Finite Group Databasebeta

← #13137888

Subgroups — #13137888

Order 1728.

441 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721758C211yesmaximal, characteristic, complemented
2#11721144C211yesmaximal, characteristic
3#63636⋊C3C212yesmaximal, complemented
4#11720308C212yesmaximal, complemented
5#11720303C211yesmaximal, characteristic, complemented
6#10590550—32nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
8C2×#6198—21no—
9C2×#6198—21no—
10C4⋊C4×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084—21no—
12C4⋊#1084C2211yescharacteristic
13#88540—21no—
14C4×#1084C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16C3×#890—32no—
17C3×#890—61no—
18#63911—32no—
19#63636D312yescomplemented
20C3×#809—32no—
21#63564—32no—
22#63562—32no—
23#63554—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
30C4⋊((C3⋊D3)⋊C3)—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32#6198—42no—
33#6198—42no—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5067—91no—
36#4764—92no—
37C2×C4⋊(C3⋊D3)C2×D312yes—
38C2×C3×D12—32no—
39C2×#626—62no—
40C2×#626—62no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4C2×D312yes—
43C2×C3⋊(C3⋊C8)C2×D312yes—
44C3×C4⋊Dic3—32no—
45C3×C4⋊Dic3—62no—
46C3×C4⋊Dic3—61no—
47C3×C4⋊Dic3—62no—
48C3×Dic3⋊C4—62no—
49C3×Dic3⋊C4—62no—
50C3×C4×Dic3—32no—
51C3×C4×Dic3—61no—
52C2×C3×C3⋊C8—32no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×(C3⋊D3)⋊C3—42no—
55C2×(C3⋊D3)⋊C3—42no—
56C2×(C3⋊D3)⋊C3—42no—
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)#4112yes—
59#1084—21no—
60#1084—21no—
61#1084—41no—
62C3×#125—91no—
63#940—92no—
64#890—32no—
65#890—61no—
66#874—91no—
67C3×#102—92no—
68#832—92no—
69#821—92no—
70#815—92no—
71#809—32no—
72#809—92no—
73#809—92no—
74#803—91no—
75C22×C3⋊D3—22no—
76C22×C3×D3—62no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—21no—
80C2×C32×C4C22×D312yes—
81C4⋊(C3⋊D3)—22no—
82C4⋊(C3⋊D3)—22no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C3×D12—62no—
90C3×D12—62no—
91#626—124no—
92#626—124no—
93C3⋊(C3⋊C8)—22no—
94C3×C3⋊C8—62no—
95#490—271noSylow
96C2×H(3,𝔽3)#12411yescharacteristic
97C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
98C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
99(C3⋊D3)⋊C3—82no—
100C2×C3×D4—92no—
101C2×#67—182no—
102C2×#67—182no—
103C2×D12—32no—
104C2×D12—92no—
105C2×D12—181no—
106C2×D12—181no—
107C2×C3×C8—92no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—61no—
110C3×C4⋊C4—91no—
111C3×C4⋊C4—181no—
112C3×C4⋊C4—181no—
113C3×C4⋊C4D3211yescharacteristic
114C3×C42—91no—
115C4⋊Dic3—32no—
116C4⋊Dic3—61no—
117C4⋊Dic3—62no—
118C4⋊Dic3—62no—
119Dic3⋊C4—62no—
120Dic3⋊C4—62no—
121C4×Dic3—32no—
122C4×Dic3—61no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—92no—
125C22×C32—21no—
126C22×C32D3×D412yes—
127C2×C3⋊D3—42no—
128C2×C3⋊D3—42no—
129C2×C3⋊D3—42no—
130C2×C3×D3—122no—
131C2×C3×D3—122no—
132C2×C3×D3—122no—
133C32×C4—22no—
134C32×C4—42no—
135C32×C4—42no—
136C32×C4#23512yes—
137C32×C4#23712yes—
138C3×Dic3—62no—
139C3×Dic3—62no—
140C3×Dic3—121no—
141C3×Dic3—121no—
142C3×Dic3—122no—
143C3×Dic3—122no—
144#127—271no—
145#125—91no—
146#102—92no—
147#102—272no—
148#97—271no—
149H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
150C23×C3—182no—
151C22×D3—62no—
152C22×D3—182no—
153C3×D4—182no—
154C3×D4—182no—
155C2×C3×C4—21no—
156C2×C3×C4—32no—
157C2×C3×C4—62no—
158C2×C3×C4—62no—
159C2×C3×C4—61no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—181no—
163C2×C3×C4C2×D3211yescharacteristic
164#67—364no—
165#67—364no—
166C2×Dic3—32no—
167C2×Dic3—32no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—62no—
171C2×Dic3—62no—
172D12—62no—
173D12—62no—
174D12—182no—
175D12—182no—
176D12—362no—
177D12—362no—
178C3×C8—182no—
179C3⋊C8—62no—
180C3⋊C8—182no—
181C2×C32—21no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32#88512yes—
185C2×C32#91112yes—
186C2×C32#91412yes—
187C3⋊D3—82no—
188C3×D3—242no—
189C2×D4—92no—
190C2×D4—541no—
191C2×D4—541no—
192C2×C8—92no—
193C4⋊C4—91no—
194C4⋊C4—181no—
195C4⋊C4—181no—
196C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
197C42—91no—
198C22×C3—32no—
199C22×C3—61no—
200C22×C3—362no—
201C22×C3—362no—
202C22×C3—362no—
203C22×C3#381111yescharacteristic
204C2×D3—122no—
205C2×D3—122no—
206C2×D3—122no—
207C2×D3—362no—
208C2×D3—362no—
209C2×D3—362no—
210C3×C4—32no—
211C3×C4—32no—
212C3×C4—41no—
213C3×C4—62no—
214C3×C4—122no—
215C3×C4—122no—
216C3×C4—181no—
217C3×C4—181no—
218C3×C4—361no—
219C3×C4#379512yes—
220Dic3—62no—
221Dic3—62no—
222Dic3—121no—
223Dic3—121no—
224Dic3—122no—
225Dic3—122no—
226C32—21no—
227C32#476412yescomplemented
228C23—182no—
229D4—182no—
230D4—182no—
231D4—1082no—
232D4—1082no—
233C2×C4—21no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—181no—
237C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C8—182no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—32no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—61no—
245C2×C3—722no—
246C2×C3#6391311yescharacteristic
247C2×C3#6392711yescharacteristic
248C2×C3#6394111yescharacteristic
249D3—242no—
250D3—722no—
251C22—362no—
252C22—362no—
253C22—362no—
254C22#8867711yescharacteristic
255C4—41no—
256C4—181no—
257C4—181no—
258C4—361no—
259C4#8864812yes—
260C3—32no—
261C3—61no—
262C3#1059055411yescharacteristic
263C2—722no—
264C2#1172114311yescharacteristic
265C2#1172119111yescharacteristic
266C2#1172122911yescharacteristic
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)#490
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. #11721758C2semidirect
3. #63636⋊C3C2semidirect
4. #11720308C2semidirect
5. #11720303C2semidirect
19. #63636D3semidirect
149. H(3,𝔽3)#490semidirect
227. C32#4764semidirect

Maximal subgroups

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

subgroupindex
#117217582
#117211442
#63636⋊C32
#117203082
#117203032
#105905503