Finite Group Databasebeta

← #13151923

Subgroups — #13151923

Order 1728.

1000 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 169 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#11722638C212yesmaximal, complemented
2#11721869C211yesmaximal, characteristic, complemented
3#11721061C216yesmaximal, complemented
4#11721060C213yesmaximal, complemented
5C2×#88432C213yesmaximal, complemented
6#10595449—32nomaximal
7#41×H(3,𝔽3)C2211yescharacteristic
8#88722—22no—
9D4×(C3⋊D3)⋊C3—26no—
10#3828⋊C3C2216yescomplemented
11C2×C4⋊((C3⋊D3)⋊C3)C2216yescomplemented
12#88538C2213yescomplemented
13C2×#6180C2213yescomplemented
14#88433C2211yescharacteristic, complemented
15#88432C2213yescomplemented
16#88432C2216yescomplemented
17#88431C2216yescomplemented
18#64366D312yescomplemented
19C3×#1009—32no—
20C3×#948—32no—
21C3×#948—61no—
22#63830—36no—
23#63829—36no—
24#63825—36no—
25C2×#3714—36no—
26C22×(C3⋊D3)⋊C3—26no—
27Q8×H(3,𝔽3)D411yescharacteristic
28D4×H(3,𝔽3)D413yes—
29C2×C4×H(3,𝔽3)C2313yes—
30#638⋊C3—26no—
31C4⋊((C3⋊D3)⋊C3)—26no—
32C4⋊((C3⋊D3)⋊C3)C2316yes—
33C4×(C3⋊D3)⋊C3—26no—
34#634⋊C3C2312yes—
35#6180C2313yes—
36#6180C2311yescharacteristic
37#5835—92no—
38#5752—91no—
39C32×#41—21no—
40C32×#41C2×D312yes—
41#3832—22no—
42D4×C3⋊D3—26no—
43#3828C2×D316yescomplemented
44C2×C4⋊(C3⋊D3)C2×D316yescomplemented
45C3×#239—62no—
46C3×D3×D4—66no—
47C3×#235—36no—
48C2×C3×D12—36no—
49C3×#208—36no—
50C3×#208—63no—
51C2×C3×C3⋊C8—36no—
52C2×C3×C3⋊C8—63no—
53#3719—32no—
54#3717—36no—
55#3716—36no—
56#3714—36no—
57#3714—312no—
58C22×H(3,𝔽3)C2×D413yes—
59C2×(C3⋊D3)⋊C3—26no—
60C2×(C3⋊D3)⋊C3—86no—
61C4×H(3,𝔽3)C2×D413yes—
62C4×H(3,𝔽3)C2411yescharacteristic
63(C3⋊Dic3)⋊C3—22no—
64C3×H(5,𝔽2)—92no—
65#1009—32no—
66#1009—92no—
67C3×#131—91no—
68#950—96no—
69#949—96no—
70#948—32no—
71#948—61no—
72#932—96no—
73C2×#213—96no—
74#908—96no—
75#904—93no—
76C2×D24—93no—
77C22×C3⋊D3—26no—
78C22×C3×D3—66no—
79C32×Q8—21no—
80C32×Q8#6712yes—
81C32×D4—23no—
82C32×D4#6716yes—
83C2×C32×C4—23no—
84C2×C32×C4C22×D316yes—
85#638—26no—
86C4⋊(C3⋊D3)—26no—
87C4⋊(C3⋊D3)C22×D316yes—
88C4×C3⋊D3—26no—
89#634C22×D312yes—
90C3×#67—66no—
91C3×D12—36no—
92C3×D12—66no—
93C3×C4×D3—66no—
94C3×Dic6—32no—
95C3×C3⋊C8—32no—
96C3×C3⋊C8—36no—
97C3×C3⋊C8—61no—
98C3×C3⋊C8—63no—
99#576—271noSylow
100C2×H(3,𝔽3)—23no—
101C2×H(3,𝔽3)C22×D411yescharacteristic
102(C3⋊D3)⋊C3—46no—
103C3×#41—32no—
104C3×#41—61no—
105C3×#41—96no—
106C3×#41—182no—
107C3×#41D3211yescharacteristic
108C2×C3×D4—96no—
109C2×C3×D4—186no—
110#239—62no—
111#239—182no—
112D3×D4—66no—
113D3×D4—186no—
114#235—36no—
115#235—96no—
116C2×D12—36no—
117C2×D12—96no—
118C3×M16—93no—
119C2×C3×C8—93no—
120#216—92no—
121#215—96no—
122#214—96no—
123#213—96no—
124#213—912no—
125#208—36no—
126#208—63no—
127C2×C3⋊C8—36no—
128C2×C3⋊C8—63no—
129Dic12—91no—
130D24—93no—
131D24—96no—
132#204—96no—
133C22×C32—23no—
134C22×C32C2×#6716yes—
135C2×C3⋊D3—26no—
136C2×C3⋊D3—86no—
137C2×C3×D3—66no—
138C2×C3×D3—246no—
139C32×C4—21no—
140C32×C4—23no—
141C32×C4C2×#6716yes—
142C32×C4C23×D312yes—
143C3⋊Dic3—22no—
144C3×Dic3—62no—
145H(5,𝔽2)—92no—
146AGL(1,ℤ/8)—276no—
147#135—273no—
148C2×D8—273no—
149#131—91no—
150H(3,𝔽3)#57611yescharacteristic, complemented, Sylow
151C23×C3—186no—
152C22×D3—66no—
153C22×D3—186no—
154C3×Q8—32no—
155C3×Q8—61no—
156C3×Q8—92no—
157C3×Q8#62511yescharacteristic
158C3×D4—36no—
159C3×D4—63no—
160C3×D4—96no—
161C3×D4—186no—
162C3×D4—186no—
163C3×D4#62513yes—
164C2×C3×C4—36no—
165C2×C3×C4—63no—
166C2×C3×C4—186no—
167C2×C3×C4C2×D3213yes—
168#67—66no—
169#67—186no—
170D12—36no—
171D12—66no—
172D12—96no—
173D12—186no—
174C4×D3—66no—
175C4×D3—186no—
176Dic6—32no—
177Dic6—92no—
178C3×C8—93no—
179C3×C8—91no—
180C3⋊C8—32no—
181C3⋊C8—36no—
182C3⋊C8—61no—
183C3⋊C8—63no—
184C2×C32—26no—
185C2×C32—21no—
186C2×C32—43no—
187C2×C32C22×#6712yes—
188C3⋊D3—46no—
189C3×D3—126no—
190#41—96no—
191#41—182no—
192#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
193C2×D4—96no—
194C2×D4—186no—
195Q16—271no—
196SD16—276no—
197D8—273no—
198D8—276no—
199M16—93no—
200C2×C8—93no—
201C22×C3—36no—
202C22×C3—63no—
203C22×C3—186no—
204C22×C3—726no—
205C22×C3C2×#62513yes—
206C2×D3—66no—
207C2×D3—186no—
208C2×D3—246no—
209C2×D3—726no—
210C3×C4—32no—
211C3×C4—36no—
212C3×C4—61no—
213C3×C4—63no—
214C3×C4—182no—
215C3×C4C2×#62513yes—
216C3×C4C22×D3211yescharacteristic
217Dic3—62no—
218Dic3—182no—
219C32—21no—
220C32#583512yescomplemented
221C23—186no—
222Q8—92no—
223Q8#619811yescharacteristic
224D4—96no—
225D4—186no—
226D4—186no—
227D4#619813yes—
228C2×C4—186no—
229C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
230C8—91no—
231C8—93no—
232C2×C3—23no—
233C2×C3—32no—
234C2×C3—66no—
235C2×C3—61no—
236C2×C3—123no—
237C2×C3—366no—
238C2×C3C22×#62511yescharacteristic
239D3—126no—
240D3—366no—
241C22—186no—
242C22—726no—
243C22C2×#619813yes—
244C4—182no—
245C4C2×#619813yes—
246C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
247C3—32no—
248C3—61no—
249C3#1059544411yescharacteristic
250C2—23no—
251C2—366no—
252C2C22×#619811yescharacteristic
subgroupG/H
centerC2C22×#6198
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3C22×#625
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#576
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11722638C2semidirect
2. #11721869C2semidirect
3. #11721061C2semidirect
4. #11721060C2semidirect
5. C2×#88432C2semidirect
10. #3828⋊C3C22semidirect
11. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
12. #88538C22semidirect
13. C2×#6180C22semidirect
14. #88433C22semidirect
15. #88432C22semidirect
16. #88432C22semidirect
17. #88431C22semidirect
18. #64366D3semidirect
43. #3828C2×D3semidirect
44. C2×C4⋊(C3⋊D3)C2×D3semidirect
150. H(3,𝔽3)#576semidirect
220. C32#5835semidirect

Maximal subgroups

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

subgroupindex
#117226382
#117218692
#117210612
#117210602
C2×#884322
#105954493