Finite Group Databasebeta

← #13156660

Subgroups — #13156660

Order 1728.

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

normal subgroups only

This group has 482 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#11723680C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic
3#11722053C212yesmaximal
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
5C3×#63575C212yesmaximal
6#10590875—31nomaximal
7#10590413—32nomaximal
8C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
9C3×C4⋊(C3⋊Dic3)C2212yes—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C32×C4⋊Dic3C2212yes—
12C2×C32×C3⋊C8C2211yescharacteristic
13#88815—22no—
14C3×Dic3⋊Dic3—22no—
15C3×#890—32no—
16#63872—31no—
17#63872D311yescharacteristic
18#63869—32no—
19C8×C3⋊Dic3—31no—
20#63635—32no—
21C3×#818—32no—
22C3×#807—32no—
23#63575D312yes—
24#63574—32no—
25#63574—32no—
26#63574—62no—
27Dic3×C3⋊C8—32no—
28Dic3×C3⋊C8—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3D412yes—
32C2×C32×Dic3—22no—
33C3⋊(C3⋊(C3⋊C8))Q812yes—
34C32×C3⋊C8Q812yes—
35#5065—91no—
36#4878—92no—
37C2×C32×C8—31no—
38C32×C4⋊C4—32no—
39C4⋊(C3⋊Dic3)C2×D312yes—
40C4×C3⋊Dic3C2×D311yescharacteristic
41C2×C3⋊(C3⋊C8)—32no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—61no—
44C2×C3⋊(C3⋊C8)—62no—
45C2×C3⋊(C3⋊C8)—121no—
46C3×C4⋊Dic3—22no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—32no—
49C3×C4⋊Dic3—62no—
50C3×C4⋊Dic3—62no—
51C3×C4⋊Dic3C2×D312yes—
52C3×C4⋊Dic3C2×D312yes—
53C3×Dic3⋊C4—62no—
54C3×C4×Dic3—32no—
55C3×C4×Dic3—61no—
56C2×C3×C3⋊C8—21no—
57C2×C3×C3⋊C8C2×D312yes—
58#3724—62no—
59Dic3⋊Dic3—22no—
60Dic3⋊Dic3—62no—
61Dic3⋊Dic3—62no—
62C22×C33C2×D411yescharacteristic
63C33×C4C2×Q811yescharacteristic
64C33×C4C2×Q811yescharacteristic
65C3×C3⋊Dic3—22no—
66C3×C3⋊Dic3—42no—
67C32×Dic3—42no—
68C3×#125—91no—
69#890—32no—
70#890—92no—
71#870—91no—
72C3×#107—92no—
73#818—32no—
74C8×Dic3—92no—
75C8×Dic3—181no—
76#808—92no—
77#808—92no—
78#808—92no—
79#808—182no—
80#807—32no—
81C4×C3⋊C8—91no—
82C2×C32×C4—21no—
83C2×C32×C4—22no—
84C2×C32×C4—41no—
85C2×C32×C4—62no—
86C2×C32×C4C22×D312yes—
87C2×C32×C4C22×D311yescharacteristic
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3#6712yes—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—32no—
93C2×C3×Dic3—32no—
94C2×C3×Dic3—42no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—122no—
99C32×C8—32no—
100C3⋊(C3⋊C8)—34no—
101C3⋊(C3⋊C8)—32no—
102C3⋊(C3⋊C8)—62no—
103C3⋊(C3⋊C8)—64no—
104C3⋊(C3⋊C8)—122no—
105C3×C3⋊C8—22no—
106C3×C3⋊C8Dic614yes—
107#499—271noSylow
108C2×C33C4⋊Q811yescharacteristic
109C2×C33#13512yes—
110C2×C3×C8—32no—
111C2×C3×C8—61no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4—32no—
114C3×C4⋊C4—62no—
115C3×C4⋊C4—92no—
116C3×C4⋊C4—182no—
117C3×C42—91no—
118C4⋊Dic3—32no—
119C4⋊Dic3—32no—
120C4⋊Dic3—62no—
121C4⋊Dic3—62no—
122C4⋊Dic3—182no—
123C4⋊Dic3D3212yes—
124Dic3⋊C4—62no—
125Dic3⋊C4—182no—
126Dic3⋊C4—182no—
127C4×Dic3—32no—
128C4×Dic3—61no—
129C2×C3⋊C8—91no—
130C2×C3⋊C8—92no—
131C2×C3⋊C8—182no—
132C2×C3⋊C8—181no—
133C2×C3⋊C8—361no—
134C2×C3⋊C8D3211yescharacteristic
135C22×C32—21no—
136C22×C32—22no—
137C22×C32—41no—
138C22×C32D3×D412yes—
139C22×C32C2×#6711yescharacteristic
140C32×C4—22no—
141C32×C4—22no—
142C32×C4—21no—
143C32×C4—21no—
144C32×C4—41no—
145C32×C4—41no—
146C32×C4—122no—
147C32×C4C2×Dic612yes—
148C32×C4D3×Q811yescharacteristic
149C32×C4D3×Q812yes—
150C32×C4D3×Q811yescharacteristic
151C3⋊Dic3—22no—
152C3⋊Dic3—42no—
153C3×Dic3—42no—
154C3×Dic3—42no—
155C3×Dic3—64no—
156C3×Dic3—62no—
157C3×Dic3—62no—
158C3×Dic3—82no—
159C3×Dic3—122no—
160C3×Dic3—122no—
161C3×Dic3—242no—
162#125—91no—
163#125—271no—
164#107—92no—
165#106—272no—
166C4×C8—271no—
167C33#49911yescharacteristic, complemented, Sylow
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—41no—
171C2×C3×C4—62no—
172C2×C3×C4—62no—
173C2×C3×C4—92no—
174C2×C3×C4—122no—
175C2×C3×C4—182no—
176C2×C3×C4C2×D3212yes—
177C2×C3×C4C2×D3211yescharacteristic
178C2×Dic3—22no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—122no—
185C3×C8—34no—
186C3×C8—62no—
187C3⋊C8—94no—
188C3⋊C8—92no—
189C3⋊C8—184no—
190C3⋊C8—182no—
191C3⋊C8—362no—
192C3⋊C8#62712yes—
193C2×C32—22no—
194C2×C32—22no—
195C2×C32—21no—
196C2×C32—22no—
197C2×C32—22no—
198C2×C32—41no—
199C2×C32—42no—
200C2×C32#88812yes—
201C2×C32#91212yes—
202C2×C32#91912yes—
203C2×C32#94411yescharacteristic
204C2×C32#95012yes—
205C2×C8—31no—
206C2×C8—271no—
207C4⋊C4—32no—
208C4⋊C4—92no—
209C4⋊C4—182no—
210C4⋊C4—542no—
211C42—91no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—41no—
215C22×C3#380211yescharacteristic
216C22×C3D3×#6712yes—
217C3×C4—21no—
218C3×C4—22no—
219C3×C4—22no—
220C3×C4—21no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—122no—
224C3×C4—122no—
225C3×C4—182no—
226C3×C4—242no—
227C3×C4—362no—
228C3×C4D3×Dic612yes—
229C3×C4#379712yes—
230C3×C4#379711yescharacteristic
231C3×C4C2×#62711yescharacteristic
232Dic3—42no—
233Dic3—64no—
234Dic3—62no—
235Dic3—62no—
236Dic3—122no—
237Dic3—122no—
238Dic3—242no—
239C32—21no—
240C32—22no—
241C32—41no—
242C32#487812yescomplemented
243C32#506511yescharacteristic, complemented
244C2×C4—62no—
245C2×C4—92no—
246C2×C4—182no—
247C2×C4D3311yescharacteristic
248C8—32no—
249C8—272no—
250C2×C3—22no—
251C2×C3—21no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—41no—
256C2×C3—42no—
257C2×C3#6381012yes—
258C2×C3#6383912yes—
259C2×C3#6391911yescharacteristic
260C2×C3#6393312yes—
261C2×C3#6394912yes—
262C22#8895311yescharacteristic
263C4—122no—
264C4—182no—
265C4—362no—
266C4D3×#62711yescharacteristic
267C4#8895811yescharacteristic
268C3—22no—
269C3—21no—
270C3—41no—
271C3#1059041312yescomplemented
272C3#1059087511yescharacteristic, complemented
273C2#1172286012yes—
274C2#1172303511yescharacteristic
subgroupG/H
centerC22#88953
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
167. C33#499semidirect
242. C32#4878semidirect
243. C32#5065semidirect
271. C3#10590413semidirect
272. C3#10590875semidirect

Maximal subgroups

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

subgroupindex
#117236802
C3×#638722
#117220532
C3⋊C8×C3⋊Dic32
C3×#635752
#105908753
#105904133