Finite Group Databasebeta

← #13138827

Subgroups — #13138827

Order 1728.

491 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721129C212yesmaximal
3#11720550C211yesmaximal, characteristic, complemented
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#11720307C212yesmaximal, complemented
6#10590685—31nomaximal
7D3×#817—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×((C3⋊D3)⋊C3)⋊C4—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2212yes—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14C8×(C3⋊D3)⋊C3C414yescomplemented
15#88448—22no—
16C2×C8×C3⋊D3D311yescharacteristic
17C2×C3×D3×C8—31no—
18#63900—32no—
19D3×C4⋊Dic3—32no—
20#63644D311yescharacteristic, complemented
21C3×#817—31no—
22C3×#817—31no—
23C3×#817—31no—
24C3×#817—61no—
25#63574—32no—
26#63573—32no—
27C22×(C3⋊D3)⋊C3D411yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31C2×(C3⋊Dic3)⋊C3D411yescharacteristic
32C4×(C3⋊D3)⋊C3C2×C412yes—
33C4×(C3⋊D3)⋊C3Q812yes—
34((C3⋊D3)⋊C3)⋊C4—44no—
35C8×H(3,𝔽3)C2×C412yescomplemented
36(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
37C2×#817—91no—
38D3×#106—91no—
39D3×#106—91no—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—31no—
42C2×#624—62no—
43C2×D3×Dic3—62no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C2×D311yescharacteristic
47C4⋊(C3⋊Dic3)C2×D312yes—
48C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
49C8×C3⋊D3Dic314yescomplemented
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—62no—
54C2×C3×C3⋊C8—31no—
55C3×D3×C8—34no—
56#3724—62no—
57Dic3⋊Dic3—62no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×(C3⋊D3)⋊C3C4⋊C412yes—
60C2×(C3⋊D3)⋊C3SD1614yes—
61C4×H(3,𝔽3)C22×C411yescharacteristic
62C4×H(3,𝔽3)C2×Q811yescharacteristic
63#1084—42no—
64(C3⋊Dic3)⋊C3—42no—
65(C3⋊Dic3)⋊C3C4⋊C412yes—
66C22×C3×C8—91no—
67C2×C4⋊Dic3—92no—
68C2×D3×C8—31no—
69C2×D3×C8—91no—
70D3×C4⋊C4—92no—
71D3×C4⋊C4—92no—
72C3×#106—91no—
73C3×#106—91no—
74#817—31no—
75#817—31no—
76#817—31no—
77#817—61no—
78#817—91no—
79#817—92no—
80#808—92no—
81#808—92no—
82C22×C3⋊D3D1211yescharacteristic
83C22×C3×D3—31no—
84C2×C32×C4—21no—
85C2×C32×C4C22×D311yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3D1211yescharacteristic
89C4×C3⋊D3Dic612yes—
90C4×C3⋊D3C2×Dic312yes—
91C2×C3×Dic3—31no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—122no—
96C3×C4×D3—32no—
97C3×C4×D3—32no—
98#624—124no—
99D3×Dic3—124no—
100C32×C8—22no—
101C32×C8C4×D312yescomplemented
102C32×C8C2×Dic312yescomplemented
103C3⋊(C3⋊C8)C2×Dic312yescomplemented
104C3×C3⋊C8—32no—
105C2×#106—271noSylow
106C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
107C2×H(3,𝔽3)C2×SD1612yes—
108(C3⋊D3)⋊C3#10614yescomplemented
109C22×C3×C4—91no—
110C22×Dic3—182no—
111C2×C4×D3—31no—
112C2×C4×D3—91no—
113C2×C4×D3—182no—
114C2×C4×D3—182no—
115C2×C3×C8—31no—
116C2×C3×C8—31no—
117C2×C3×C8—61no—
118C2×C3×C8—94no—
119C2×C3×C8—91no—
120C2×C3×C8D3211yescharacteristic
121C3×C4⋊C4—92no—
122C3×C4⋊C4—92no—
123C4⋊Dic3—32no—
124C4⋊Dic3—32no—
125C4⋊Dic3—32no—
126C4⋊Dic3—62no—
127C4⋊Dic3—92no—
128C4⋊Dic3—182no—
129Dic3⋊C4—182no—
130Dic3⋊C4—182no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—91no—
133D3×C8—34no—
134D3×C8—94no—
135C22×C32—21no—
136C22×C32C2×D1211yescharacteristic
137C22×C32D3×D411yescharacteristic
138C2×C3⋊D3#20414yes—
139C2×C3⋊D3C4⋊Dic312yes—
140C2×C3×D3—32no—
141C2×C3×D3—34no—
142C32×C4—21no—
143C32×C4—21no—
144C32×C4C2×Dic611yescharacteristic
145C32×C4C2×C4×D311yescharacteristic
146C32×C4D3×Q811yescharacteristic
147C32×C4C22×Dic311yescharacteristic
148C3⋊Dic3—42no—
149C3⋊Dic3C4⋊Dic312yes—
150C3×Dic3—32no—
151C3×Dic3—122no—
152C3×Dic3—122no—
153C3×Dic3—122no—
154C3×Dic3—242no—
155C22×C8—91no—
156C2×C4⋊C4—272no—
157#106—91no—
158#106—91no—
159#106—272no—
160H(3,𝔽3)C2×#10611yescharacteristic, complemented, Sylow
161C23×C3—91no—
162C22×D3—31no—
163C22×D3—91no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—61no—
167C2×C3×C4—92no—
168C2×C3×C4—92no—
169C2×C3×C4—91no—
170C2×C3×C4—182no—
171C2×C3×C4—182no—
172C2×C3×C4C2×D3211yescharacteristic
173C2×Dic3—31no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—91no—
178C2×Dic3—122no—
179C2×Dic3—182no—
180C2×Dic3—364no—
181C4×D3—32no—
182C4×D3—32no—
183C4×D3—92no—
184C4×D3—92no—
185C4×D3—364no—
186C4×D3—364no—
187C3×C8—32no—
188C3×C8—32no—
189C3×C8—62no—
190C3×C8—92no—
191C3×C8D3×Dic312yes—
192C3⋊C8—32no—
193C3⋊C8—92no—
194C2×C32—22no—
195C2×C32—21no—
196C2×C32D3×C4⋊C411yescharacteristic
197C2×C32C2×#20412yes—
198C2×C32D3×SD1612yes—
199C2×C32C2×C4⋊Dic311yescharacteristic
200C3⋊D3#81714yescomplemented
201C3×D3—34no—
202C22×C4—91no—
203C22×C4—542no—
204C2×C8—91no—
205C2×C8—94no—
206C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
207C4⋊C4—92no—
208C4⋊C4—92no—
209C4⋊C4—542no—
210C22×C3—31no—
211C22×C3—31no—
212C22×C3—61no—
213C22×C3—94no—
214C22×C3—92no—
215C22×C3D3×D1211yescharacteristic
216C2×D3—34no—
217C2×D3—32no—
218C2×D3—92no—
219C2×D3—94no—
220C3×C4—31no—
221C3×C4—31no—
222C3×C4—31no—
223C3×C4—31no—
224C3×C4—61no—
225C3×C4—61no—
226C3×C4—92no—
227C3×C4—362no—
228C3×C4—362no—
229C3×C4D3×Dic611yescharacteristic
230C3×C4C2×D3×Dic311yescharacteristic
231Dic3—32no—
232Dic3—92no—
233Dic3—122no—
234Dic3—122no—
235Dic3—122no—
236Dic3—242no—
237Dic3—362no—
238C32—21no—
239C32D3×#10611yescharacteristic, complemented
240C32C2×#81711yescharacteristic, complemented
241C23—91no—
242C2×C4—91no—
243C2×C4—92no—
244C2×C4—92no—
245C2×C4—182no—
246C2×C4—182no—
247C2×C4—1084no—
248C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
249C8—92no—
250C8((C3⋊D3)⋊C3)⋊C412yescomplemented
251C2×C3—32no—
252C2×C3—32no—
253C2×C3—31no—
254C2×C3—31no—
255C2×C3—62no—
256C2×C3—61no—
257C2×C3—94no—
258C2×C3D3×#20412yes—
259C2×C3D3×C4⋊Dic311yescharacteristic
260D3—34no—
261D3—94no—
262C22—94no—
263C22—92no—
264C22#8865411yescharacteristic
265C4—92no—
266C4—362no—
267C4—362no—
268C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
269C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
270C3—31no—
271C3—31no—
272C3—61no—
273C3D3×#81711yescharacteristic
274C2—94no—
275C2#1172084612yes—
276C2#1172112911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#106
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11720550C2semidirect
4. #63644⋊C3C2semidirect
5. #11720307C2semidirect
14. C8×(C3⋊D3)⋊C3C4semidirect
20. #63644D3semidirect
35. C8×H(3,𝔽3)C2×C4semidirect
36. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
49. C8×C3⋊D3Dic3semidirect
101. C32×C8C4×D3semidirect
102. C32×C8C2×Dic3semidirect
103. C3⋊(C3⋊C8)C2×Dic3semidirect
108. (C3⋊D3)⋊C3#106semidirect
160. H(3,𝔽3)C2×#106semidirect
200. C3⋊D3#817semidirect
239. C32D3×#106semidirect
240. C32C2×#817semidirect
250. C8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C8×(C3⋊D3)⋊C32
#117211292
#117205502
#63644⋊C32
#117203072
#105906853
D3×#8173