Finite Group Databasebeta

← #13156662

Subgroups — #13156662

Order 1728.

530 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 108 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#11723450C212yesmaximal
2#11722079C211yesmaximal, characteristic
3#11722018C211yesmaximal, characteristic
4C3×#63575C212yesmaximal
5C3×#63574C211yesmaximal, characteristic
6#10590875—31nomaximal
7#10590417—32nomaximal
8C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C3×C4⋊(C3⋊Dic3)C2212yes—
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×C4⋊Dic3C2212yes—
12C2×C32×C3⋊C8C2211yescharacteristic
13#88807—22no—
14Dic3⋊(C3⋊Dic3)—22no—
15#64099—32no—
16#63872—32no—
17#63872—32no—
18#63872—62no—
19C8×C3⋊Dic3—31no—
20C3×#818—32no—
21C3×#808—32no—
22C3×#807—32no—
23#63575D312yes—
24#63574—31no—
25#63574D311yescharacteristic
26#63573—32no—
27#63553—32no—
28#63553—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3⋊(C3⋊Dic3)D412yes—
31C2×C3×C3⋊Dic3—22no—
32C2×C32×Dic3—22no—
33C3×C3⋊(C3⋊C8)Q812yes—
34C32×C3⋊C8Q812yes—
35#4878—92no—
36#4858—91no—
37C2×C32×C8—31no—
38C32×C4⋊C4—32no—
39C4⋊(C3⋊Dic3)—62no—
40C4⋊(C3⋊Dic3)C2×D312yes—
41#3750—62no—
42C4×C3⋊Dic3—32no—
43C4×C3⋊Dic3—31no—
44C4×C3⋊Dic3—61no—
45C4×C3⋊Dic3—62no—
46C4×C3⋊Dic3—121no—
47C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
48C3×C4⋊Dic3—22no—
49C3×C4⋊Dic3—32no—
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—62no—
52C3×C4⋊Dic3C2×D312yes—
53C3×C4⋊Dic3C2×D312yes—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—32no—
56C2×C3×C3⋊C8—61no—
57C2×C3×C3⋊C8C2×D312yes—
58Dic3⋊Dic3—62no—
59Dic3⋊Dic3—62no—
60Dic3⋊Dic3—62no—
61Dic3⋊Dic3—62no—
62Dic3⋊Dic3—64no—
63Dic3⋊Dic3—64no—
64C22×C33C2×D411yescharacteristic
65C33×C4C2×Q811yescharacteristic
66C33×C4C2×Q811yescharacteristic
67C3⋊(C3⋊Dic3)—22no—
68C3×C3⋊Dic3—42no—
69C32×Dic3—42no—
70#890—92no—
71#890—92no—
72#890—92no—
73#890—182no—
74C3×#107—92no—
75C3×#106—91no—
76#818—32no—
77#817—91no—
78C8×Dic3—92no—
79C8×Dic3—91no—
80C8×Dic3—181no—
81#808—32no—
82#808—92no—
83#807—32no—
84C2×C32×C4—21no—
85C2×C32×C4—22no—
86C2×C32×C4—41no—
87C2×C32×C4—62no—
88C2×C32×C4C22×D312yes—
89C2×C32×C4C22×D311yescharacteristic
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3—32no—
92C2×C3⋊Dic3—32no—
93C2×C3⋊Dic3—32no—
94C2×C3⋊Dic3—62no—
95C2×C3⋊Dic3—62no—
96C2×C3⋊Dic3—62no—
97C2×C3⋊Dic3—122no—
98C2×C3×Dic3—24no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C32×C8—32no—
105C3⋊(C3⋊C8)Dic612yes—
106C3×C3⋊C8—22no—
107C3×C3⋊C8—34no—
108C3×C3⋊C8—62no—
109C3×C3⋊C8Dic614yes—
110#499—271noSylow
111C2×C33C4⋊Q811yescharacteristic
112C2×C33#13512yes—
113C2×C3×C8—32no—
114C2×C3×C8—61no—
115C2×C3×C8—91no—
116C3×C4⋊C4—32no—
117C3×C4⋊C4—32no—
118C3×C4⋊C4—62no—
119C3×C4⋊C4—92no—
120C4⋊Dic3—32no—
121C4⋊Dic3—32no—
122C4⋊Dic3—62no—
123C4⋊Dic3—182no—
124C4⋊Dic3—182no—
125C4⋊Dic3—182no—
126C4⋊Dic3—184no—
127C4⋊Dic3D3212yes—
128Dic3⋊C4—182no—
129Dic3⋊C4—182no—
130Dic3⋊C4—182no—
131Dic3⋊C4—184no—
132C4×Dic3—91no—
133C4×Dic3—92no—
134C4×Dic3—181no—
135C4×Dic3—182no—
136C4×Dic3—361no—
137C2×C3⋊C8—32no—
138C2×C3⋊C8—61no—
139C2×C3⋊C8D3211yescharacteristic
140C22×C32—21no—
141C22×C32—22no—
142C22×C32—41no—
143C22×C32D3×D412yes—
144C22×C32D3×D411yescharacteristic
145C32×C4—22no—
146C32×C4—22no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—122no—
152C32×C4C2×Dic611yescharacteristic
153C32×C4C2×Dic612yes—
154C32×C4D3×Q811yescharacteristic
155C32×C4D3×Q812yes—
156C3⋊Dic3—42no—
157C3⋊Dic3—62no—
158C3⋊Dic3—64no—
159C3⋊Dic3—62no—
160C3⋊Dic3—62no—
161C3⋊Dic3—122no—
162C3⋊Dic3—122no—
163C3⋊Dic3—124no—
164C3×Dic3—44no—
165C3×Dic3—42no—
166C3×Dic3—42no—
167C3×Dic3—124no—
168C3×Dic3—122no—
169C3×Dic3—122no—
170#125—272no—
171#107—92no—
172#106—91no—
173#106—271no—
174C4×C8—271no—
175C33#49911yescharacteristic, complemented, Sylow
176C2×C3×C4—21no—
177C2×C3×C4—22no—
178C2×C3×C4—41no—
179C2×C3×C4—64no—
180C2×C3×C4—62no—
181C2×C3×C4—62no—
182C2×C3×C4—182no—
183C2×C3×C4C2×D3212yes—
184C2×C3×C4C2×D3211yescharacteristic
185C2×Dic3—22no—
186C2×Dic3—64no—
187C2×Dic3—62no—
188C2×Dic3—62no—
189C2×Dic3—92no—
190C2×Dic3—92no—
191C2×Dic3—92no—
192C2×Dic3—182no—
193C2×Dic3—182no—
194C2×Dic3—182no—
195C2×Dic3—362no—
196C3×C8—34no—
197C3×C8—62no—
198C3×C8—92no—
199C3⋊C8—34no—
200C3⋊C8—62no—
201C3⋊C8#62712yes—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—21no—
205C2×C32—22no—
206C2×C32—22no—
207C2×C32—41no—
208C2×C32—42no—
209C2×C32#88811yescharacteristic
210C2×C32#88812yes—
211C2×C32#91212yes—
212C2×C32#91612yes—
213C2×C32#91912yes—
214C2×C8—31no—
215C2×C8—91no—
216C4⋊C4—32no—
217C4⋊C4—92no—
218C4⋊C4—542no—
219C4⋊C4—542no—
220C42—271no—
221C22×C3—22no—
222C22×C3—21no—
223C22×C3—41no—
224C22×C3#380211yescharacteristic
225C22×C3#381112yes—
226C3×C4—21no—
227C3×C4—22no—
228C3×C4—21no—
229C3×C4—22no—
230C3×C4—41no—
231C3×C4—41no—
232C3×C4—124no—
233C3×C4—122no—
234C3×C4—122no—
235C3×C4—362no—
236C3×C4D3×Dic612yes—
237C3×C4D3×Dic612yes—
238C3×C4#379711yescharacteristic
239C3×C4C2×#62711yescharacteristic
240Dic3—42no—
241Dic3—124no—
242Dic3—122no—
243Dic3—122no—
244Dic3—184no—
245Dic3—182no—
246Dic3—182no—
247Dic3—182no—
248Dic3—362no—
249Dic3—362no—
250Dic3—364no—
251C32—21no—
252C32—22no—
253C32—41no—
254C32#485811yescharacteristic, complemented
255C32#487812yescomplemented
256C2×C4—62no—
257C2×C4—182no—
258C2×C4—272no—
259C2×C4D3311yescharacteristic
260C8—32no—
261C8—92no—
262C2×C3—22no—
263C2×C3—21no—
264C2×C3—22no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—41no—
268C2×C3—42no—
269C2×C3#6381012yes—
270C2×C3#6384012yes—
271C2×C3#6391911yescharacteristic
272C2×C3#6393612yes—
273C2×C3#6395012yes—
274C22#8895411yescharacteristic
275C4—122no—
276C4—362no—
277C4—542no—
278C4D3×#62711yescharacteristic
279C4#8895711yescharacteristic
280C3—22no—
281C3—21no—
282C3—41no—
283C3#1059041712yescomplemented
284C3#1059087511yescharacteristic, complemented
285C2#1172286212yes—
286C2#1172303711yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#499
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
175. C33#499semidirect
254. C32#4858semidirect
255. C32#4878semidirect
283. C3#10590417semidirect
284. C3#10590875semidirect

Maximal subgroups

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

subgroupindex
#117234502
#117220792
#117220182
C3×#635752
C3×#635742
#105908753
#105904173