541no—
263C3—41no—
264C3—81no—
265C3#5576⋊C311yescharacteristic
266C2—182no—
267C2—1081no—
268C2A4×D3611yescharacteristic
269C2#1172241011yescharacteristic
270C2D9×#41⋊C311yescharacteristic
subgroupG/H
centerC22C2×A4×D9
commutator, cosocleC2×Q8×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC4×Q8×C9C2×C3
hypercenterC2×C4A4×D9
hypocenterQ8×C9C3×D4
LCSG ▷ C2×Q8×C9 ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×D9×SL(2,𝔽3)C2semidirect
2. C2×#88615C2semidirect
3. C4×C9×SL(2,𝔽3)C2semidirect
5. #10589678C3semidirect
13. C2×#3701C2×C3semidirect
15. C4×Q8×C9C2×C3semidirect
24. C9×SL(2,𝔽3)D4semidirect
64. C4×SL(2,𝔽3)D9semidirect
128. C2×D9#41⋊C3semidirect
141. C4×Q8C3×D9semidirect
167. SL(2,𝔽3)D36semidirect
219. C9#548⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D9×SL(2,𝔽3)2
C2×#886152
C4×C9×SL(2,𝔽3)2
#5576⋊C33
#105896783
C3×(C2×D9)⋊C44