Finite Group Databasebeta

← #13152099

Subgroups — #13152099

Order 1728.

1040 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 129 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#11721876C211yesmaximal, characteristic, complemented
2#11721702C212yesmaximal, complemented
3#11721324C213yesmaximal, complemented
4#11721304C216yesmaximal, complemented
5#11721294C213yesmaximal, complemented
6#10595529—32nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic, complemented
8C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic, complemented
9C2×#6235C2213yes—
10C2×#638⋊C3C2216yes—
11C2×#6196C2213yescomplemented
12#88552C2213yescomplemented
13#3757⋊C3C2216yescomplemented
14#88451C2213yescomplemented
15#88450C2216yescomplemented
16#88447C2213yescomplemented
17#64162D312yescomplemented
18C3×#953—32no—
19C3×#953—32no—
20C3×#953—61no—
21#63983—36no—
22#63959—36no—
23#63959—36no—
24#63953—36no—
25C23×H(3,𝔽3)—43no—
26C23×H(3,𝔽3)C2313yes—
27C22×#272C2311yescharacteristic
28C22×(C3⋊D3)⋊C3C2312yes—
29C2×((C3⋊D3)⋊C3)⋊C2—43no—
30#6235—43no—
31C2×#1084C2313yes—
32#638⋊C3—46no—
33C2×(C3⋊Dic3)⋊C3C2316yes—
34#6196—43no—
35#5594—91no—
36#5594—92no—
37C24×C32—21no—
38C24×C32C2×D312yescomplemented
39C22×D32—32no—
40C2×#638C2×D316yes—
41C2×C3×#67—36no—
42C2×C3×#67—36no—
43C2×C3×#67—63no—
44C2×#627—36no—
45#3757C2×D316yescomplemented
46C3×#217—36no—
47C3×#217—36no—
48C3×#217—63no—
49#3722—36no—
50(C2×D3)⋊Dic3—36no—
51(C2×D3)⋊Dic3—36no—
52Dic32—36no—
53C22×H(3,𝔽3)—23no—
54C22×H(3,𝔽3)—26no—
55C22×H(3,𝔽3)—44no—
56C22×H(3,𝔽3)C2411yescharacteristic
57C2×#272—23no—
58C2×(C3⋊D3)⋊C3—26no—
59((C3⋊D3)⋊C3)⋊C2—44no—
60#1084—23no—
61(C3⋊Dic3)⋊C3—26no—
62C3×#120—92no—
63C3×#120—91no—
64#953—32no—
65#953—32no—
66#953—61no—
67#953—92no—
68#937—93no—
69#937—96no—
70#936—93no—
71#936—96no—
72#885—96no—
73#885—96no—
74#885—96no—
75C23×C32—23no—
76C23×C32—46no—
77C23×C32—46no—
78C23×C32C22×D316yes—
79C22×C3⋊D3C22×D312yes—
80C22×C3×D3—32no—
81C22×C3×D3—32no—
82C22×C3×D3—61no—
83C2×D32—126no—
84#638—46no—
85C2×C3⋊Dic3C22×D316yes—
86C3×#67—126no—
87C3×#67—126no—
88C3×#67—126no—
89C2×C3×Dic3—36no—
90C2×C3×Dic3—36no—
91C2×C3×Dic3—63no—
92#627—126no—
93H(3,𝔽4)—271noSylow
94C2×H(3,𝔽3)—43no—
95C2×H(3,𝔽3)H(5,𝔽2)13yes—
96#272—41no—
97(C3⋊D3)⋊C3—42no—
98C24×C3—32no—
99C24×C3—61no—
100C24×C3D3211yescharacteristic
101C23×D3—91no—
102C23×D3—92no—
103C2×C3×D4—96no—
104C2×C3×D4—93no—
105C2×#67—36no—
106C2×#67—36no—
107C2×#67—63no—
108C2×#67—96no—
109C2×Dic6—93no—
110C2×Dic6—96no—
111C3×C22⋊C4—96no—
112C3×C22⋊C4—93no—
113#217—36no—
114#217—36no—
115#217—63no—
116#217—96no—
117#217—93no—
118#217—96no—
119(C2×D3)⋊C4—96no—
120(C2×D3)⋊C4—96no—
121(C2×D3)⋊C4—96no—
122C4×Dic3—93no—
123C4×Dic3—96no—
124C22×C32—26no—
125C22×C32—26no—
126C22×C32—26no—
127C22×C32—26no—
128C22×C32—21no—
129C22×C32—46no—
130C22×C32—48no—
131C22×C32—48no—
132C22×C32C23×D312yes—
133C2×C3⋊D3—26no—
134C2×C3×D3—66no—
135C2×C3×D3—66no—
136C2×C3×D3—123no—
137D32—128no—
138C3⋊Dic3—26no—
139C3×Dic3—66no—
140C3×Dic3—66no—
141C3×Dic3—123no—
142#124—276no—
143#124—273no—
144#120—91no—
145#120—92no—
146#120—273no—
147H(3,𝔽3)H(3,𝔽4)11yescharacteristic, complemented, Sylow
148C23×C3—36no—
149C23×C3—43no—
150C23×C3—63no—
151C23×C3—92no—
152C23×C3—91no—
153C23×C3—126no—
154C23×C3—126no—
155C23×C3C2×D3213yes—
156C22×D3—32no—
157C22×D3—32no—
158C22×D3—61no—
159C22×D3—92no—
160C22×D3—363no—
161C22×D3—366no—
162C3×D4—363no—
163C3×D4—366no—
164C2×C3×C4—96no—
165C2×C3×C4—93no—
166#67—126no—
167#67—126no—
168#67—126no—
169#67—366no—
170C2×Dic3—36no—
171C2×Dic3—36no—
172C2×Dic3—63no—
173C2×Dic3—96no—
174Dic6—363no—
175Dic6—366no—
176C2×C32—23no—
177C2×C32—46no—
178C2×C32—46no—
179C2×C32#100416yes—
180C3⋊D3—42no—
181C3×D3—122no—
182C3×D3—122no—
183C3×D3—241no—
184C24—271no—
185C24((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
186C2×Q8—273no—
187C2×D4—93no—
188C2×D4—96no—
189C22⋊C4—93no—
190C22⋊C4—96no—
191C22⋊C4—276no—
192C22⋊C4—273no—
193C42—273no—
194C22×C3—26no—
195C22×C3—23no—
196C22×C3—32no—
197C22×C3—44no—
198C22×C3—66no—
199C22×C3—66no—
200C22×C3—66no—
201C22×C3—66no—
202C22×C3—61no—
203C22×C3—126no—
204C22×C3—128no—
205C22×C3—128no—
206C22×C3—186no—
207C22×C3—183no—
208C22×C3C22×D3211yescharacteristic
209C2×D3—66no—
210C2×D3—66no—
211C2×D3—123no—
212C2×D3—186no—
213C2×D3—364no—
214C2×D3—368no—
215C3×C4—186no—
216C3×C4—183no—
217Dic3—66no—
218Dic3—66no—
219Dic3—123no—
220Dic3—186no—
221C32—21no—
222C32#559412yescomplemented
223C23—43no—
224C23—91no—
225C23—92no—
226C23—1083no—
227C23C2×((C3⋊D3)⋊C3)⋊C213yes—
228Q8—1083no—
229D4—363no—
230D4—366no—
231C2×C4—93no—
232C2×C4—96no—
233C2×C3—36no—
234C2×C3—43no—
235C2×C3—63no—
236C2×C3—126no—
237C2×C3—126no—
238C2×C3—362no—
239C2×C3—361no—
240C2×C3#6433013yes—
241D3—122no—
242D3—122no—
243D3—241no—
244D3—362no—
245C22—23no—
246C22—26no—
247C22—44no—
248C22—183no—
249C22—186no—
250C22—1084no—
251C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C4—183no—
253C4—186no—
254C3—32no—
255C3—61no—
256C3#1059552911yescharacteristic
257C2—43no—
258C2—361no—
259C2—362no—
260C2#1172254513yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)H(3,𝔽4)
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. #11721876C2semidirect
2. #11721702C2semidirect
3. #11721324C2semidirect
4. #11721304C2semidirect
5. #11721294C2semidirect
7. C24×H(3,𝔽3)C22semidirect
8. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
11. C2×#6196C22semidirect
12. #88552C22semidirect
13. #3757⋊C3C22semidirect
14. #88451C22semidirect
15. #88450C22semidirect
16. #88447C22semidirect
17. #64162D3semidirect
38. C24×C32C2×D3semidirect
45. #3757C2×D3semidirect
147. H(3,𝔽3)H(3,𝔽4)semidirect
185. C24((C3⋊D3)⋊C3)⋊C2semidirect
222. C32#5594semidirect

Maximal subgroups

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

subgroupindex
#117218762
#117217022
#117213242
#117213042
#117212942
#105955293