Finite Group Databasebeta

← #13153278

Subgroups — #13153278

Order 1728.

990 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 202 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
1H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
2#64148⋊C3C216yesmaximal, complemented
3#64146⋊C3C213yesmaximal, complemented
4#64144⋊C3C213yesmaximal, complemented
5#11721387C212yesmaximal, complemented
6#10596191C311yesmaximal, characteristic, complemented
7C3×#5720—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×D4×H(3,𝔽3)C2213yes—
10C2×D4×H(3,𝔽3)C2213yescomplemented
11C2×#638⋊C3C2216yes—
12C22×(C3⋊Dic3)⋊C3C2213yescomplemented
13C42×H(3,𝔽3)C2211yescharacteristic, complemented
14#3757⋊C3C2213yes—
15#3757⋊C3C2213yescomplemented
16#3752⋊C3C2216yes—
17#3750⋊C3C2216yescomplemented
18C32×#127—21no—
19C32×#127D311yescharacteristic, complemented
20C32×#127C2×C311yescharacteristic, complemented
21#64148C2×C316yescomplemented
22#64146C2×C313yescomplemented
23#64144C2×C313yescomplemented
24#64085C2×C312yescomplemented
25C3×#939—36no—
26C3×#937—33no—
27C3×#935—33no—
28C3×#876—32no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C23×H(3,𝔽3)C2313yes—
31C22×(C3⋊D3)⋊C3—43no—
32C22×(C3⋊D3)⋊C3C2312yes—
33D4×H(3,𝔽3)—43no—
34D4×H(3,𝔽3)—43no—
35C2×C4×H(3,𝔽3)C2313yes—
36#638⋊C3—46no—
37C2×(C3⋊Dic3)⋊C3—43no—
38C2×(C3⋊Dic3)⋊C3C2316yes—
39C3×#560—91no—
40#5720—31no—
41#5720—91no—
42C23×C3⋊D3C22×C311yescharacteristic
43C23×C3×D3—31no—
44C2×C32×D4—23no—
45C2×C32×D4—23no—
46C2×C32×D4C2×D313yes—
47C2×C32×D4C2×D313yescomplemented
48C2×C32×D4C22×C313yes—
49C2×C32×D4C22×C313yescomplemented
50C2×#638C22×C316yes—
51C22×C3⋊Dic3C22×C313yescomplemented
52C2×C3×#67—36no—
53C22×C3×Dic3—33no—
54C32×C42—21no—
55C32×C42C2×D311yescharacteristic, complemented
56C32×C42C22×C311yescharacteristic, complemented
57#3757C22×C313yes—
58#3757C22×C313yescomplemented
59#3752C22×C316yes—
60#3750C22×C316yescomplemented
61C3×#217—33no—
62C3×#217—33no—
63C3×(C2×D3)⋊C4—36no—
64C3×Dic3⋊C4—36no—
65C22×H(3,𝔽3)—23no—
66C22×H(3,𝔽3)—26no—
67C22×H(3,𝔽3)—23no—
68C22×H(3,𝔽3)C2411yescharacteristic
69C2×(C3⋊D3)⋊C3—26no—
70C2×(C3⋊D3)⋊C3—44no—
71C4×H(3,𝔽3)—23no—
72(C3⋊Dic3)⋊C3—26no—
73C3×#127—31no—
74C3×#127—31no—
75C3×#127—61no—
76C3×#127C3×D311yescharacteristic
77C3×#126—92no—
78C3×#123—93no—
79C3×#121—96no—
80C3×#120—93no—
81#939—36no—
82#939—96no—
83#937—33no—
84#937—93no—
85#935—33no—
86#935—93no—
87#876—32no—
88#876—92no—
89C23×C32—21no—
90C23×C32—23no—
91C23×C32C22×D311yescharacteristic
92C23×C32C22×D313yes—
93C23×C32C23×C311yescharacteristic
94C23×C32C23×C313yes—
95C22×C3⋊D3—43no—
96C22×C3⋊D3C23×C312yes—
97C22×C3×D3—32no—
98C22×C3×D3—123no—
99C32×D4—43no—
100C32×D4—43no—
101C32×D4—43no—
102C32×D4—43no—
103C32×D4—46no—
104C32×D4—46no—
105C2×C32×C4—23no—
106C2×C32×C4C22×D313yes—
107C2×C32×C4C23×C313yes—
108#638—46no—
109C2×C3⋊Dic3—43no—
110C2×C3⋊Dic3C23×C316yes—
111C3×#67—126no—
112C2×C3×Dic3—36no—
113C2×C3×Dic3—123no—
114#560—91noSylow
115C2×H(3,𝔽3)—41no—
116C2×H(3,𝔽3)—43no—
117C2×H(3,𝔽3)H(5,𝔽2)13yes—
118(C3⋊D3)⋊C3—42no—
119C24×C3—91no—
120C23×D3—31no—
121C23×D3—91no—
122C2×C3×D4—33no—
123C2×C3×D4—33no—
124C2×C3×D4—33no—
125C2×C3×D4—33no—
126C2×C3×D4—63no—
127C2×C3×D4—63no—
128C2×C3×D4—96no—
129C2×C3×D4C2×C3×D313yes—
130C2×C3×D4C2×C3×D313yes—
131C22×C3×C4—93no—
132C2×#67—36no—
133C2×#67—96no—
134C22×Dic3—33no—
135C22×Dic3—93no—
136C3×C4⋊C4—96no—
137C3×C22⋊C4—96no—
138C3×C22⋊C4—93no—
139C3×C22⋊C4—93no—
140C3×C42—31no—
141C3×C42—31no—
142C3×C42—61no—
143C3×C42C2×C3×D311yescharacteristic
144#217—33no—
145#217—33no—
146#217—93no—
147#217—93no—
148(C2×D3)⋊C4—36no—
149(C2×D3)⋊C4—96no—
150Dic3⋊C4—36no—
151Dic3⋊C4—96no—
152C22×C32—23no—
153C22×C32—26no—
154C22×C32—23no—
155C22×C32—23no—
156C22×C32—26no—
157C22×C32—23no—
158C22×C32—21no—
159C22×C32—43no—
160C22×C32—46no—
161C22×C32—43no—
162C22×C32C23×D311yescharacteristic
163C22×C32C24×C311yescharacteristic
164C2×C3⋊D3—26no—
165C2×C3⋊D3—44no—
166C2×C3×D3—66no—
167C2×C3×D3—124no—
168C32×C4—26no—
169C32×C4—23no—
170C32×C4—23no—
171C3⋊Dic3—26no—
172C3×Dic3—66no—
173#127(C3⋊D3)⋊C311yescharacteristic, complemented
174#126—92no—
175#123—93no—
176#121—96no—
177#120—93no—
178H(3,𝔽3)#56011yescharacteristic, complemented, Sylow
179C23×C3—31no—
180C23×C3—33no—
181C23×C3—31no—
182C23×C3—33no—
183C23×C3—61no—
184C23×C3—63no—
185C23×C3—92no—
186C23×C3—363no—
187C23×C3C22×C3×D311yescharacteristic
188C23×C3C22×C3×D313yes—
189C22×D3—32no—
190C22×D3—92no—
191C22×D3—123no—
192C22×D3—363no—
193C3×D4—43no—
194C3×D4—43no—
195C3×D4—123no—
196C3×D4—123no—
197C3×D4—123no—
198C3×D4—123no—
199C3×D4—126no—
200C3×D4—126no—
201C3×D4—366no—
202C2×C3×C4—33no—
203C2×C3×C4—33no—
204C2×C3×C4—63no—
205C2×C3×C4—96no—
206C2×C3×C4—363no—
207C2×C3×C4C22×C3×D313yes—
208#67—126no—
209#67—366no—
210C2×Dic3—36no—
211C2×Dic3—96no—
212C2×Dic3—123no—
213C2×Dic3—363no—
214C2×C32—23no—
215C2×C32—41no—
216C2×C32—43no—
217C2×C32—41no—
218C2×C32—43no—
219C2×C32—81no—
220C2×C32—83no—
221C2×C32#100413yes—
222C2×C32C3×H(5,𝔽2)13yes—
223C3⋊D3—42no—
224C3×D3—122no—
225C24—91no—
226C2×D4—96no—
227C2×D4C2×(C3⋊D3)⋊C313yes—
228C2×D4C2×(C3⋊D3)⋊C313yescomplemented
229C22×C4—93no—
230C4⋊C4—96no—
231C22⋊C4—93no—
232C22⋊C4—93no—
233C22⋊C4—96no—
234C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
235C22×C3—23no—
236C22×C3—26no—
237C22×C3—23no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—63no—
241C22×C3—66no—
242C22×C3—63no—
243C22×C3—66no—
244C22×C3—63no—
245C22×C3—63no—
246C22×C3—61no—
247C22×C3—123no—
248C22×C3—126no—
249C22×C3—123no—
250C22×C3—186no—
251C22×C3—364no—
252C22×C3C23×C3×D311yescharacteristic
253C2×D3—66no—
254C2×D3—124no—
255C2×D3—186no—
256C2×D3—364no—
257C3×C4—23no—
258C3×C4—63no—
259C3×C4—63no—
260C3×C4—66no—
261C3×C4—186no—
262Dic3—66no—
263Dic3—186no—
264C32—21no—
265C32#572011yescharacteristic, complemented
266C32C3×#56011yescharacteristic, complemented
267C23—92no—
268C23—363no—
269C23C22×(C3⋊D3)⋊C311yescharacteristic
270C23C22×(C3⋊D3)⋊C313yes—
271D4—43no—
272D4—43no—
273D4—366no—
274C2×C4—96no—
275C2×C4—363no—
276C2×C4C22×(C3⋊D3)⋊C313yes—
277C2×C3—33no—
278C2×C3—33no—
279C2×C3—41no—
280C2×C3—43no—
281C2×C3—63no—
282C2×C3—121no—
283C2×C3—121no—
284C2×C3—123no—
285C2×C3—123no—
286C2×C3—241no—
287C2×C3—243no—
288C2×C3—362no—
289C2×C3C3×#100413yes—
290D3—122no—
291D3—362no—
292C22—23no—
293C22—26no—
294C22—23no—
295C22—186no—
296C22—364no—
297C22C23×(C3⋊D3)⋊C311yescharacteristic
298C4—23no—
299C4—186no—
300C3—31no—
301C3—31no—
302C3—61no—
303C3C3×#572011yescharacteristic
304C2—41no—
305C2—43no—
306C2—362no—
307C2#64361⋊C313yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#127C2
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#127C2semidirect
2. #64148⋊C3C2semidirect
3. #64146⋊C3C2semidirect
4. #64144⋊C3C2semidirect
5. #11721387C2semidirect
6. #10596191C3semidirect
10. C2×D4×H(3,𝔽3)C22semidirect
12. C22×(C3⋊Dic3)⋊C3C22semidirect
13. C42×H(3,𝔽3)C22semidirect
15. #3757⋊C3C22semidirect
17. #3750⋊C3C22semidirect
19. C32×#127D3semidirect
20. C32×#127C2×C3semidirect
21. #64148C2×C3semidirect
22. #64146C2×C3semidirect
23. #64144C2×C3semidirect
24. #64085C2×C3semidirect
47. C2×C32×D4C2×D3semidirect
49. C2×C32×D4C22×C3semidirect
51. C22×C3⋊Dic3C22×C3semidirect
55. C32×C42C2×D3semidirect
56. C32×C42C22×C3semidirect
58. #3757C22×C3semidirect
60. #3750C22×C3semidirect
173. #127(C3⋊D3)⋊C3semidirect
178. H(3,𝔽3)#560semidirect
228. C2×D4C2×(C3⋊D3)⋊C3semidirect
234. C42C2×(C3⋊D3)⋊C3semidirect
265. C32#5720semidirect
266. C32C3×#560semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1272
#64148⋊C32
#64146⋊C32
#64144⋊C32
#117213872
#105961913
C3×#57203