Finite Group Databasebeta

← #13147904

Subgroups — #13147904

Order 1728.

1606 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 181 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×#88654C214yesmaximal, complemented
2#11721720C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
4#11721163C214yesmaximal, complemented
5#11721156C214yesmaximal, complemented
6#10594004—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
8C2×C4⋊#272C2214yescomplemented
9C2×C4⋊#272C2212yescomplemented
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
12C2×#6200C2218yescomplemented
13#88654—216no—
14C42×H(3,𝔽3)C2211yescharacteristic, complemented
15#3752⋊C3C2214yescomplemented
16C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
17#88451C2214yescomplemented
18C2×#3802—34no—
19C2×D3×D12—34no—
20C4×C4⋊(C3⋊D3)D312yescomplemented
21C3×#874—32no—
22C3×#874—61no—
23C3×C4×D12—32no—
24#63922—38no—
25#63916—34no—
26#63908—34no—
27C22×#272C2314yes—
28C22×(C3⋊D3)⋊C3C2314yes—
29C2×((C3⋊D3)⋊C3)⋊C2—216no—
30C2×((C3⋊D3)⋊C3)⋊C2—44no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×C4×H(3,𝔽3)C2312yes—
33C4⋊#272—28no—
34C4⋊#272—42no—
35C2×(C3⋊Dic3)⋊C3C2314yes—
36C4⋊((C3⋊D3)⋊C3)D418yescomplemented
37C4×(C3⋊D3)⋊C3—28no—
38#6200—216no—
39#5717—91no—
40D4×D12—92no—
41C22×D32—38no—
42C2×C4⋊(C3⋊D3)C2×D312yes—
43C2×C4×C3⋊D3C2×D314yescomplemented
44C2×C3×D12—32no—
45C2×C3×D12—34no—
46C2×C3×D12—34no—
47C2×C3×D12—34no—
48C2×C3×D12—64no—
49C2×C3×D12—62no—
50C2×C3×C4×D3—34no—
51C2×#626—38no—
52C2×#625—38no—
53#3802—616no—
54D3×D12—616no—
55C32×C42—21no—
56C32×C42C2×D312yescomplemented
57#3752C2×D314yescomplemented
58C4⋊(C3⋊Dic3)C2×D312yescomplemented
59C3×(C2×D3)⋊C4—34no—
60C3×C4⋊Dic3—32no—
61(C2×D3)⋊Dic3—38no—
62C22×H(3,𝔽3)C2411yescharacteristic
63C2×#272—28no—
64C2×#272—24no—
65C2×(C3⋊D3)⋊C3—24no—
66C2×(C3⋊D3)⋊C3—24no—
67C2×(C3⋊D3)⋊C3C2×D418yescomplemented
68((C3⋊D3)⋊C3)⋊C2—416no—
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)C2×D414yescomplemented
71(C3⋊Dic3)⋊C3—24no—
72C2×D3×D4—94no—
73C2×D3×D4—94no—
74C22×D12—94no—
75C3×#127—91no—
76C3×C4×D4—92no—
77#938—94no—
78#937—94no—
79#930—94no—
80#895—94no—
81#882—98no—
82#874—32no—
83#874—61no—
84C4×D12—32no—
85C4×D12—92no—
86C22×C3⋊D3C22×D314yes—
87C22×C3×D3—34no—
88C22×C3×D3—38no—
89C22×C3×D3—64no—
90C2×D32—616no—
91C2×D32—616no—
92C2×D32—128no—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4C22×D312yes—
96C2×C32×C4C22×D312yes—
97C2×C32×C4C22×D312yes—
98C2×C3⋊Dic3C22×D314yes—
99C4⋊(C3⋊D3)D1218yescomplemented
100C4×C3⋊D3—28no—
101C2×C3×Dic3—34no—
102C3×D12—38no—
103C3×D12—68no—
104C3×D12—68no—
105C3×D12—128no—
106C3×D12—124no—
107C3×D12—124no—
108C3×C4×D3—68no—
109#626—616no—
110#625—616no—
111D42—271noSylow
112C2×H(3,𝔽3)C22×D412yes—
113C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
114#272—44no—
115(C3⋊D3)⋊C3—28no—
116C23×D3—94no—
117C23×D3—98no—
118C2×C3×D4—92no—
119C2×C3×D4—94no—
120C2×C3×D4—92no—
121C22×C3×C4—94no—
122C2×#67—98no—
123C2×#67—98no—
124D3×D4—1816no—
125D3×D4—1816no—
126C2×D12—32no—
127C2×D12—34no—
128C2×D12—34no—
129C2×D12—34no—
130C2×D12—64no—
131C2×D12—62no—
132C2×D12—92no—
133C2×D12—98no—
134C2×D12—1816no—
135C2×C4×D3—34no—
136C2×C4×D3—94no—
137C3×C4⋊C4—92no—
138C3×C22⋊C4—94no—
139C3×C42—32no—
140C3×C42—61no—
141C3×C42D3211yescharacteristic
142#217—94no—
143(C2×D3)⋊C4—34no—
144(C2×D3)⋊C4—94no—
145(C2×D3)⋊C4—98no—
146C4⋊Dic3—32no—
147C4⋊Dic3—92no—
148C22×C32—21no—
149C22×C32C23×D312yes—
150C2×C3⋊D3—24no—
151C2×C3⋊D3—24no—
152C2×C3⋊D3C2×D1218yescomplemented
153C2×C3×D3—38no—
154C2×C3×D3—64no—
155C2×C3×D3—64no—
156C2×C3×D3—68no—
157C2×C3×D3—68no—
158C2×C3×D3—68no—
159C2×C3×D3—128no—
160C2×C3×D3—124no—
161D32—1232no—
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—24no—
165C32×C4C2×D1214yescomplemented
166C32×C4D3×D414yescomplemented
167C3⋊Dic3—24no—
168C3×Dic3—64no—
169C22×D4—274no—
170#127—91no—
171#121—274no—
172#120—274no—
173C4×D4—92no—
174H(3,𝔽3)D4211yescharacteristic, complemented, Sylow
175C23×C3—94no—
176C23×C3—94no—
177C22×D3—34no—
178C22×D3—38no—
179C22×D3—64no—
180C22×D3—94no—
181C22×D3—1816no—
182C22×D3—1816no—
183C22×D3—1816no—
184C22×D3—364no—
185C22×D3—368no—
186C3×D4—98no—
187C3×D4—188no—
188C3×D4—362no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—61no—
193C2×C3×C4—62no—
194C2×C3×C4—94no—
195C2×C3×C4—188no—
196C2×C3×C4C2×D3211yescharacteristic
197C2×C3×C4C2×D3212yes—
198#67—1816no—
199#67—1816no—
200C2×Dic3—34no—
201C2×Dic3—94no—
202D12—38no—
203D12—68no—
204D12—68no—
205D12—98no—
206D12—128no—
207D12—124no—
208D12—124no—
209D12—1816no—
210C4×D3—68no—
211C4×D3—188no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32C22×D1212yes—
215C2×C32C2×D3×D412yes—
216C2×C32#100912yes—
217C3⋊D3—28no—
218C3×D3—68no—
219C3×D3—128no—
220C3×D3—244no—
221C24—274no—
222C2×D4—92no—
223C2×D4—94no—
224C2×D4—92no—
225C2×D4—278no—
226C2×D4—5416no—
227C22×C4—94no—
228C4⋊C4—92no—
229C22⋊C4—94no—
230C22⋊C4—274no—
231C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
232C22×C3—32no—
233C22×C3—61no—
234C22×C3—98no—
235C22×C3—184no—
236C22×C3—184no—
237C22×C3—188no—
238C22×C3—184no—
239C22×C3C22×D3211yescharacteristic
240C2×D3—38no—
241C2×D3—64no—
242C2×D3—64no—
243C2×D3—68no—
244C2×D3—68no—
245C2×D3—68no—
246C2×D3—98no—
247C2×D3—128no—
248C2×D3—124no—
249C2×D3—184no—
250C2×D3—184no—
251C2×D3—3616no—
252C2×D3—3632no—
253C3×C4—21no—
254C3×C4—34no—
255C3×C4—34no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4—64no—
259C3×C4—184no—
260C3×C4D3×D1214yes—
261Dic3—64no—
262Dic3—184no—
263C32—21no—
264C32D4×D1212yescomplemented
265C23—94no—
266C23—94no—
267C23—5416no—
268C23—1084no—
269D4—98no—
270D4—188no—
271D4—362no—
272D4—5416no—
273C2×C4—94no—
274C2×C4—188no—
275C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
277C2×C3—32no—
278C2×C3—32no—
279C2×C3—32no—
280C2×C3—62no—
281C2×C3—61no—
282C2×C3—188no—
283C2×C3—364no—
284C2×C3C2×D3×D1212yes—
285C2×C3#6430811yescharacteristic
286D3—68no—
287D3—128no—
288D3—188no—
289D3—244no—
290C22—98no—
291C22—184no—
292C22—184no—
293C22—188no—
294C22—184no—
295C22—10816no—
296C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
297C4—21no—
298C4—184no—
299C4#8865414yescomplemented
300C3—32no—
301C3—61no—
302C3D12211yescharacteristic
303C2—188no—
304C2—364no—
305C2C2×#8865412yes—
306C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)D42
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
1. C2×#88654C2semidirect
2. #11721720C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #11721163C2semidirect
5. #11721156C2semidirect
7. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
8. C2×C4⋊#272C22semidirect
9. C2×C4⋊#272C22semidirect
11. C2×C4×(C3⋊D3)⋊C3C22semidirect
12. C2×#6200C22semidirect
14. C42×H(3,𝔽3)C22semidirect
15. #3752⋊C3C22semidirect
16. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
17. #88451C22semidirect
20. C4×C4⋊(C3⋊D3)D3semidirect
36. C4⋊((C3⋊D3)⋊C3)D4semidirect
43. C2×C4×C3⋊D3C2×D3semidirect
56. C32×C42C2×D3semidirect
57. #3752C2×D3semidirect
58. C4⋊(C3⋊Dic3)C2×D3semidirect
67. C2×(C3⋊D3)⋊C3C2×D4semidirect
70. C4×H(3,𝔽3)C2×D4semidirect
99. C4⋊(C3⋊D3)D12semidirect
152. C2×C3⋊D3C2×D12semidirect
165. C32×C4C2×D12semidirect
166. C32×C4D3×D4semidirect
174. H(3,𝔽3)D42semidirect
231. C42((C3⋊D3)⋊C3)⋊C2semidirect
264. C32D4×D12semidirect
299. C4#88654semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
#117217202
C4×C4⋊((C3⋊D3)⋊C3)2
#117211632
#117211562
#105940043