Finite Group Databasebeta

← #13144449

Subgroups — #13144449

Order 1728.

1089 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#11722664C211yesmaximal, characteristic, complemented
2H(3,𝔽3)×#120C211yesmaximal, characteristic, complemented
3#11721876C212yesmaximal, complemented
4#11720580C211yesmaximal, characteristic, complemented
5#11720524C212yesmaximal, complemented
6C3×#5032—34nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic, complemented
8C2×D4×H(3,𝔽3)—21no—
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×#6235—22no—
12C2×#6235C2212yes—
13#88744—22no—
14D4×#272—22no—
15#88740—21no—
16C22⋊C4×H(3,𝔽3)—21no—
17C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
18#88552—22no—
19#88552C2212yescomplemented
20C3×#1004—34no—
21C32×#120D314yescomplemented
22C3×#953—38no—
23C3×#834—34no—
24C3×#806—38no—
25C23×H(3,𝔽3)—22no—
26C23×H(3,𝔽3)—21no—
27C23×H(3,𝔽3)—42no—
28C23×H(3,𝔽3)D411yescharacteristic, complemented
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×#272—21no—
31C22×#272D412yescomplemented
32D4×H(3,𝔽3)—22no—
33D4×H(3,𝔽3)—42no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)D411yescharacteristic, complemented
36#6235—22no—
37#6235—22no—
38#6235—22no—
39#6235—44no—
40C2×#1084—21no—
41C2×#1084—22no—
42C2×#1084D412yescomplemented
43C4⋊#272—21no—
44C4×#272—22no—
45#6229—21no—
46C3×B(4,𝔽2)—91no—
47#5032—34no—
48C24×C32C2×D314yescomplemented
49C2×C32×D4—24no—
50C2×C32×D4C2×D314yes—
51C2×C3×#67—38no—
52C2×C3×#67—64no—
53C2×C3×#67—68no—
54C3×#237—68no—
55C3×D3×D4—68no—
56C3×#235—64no—
57C32×C22⋊C4—24no—
58C32×C22⋊C4C2×D314yescomplemented
59C3×#217—38no—
60C3×#217—68no—
61C22×H(3,𝔽3)—24no—
62C22×H(3,𝔽3)—22no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)—42no—
66C22×H(3,𝔽3)—41no—
67C22×H(3,𝔽3)—41no—
68C22×H(3,𝔽3)—41no—
69C22×H(3,𝔽3)C2×D412yes—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C2×#272—22no—
72C2×#272—42no—
73C2×#272—81no—
74C4×H(3,𝔽3)—21no—
75C4×H(3,𝔽3)—41no—
76#1084—22no—
77#1084—42no—
78C3×H(5,𝔽2)—91no—
79#1004—34no—
80C3×#120—34no—
81C3×#120—92no—
82C3×#120C3⋊D311yescharacteristic
83#953—38no—
84C3×#99—91no—
85C3×#99—92no—
86#834—34no—
87#806—38no—
88C23×C32—28no—
89C23×C32—24no—
90C23×C32—48no—
91C23×C32#6714yescomplemented
92C23×C32C22×D314yes—
93C22×C3×D3—38no—
94C22×C3×D3—64no—
95C32×D4—28no—
96C32×D4—48no—
97C2×C32×C4—24no—
98C2×C32×C4#6714yescomplemented
99C3×#67—68no—
100C3×#67—68no—
101C3×#67—68no—
102C3×#67—1216no—
103C2×C3×Dic3—38no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—68no—
106C3×D12—64no—
107C3×C4×D3—68no—
108C3×Dic6—64no—
109B(4,𝔽2)—91noSylow
110C2×H(3,𝔽3)—22no—
111C2×H(3,𝔽3)—21no—
112C2×H(3,𝔽3)—41no—
113C2×H(3,𝔽3)—42no—
114C2×H(3,𝔽3)#12011yescharacteristic
115#272—42no—
116C24×C3—34no—
117C24×C3C2×C3⋊D311yescharacteristic
118C3×#41—182no—
119C3×#41—181no—
120C2×C3×D4—21no—
121C2×C3×D4—34no—
122C2×C3×D4—64no—
123C2×C3×D4—92no—
124C2×C3×D4—182no—
125C2×C3×D4—181no—
126C2×C3×D4—182no—
127C2×C3×D4C2×C3⋊D311yescharacteristic
128C2×#67—38no—
129C2×#67—64no—
130C2×#67—68no—
131#237—68no—
132D3×D4—68no—
133#235—64no—
134C3×C22⋊C4—21no—
135C3×C22⋊C4—34no—
136C3×C22⋊C4—64no—
137C3×C22⋊C4—92no—
138C3×C22⋊C4—182no—
139C3×C22⋊C4C2×C3⋊D311yescharacteristic
140#217—38no—
141#217—68no—
142C22×C32—216no—
143C22×C32—28no—
144C22×C32—24no—
145C22×C32—44no—
146C22×C32—44no—
147C22×C32—48no—
148C22×C32—44no—
149C22×C32—44no—
150C22×C32D3×D418yes—
151C22×C32C2×#6714yes—
152C2×C3×D3—68no—
153C2×C3×D3—128no—
154C2×C3×D3—244no—
155C32×C4—24no—
156C32×C4—44no—
157C3×Dic3—68no—
158C3×Dic3—128no—
159H(5,𝔽2)—91no—
160#120—92no—
161#120#27211yescharacteristic, complemented
162#99—91no—
163#99—92no—
164H(3,𝔽3)B(4,𝔽2)11yescharacteristic, complemented, Sylow
165C23×C3—22no—
166C23×C3—21no—
167C23×C3—34no—
168C23×C3—34no—
169C23×C3—42no—
170C23×C3—68no—
171C23×C3—64no—
172C23×C3—92no—
173C23×C3—128no—
174C23×C3—181no—
175C23×C3#63811yescharacteristic
176C23×C3C22×C3⋊D311yescharacteristic
177C22×D3—38no—
178C22×D3—64no—
179C3×Q8—181no—
180C3×D4—22no—
181C3×D4—42no—
182C3×D4—68no—
183C3×D4—128no—
184C3×D4—181no—
185C3×D4—182no—
186C3×D4—182no—
187C3×D4—182no—
188C3×D4—364no—
189C2×C3×C4—21no—
190C2×C3×C4—34no—
191C2×C3×C4—64no—
192C2×C3×C4—92no—
193C2×C3×C4—182no—
194C2×C3×C4—181no—
195C2×C3×C4—182no—
196C2×C3×C4#63811yescharacteristic
197#67—68no—
198#67—68no—
199#67—68no—
200#67—1216no—
201C2×Dic3—38no—
202C2×Dic3—64no—
203C2×Dic3—68no—
204D12—64no—
205C4×D3—68no—
206Dic6—64no—
207C2×C32—28no—
208C2×C32—24no—
209C2×C32—44no—
210C2×C32—48no—
211C2×C32#93714yes—
212C3×D3—128no—
213C24C2×#27211yescharacteristic, complemented
214#41—182no—
215#41—181no—
216C2×D4—21no—
217C2×D4—92no—
218C2×D4—182no—
219C2×D4—181no—
220C2×D4—182no—
221C2×D4C2×#27211yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—92no—
224C22⋊C4—182no—
225C22⋊C4C2×#27211yescharacteristic, complemented
226C22×C3—22no—
227C22×C3—24no—
228C22×C3—21no—
229C22×C3—38no—
230C22×C3—34no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—42no—
235C22×C3—41no—
236C22×C3—64no—
237C22×C3—616no—
238C22×C3—68no—
239C22×C3—124no—
240C22×C3—124no—
241C22×C3—124no—
242C22×C3—128no—
243C22×C3—124no—
244C22×C3—182no—
245C22×C3—362no—
246C22×C3—721no—
247C22×C3D4×C3⋊D312yes—
248C22×C3C2×#63811yescharacteristic
249C2×D3—68no—
250C2×D3—128no—
251C2×D3—244no—
252C3×C4—21no—
253C3×C4—41no—
254C3×C4—64no—
255C3×C4—124no—
256C3×C4—182no—
257C3×C4—362no—
258Dic3—68no—
259Dic3—128no—
260C32#503214yescomplemented
261C23—22no—
262C23—21no—
263C23—42no—
264C23—92no—
265C23—181no—
266C23#623511yescharacteristic, complemented
267C23C22×#27211yescharacteristic
268Q8—181no—
269D4—22no—
270D4—42no—
271D4—181no—
272D4—182no—
273D4—182no—
274D4—182no—
275D4—364no—
276C2×C4—21no—
277C2×C4—92no—
278C2×C4—182no—
279C2×C4—181no—
280C2×C4—182no—
281C2×C4#623511yescharacteristic, complemented
282C2×C3—21no—
283C2×C3—22no—
284C2×C3—34no—
285C2×C3—42no—
286C2×C3—41no—
287C2×C3—68no—
288C2×C3—64no—
289C2×C3—124no—
290C2×C3—128no—
291C2×C3—362no—
292C2×C3#6414611yescharacteristic
293D3—128no—
294C22—22no—
295C22—24no—
296C22—21no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22—41no—
302C22—182no—
303C22—362no—
304C22—721no—
305C22D4×#27212yes—
306C22C2×#623511yescharacteristic
307C4—21no—
308C4—41no—
309C4—182no—
310C4—362no—
311C3—34no—
312C3#1059273111yescharacteristic
313C2—21no—
314C2—22no—
315C2—42no—
316C2—41no—
317C2—362no—
318C2#1172184711yescharacteristic
subgroupG/H
center, socleC2×C3#64146
commutator, cosocleC23×H(3,𝔽3)C23
FrattiniC23×C3C22×C3⋊D3
FittingH(3,𝔽3)×#120C2
hypercenterC3×#120C3⋊D3
hypocenterH(3,𝔽3)B(4,𝔽2)
LCSG ▷ C23×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. #11722664C2semidirect
2. H(3,𝔽3)×#120C2semidirect
3. #11721876C2semidirect
4. #11720580C2semidirect
5. #11720524C2semidirect
7. C24×H(3,𝔽3)C22semidirect
17. C22⋊C4×H(3,𝔽3)C22semidirect
19. #88552C22semidirect
21. C32×#120D3semidirect
28. C23×H(3,𝔽3)D4semidirect
31. C22×#272D4semidirect
35. C2×C4×H(3,𝔽3)D4semidirect
42. C2×#1084D4semidirect
48. C24×C32C2×D3semidirect
58. C32×C22⋊C4C2×D3semidirect
91. C23×C32#67semidirect
98. C2×C32×C4#67semidirect
161. #120#272semidirect
164. H(3,𝔽3)B(4,𝔽2)semidirect
213. C24C2×#272semidirect
225. C22⋊C4C2×#272semidirect
260. C32#5032semidirect
266. C23#6235semidirect
281. C2×C4#6235semidirect

Maximal subgroups

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

subgroupindex
#117226642
H(3,𝔽3)×#1202
#117218762
#117205802
#117205242
C3×#50323