Finite Group Databasebeta

← #13137886

Subgroups — #13137886

Order 1728.

481 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721756C211yesmaximal, characteristic, complemented
2#11721144C211yesmaximal, characteristic
3#63636⋊C3C212yesmaximal, complemented
4#11720308C212yesmaximal, complemented
5#11720302C211yesmaximal, characteristic, complemented
6#10590548—32nomaximal
7C2×#6229—21no—
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×#6198—21no—
10C2×#6198—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13#88540—21no—
14C4×#1084C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16C3×#888—32no—
17C3×#888—61no—
18#63911—32no—
19#63636D312yescomplemented
20C3×#809—32no—
21#63564—32no—
22#63562—32no—
23Dic3×C3⋊C8—32no—
24C22×(C3⋊D3)⋊C3—22no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—21no—
28C2×#1084D411yescharacteristic
29C2×#1084D411yescharacteristic
30#6229—42no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33#6198—42no—
34#6198—42no—
35(C3⋊(C3⋊C8))⋊C3—22no—
36#5088—91no—
37#4760—92no—
38C2×C4⋊(C3⋊D3)C2×D312yes—
39C2×C3×D12—32no—
40C2×C3×Dic6—62no—
41C2×C3×Dic6—62no—
42C2×#626—62no—
43C2×#626—62no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4C2×D312yes—
46C2×C3⋊(C3⋊C8)C2×D312yes—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—61no—
49C3×Dic3⋊C4—62no—
50C3×Dic3⋊C4—62no—
51C3×C4×Dic3—32no—
52C3×C4×Dic3—61no—
53C2×C3×C3⋊C8—32no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×(C3⋊D3)⋊C3—42no—
56C2×(C3⋊D3)⋊C3—42no—
57C2×(C3⋊D3)⋊C3—42no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)#4112yes—
60#1084—41no—
61#1084D812yes—
62#1084SD1612yes—
63C3×C4⋊Q8—91no—
64#940—92no—
65#888—32no—
66#888—61no—
67#874—91no—
68C3×#102—92no—
69#832—92no—
70#821—92no—
71C8×Dic3—92no—
72#809—32no—
73#809—92no—
74#809—92no—
75C4×C3⋊C8—91no—
76C22×C3⋊D3—22no—
77C22×C3×D3—62no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—21no—
81C2×C32×C4C22×D312yes—
82C4⋊(C3⋊D3)—22no—
83C4⋊(C3⋊D3)—22no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—32no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C3×D12—62no—
91C3×D12—62no—
92C3×Dic6—124no—
93C3×Dic6—124no—
94#626—124no—
95#626—124no—
96C3⋊(C3⋊C8)—22no—
97C3×C3⋊C8—62no—
98#486—271noSylow
99C2×H(3,𝔽3)#12411yescharacteristic
100C2×H(3,𝔽3)C2×D811yescharacteristic
101C2×H(3,𝔽3)C2×SD1611yescharacteristic
102(C3⋊D3)⋊C3—82no—
103C2×C3×Q8—181no—
104C2×C3×D4—92no—
105C2×#67—182no—
106C2×#67—182no—
107C2×D12—32no—
108C2×D12—92no—
109C2×D12—181no—
110C2×D12—181no—
111C2×Dic6—62no—
112C2×Dic6—62no—
113C2×C3×C8—92no—
114C3×C4⋊C4—32no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—91no—
117C3×C4⋊C4—181no—
118C3×C4⋊C4D3211yescharacteristic
119C3×C42—91no—
120C4⋊Dic3—32no—
121C4⋊Dic3—61no—
122Dic3⋊C4—62no—
123Dic3⋊C4—62no—
124C4×Dic3—32no—
125C4×Dic3—61no—
126C2×C3⋊C8—32no—
127C2×C3⋊C8—92no—
128C22×C32—21no—
129C22×C32D3×D412yes—
130C2×C3⋊D3—42no—
131C2×C3⋊D3—42no—
132C2×C3⋊D3—42no—
133C2×C3×D3—122no—
134C2×C3×D3—122no—
135C2×C3×D3—122no—
136C32×C4—22no—
137C32×C4—42no—
138C32×C4—42no—
139C32×C4#23512yes—
140C32×C4#23712yes—
141C3×Dic3—34no—
142C3×Dic3—34no—
143C3×Dic3—62no—
144C3×Dic3—62no—
145C3×Dic3—122no—
146C3×Dic3—122no—
147C4⋊Q8—91no—
148#127—271no—
149#102—92no—
150#102—272no—
151C4×C8—271no—
152H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
153C23×C3—182no—
154C22×D3—62no—
155C22×D3—182no—
156C3×Q8—362no—
157C3×D4—182no—
158C3×D4—182no—
159C2×C3×C4—21no—
160C2×C3×C4—32no—
161C2×C3×C4—62no—
162C2×C3×C4—62no—
163C2×C3×C4—61no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—181no—
167C2×C3×C4C2×D3211yescharacteristic
168#67—364no—
169#67—364no—
170C2×Dic3—32no—
171C2×Dic3—32no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176D12—62no—
177D12—62no—
178D12—182no—
179D12—182no—
180D12—362no—
181D12—362no—
182Dic6—124no—
183Dic6—124no—
184C3×C8—182no—
185C3⋊C8—62no—
186C3⋊C8—182no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32#88512yes—
191C2×C32D3×D812yes—
192C2×C32D3×SD1612yes—
193C3⋊D3—82no—
194C3×D3—242no—
195C2×Q8—181no—
196C2×D4—92no—
197C2×D4—541no—
198C2×D4—541no—
199C2×C8—92no—
200C4⋊C4—91no—
201C4⋊C4—181no—
202C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
203C42—91no—
204C22×C3—32no—
205C22×C3—61no—
206C22×C3—362no—
207C22×C3—362no—
208C22×C3—362no—
209C22×C3#381111yescharacteristic
210C2×D3—122no—
211C2×D3—122no—
212C2×D3—122no—
213C2×D3—362no—
214C2×D3—362no—
215C2×D3—362no—
216C3×C4—32no—
217C3×C4—32no—
218C3×C4—41no—
219C3×C4—62no—
220C3×C4—92no—
221C3×C4—92no—
222C3×C4—122no—
223C3×C4—122no—
224C3×C4—361no—
225C3×C4#379512yes—
226Dic3—34no—
227Dic3—34no—
228Dic3—62no—
229Dic3—62no—
230Dic3—122no—
231Dic3—122no—
232C32—21no—
233C32#476012yescomplemented
234C23—182no—
235Q8—362no—
236D4—182no—
237D4—182no—
238D4—1082no—
239D4—1082no—
240C2×C4—21no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—181no—
244C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
245C8—182no—
246C2×C3—32no—
247C2×C3—32no—
248C2×C3—32no—
249C2×C3—61no—
250C2×C3—61no—
251C2×C3—61no—
252C2×C3—722no—
253C2×C3#6391311yescharacteristic
254C2×C3#6392611yescharacteristic
255C2×C3#6394011yescharacteristic
256D3—242no—
257D3—722no—
258C22—362no—
259C22—362no—
260C22—362no—
261C22#8867711yescharacteristic
262C4—41no—
263C4—92no—
264C4—92no—
265C4—361no—
266C4#8864812yes—
267C3—32no—
268C3—61no—
269C3#1059055211yescharacteristic
270C2—722no—
271C2#1172114311yescharacteristic
272C2#1172119011yescharacteristic
273C2#1172122811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721144C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#486
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721756C2semidirect
3. #63636⋊C3C2semidirect
4. #11720308C2semidirect
5. #11720302C2semidirect
19. #63636D3semidirect
152. H(3,𝔽3)#486semidirect
233. C32#4760semidirect

Maximal subgroups

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

subgroupindex
#117217562
#117211442
#63636⋊C32
#117203082
#117203022
#105905483