Finite Group Databasebeta

← #13143957

Subgroups — #13143957

Order 1728.

737 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 121 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
1D4×#1084C211yesmaximal, characteristic, complemented
2C2×#88545C211yesmaximal, characteristic
3#11721761C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic, complemented
6C8×#1084C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#4756—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88552—21no—
16#88545C414yescomplemented
17#88542—21no—
18C4⋊#1084C2211yescharacteristic
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C3×D4×Dic3—34no—
22C2×C3×#213—34no—
23C3×#893—34no—
24C32×#102D314yescomplemented
25C3×#821—34no—
26C3×C8×Dic3—34no—
27C3×#807—34no—
28C23×H(3,𝔽3)—21no—
29C22×#272—21no—
30D4×H(3,𝔽3)—21no—
31D4×H(3,𝔽3)C2×C412yescomplemented
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084—42no—
36C2×#1084D411yescharacteristic
37C2×#1084D411yescharacteristic
38C4⋊#272—21no—
39C4⋊#272C2×C412yescomplemented
40C4×#272—42no—
41C8×H(3,𝔽3)—21no—
42#6180C2×C412yescomplemented
43C3×C4×D8—91no—
44#4756—34no—
45C2×C32×D4C2×D314yes—
46C22×C3×Dic3—64no—
47C2×C3×D12—34no—
48C2×C3×C4×D3—64no—
49C2×C32×C8C2×D314yes—
50C32×C4⋊C4C2×D314yes—
51C3×#217—64no—
52C3×#213—316no—
53C3×(C2×D3)⋊C4—64no—
54C3×C4⋊Dic3—34no—
55C3×C4×Dic3—34no—
56C2×C3×C3⋊C8—34no—
57C22×H(3,𝔽3)—22no—
58C22×H(3,𝔽3)—41no—
59C22×H(3,𝔽3)—41no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C2×#272—22no—
62C2×#272—41no—
63C2×#272—41no—
64C4×H(3,𝔽3)—41no—
65C4×H(3,𝔽3)C22×C411yescharacteristic
66C4×H(3,𝔽3)#4111yescharacteristic
67#1084—21no—
68#1084—41no—
69#1084D812yescomplemented
70C2×C3×D8—91no—
71C3×C4×D4—91no—
72C3×C4×D4—91no—
73D4×Dic3—34no—
74C2×#213—34no—
75#893—34no—
76C3×#107—91no—
77C3×#102—34no—
78C3×#102—91no—
79C3×#102C3⋊D311yescharacteristic
80C3×C4×C8—91no—
81#821—34no—
82C8×Dic3—34no—
83#807—34no—
84C23×C32—24no—
85C22×C3×D3—64no—
86C32×D4—24no—
87C32×D4C4×D318yescomplemented
88C2×C32×C4—24no—
89C2×C32×C4C22×D314yes—
90C2×C3×Dic3—34no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—64no—
93C2×C3×Dic3—128no—
94C3×D12—38no—
95C3×D12—64no—
96C3×C4×D3—128no—
97C32×C8—24no—
98C3×C3⋊C8—38no—
99C4×D8—91noSylow
100C2×H(3,𝔽3)—42no—
101C2×H(3,𝔽3)C4×D411yescharacteristic
102C2×H(3,𝔽3)C2×D811yescharacteristic
103C2×H(3,𝔽3)#13511yescharacteristic
104#272—42no—
105C2×C3×D4—34no—
106C2×C3×D4—91no—
107C2×C3×D4C2×C3⋊D311yescharacteristic
108C22×C3×C4—181no—
109C22×C3×C4—181no—
110C22×Dic3—64no—
111C2×D12—34no—
112C2×C4×D3—64no—
113C3×D8—94no—
114C2×C3×C8—34no—
115C2×C3×C8—91no—
116C2×C3×C8C2×C3⋊D311yescharacteristic
117C3×C4⋊C4—34no—
118C3×C4⋊C4—91no—
119C3×C4⋊C4C2×C3⋊D311yescharacteristic
120C3×C22⋊C4—181no—
121C3×C22⋊C4—181no—
122C3×C42—91no—
123#217—64no—
124#213—316no—
125(C2×D3)⋊C4—64no—
126C4⋊Dic3—34no—
127C4×Dic3—34no—
128C2×C3⋊C8—34no—
129C22×C32—28no—
130C22×C32—44no—
131C22×C32—44no—
132C22×C32D3×D414yes—
133C2×C3×D3—68no—
134C2×C3×D3—124no—
135C2×C3×D3—124no—
136C32×C4—44no—
137C32×C4C2×C4×D314yes—
138C32×C4#23714yes—
139C3×Dic3—38no—
140C3×Dic3—64no—
141C3×Dic3—124no—
142C2×D8—91no—
143C4×D4—91no—
144C4×D4—91no—
145#107—91no—
146#102—91no—
147#102#27211yescharacteristic, complemented
148C4×C8—91no—
149H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
150C23×C3—21no—
151C23×C3—64no—
152C23×C3—181no—
153C22×D3—64no—
154C3×D4—21no—
155C3×D4—38no—
156C3×D4—64no—
157C3×D4—92no—
158C3×D4—181no—
159C3×D4C4×C3⋊D312yes—
160C2×C3×C4—21no—
161C2×C3×C4—34no—
162C2×C3×C4—64no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165C2×C3×C4—181no—
166C2×C3×C4—362no—
167C2×C3×C4—362no—
168C2×C3×C4C22×C3⋊D311yescharacteristic
169C2×Dic3—34no—
170C2×Dic3—34no—
171C2×Dic3—64no—
172C2×Dic3—128no—
173D12—38no—
174D12—64no—
175C4×D3—128no—
176C3×C8—21no—
177C3×C8—64no—
178C3×C8—92no—
179C3⋊C8—38no—
180C2×C32—48no—
181C2×C32#67⋊C414yes—
182C2×C32D3×D814yes—
183C2×C32#91614yes—
184C3×D3—128no—
185C2×D4—91no—
186C2×D4C2×#27211yescharacteristic
187C22×C4—181no—
188C22×C4—181no—
189D8—94no—
190C2×C8—91no—
191C2×C8C2×#27211yescharacteristic
192C4⋊C4—91no—
193C4⋊C4C2×#27211yescharacteristic
194C22⋊C4—181no—
195C22⋊C4—181no—
196C42—91no—
197C22×C3—22no—
198C22×C3—34no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—68no—
202C22×C3—124no—
203C22×C3—124no—
204C22×C3—182no—
205C22×C3—361no—
206C22×C3—361no—
207C22×C3D4×C3⋊D311yescharacteristic
208C2×D3—68no—
209C2×D3—124no—
210C2×D3—124no—
211C3×C4—34no—
212C3×C4—34no—
213C3×C4—41no—
214C3×C4—92no—
215C3×C4—124no—
216C3×C4—181no—
217C3×C4—361no—
218C3×C4C2×C4×C3⋊D311yescharacteristic
219C3×C4#383011yescharacteristic
220Dic3—38no—
221Dic3—64no—
222Dic3—124no—
223C32#475614yescomplemented
224C23—21no—
225C23—181no—
226D4—21no—
227D4—92no—
228D4—181no—
229D4C4×#27212yescomplemented
230C2×C4—21no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—181no—
234C2×C4—362no—
235C2×C4—362no—
236C2×C4C22×#27211yescharacteristic
237C8—21no—
238C8—92no—
239C2×C3—34no—
240C2×C3—34no—
241C2×C3—34no—
242C2×C3—42no—
243C2×C3—128no—
244C2×C3—362no—
245C2×C3#6409011yescharacteristic
246C2×C3D8×C3⋊D311yescharacteristic
247C2×C3#6412511yescharacteristic
248D3—128no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—182no—
253C22—361no—
254C22—361no—
255C22D4×#27211yescharacteristic
256C4—41no—
257C4—92no—
258C4—181no—
259C4—361no—
260C4C2×C4×#27211yescharacteristic
261C4#8874411yescharacteristic
262C3—34no—
263C3#1059245511yescharacteristic
264C2—42no—
265C2—362no—
266C2#1172173711yescharacteristic
267C2D8×#27211yescharacteristic
268C2#1172180211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radicalC2×#88545C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)C4×D8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
1. D4×#1084C2semidirect
3. #11721761C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #11720554C2semidirect
6. C8×#1084C2semidirect
7. #11720529C2semidirect
16. #88545C4semidirect
24. C32×#102D3semidirect
31. D4×H(3,𝔽3)C2×C4semidirect
39. C4⋊#272C2×C4semidirect
42. #6180C2×C4semidirect
69. #1084D8semidirect
87. C32×D4C4×D3semidirect
147. #102#272semidirect
149. H(3,𝔽3)C4×D8semidirect
223. C32#4756semidirect
229. D4C4×#272semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
C2×#885452
#117217612
H(3,𝔽3)×#1022
#117205542
C8×#10842
#117205292
C3×#47563