Finite Group Databasebeta

← #13156648

Subgroups — #13156648

Order 1728.

530 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C3×#63918C212yesmaximal
2#11722079C211yesmaximal, characteristic
3#11722053C212yesmaximal
4Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
5C3×#63574C211yesmaximal, characteristic
6#10590412—32nomaximal
7#10590309—31nomaximal
8C2×C3×#627—22no—
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C32×C4×Dic3C2211yescharacteristic
14C3×Dic3⋊Dic3—22no—
15C3×#888—32no—
16C3×#869—32no—
17#63918D312yes—
18#63635—32no—
19C4×C3⋊(C3⋊C8)—31no—
20C3×#808—32no—
21#63574—31no—
22#63574—32no—
23#63574—32no—
24#63574—62no—
25#63574D311yescharacteristic
26#63573—32no—
27Dic3×C3⋊C8—32no—
28Dic3×C3⋊C8—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C32×Dic3—22no—
32C2×C32×Dic3D412yes—
33C3×#627—44no—
34C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
35C3×C3⋊(C3⋊C8)Q812yescomplemented
36#5080—92no—
37#4855—91no—
38C2×C3×Dic6—62no—
39C2×C3×Dic6—62no—
40C2×#627—22no—
41C32×C4⋊C4—32no—
42C32×C42—31no—
43C4⋊(C3⋊Dic3)C2×D312yes—
44C2×C3⋊(C3⋊C8)—32no—
45C2×C3⋊(C3⋊C8)—31no—
46C2×C3⋊(C3⋊C8)—61no—
47C2×C3⋊(C3⋊C8)—62no—
48C2×C3⋊(C3⋊C8)—121no—
49C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
50C3×C4⋊Dic3—22no—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—62no—
54C3×C4⋊Dic3—62no—
55C3×C4⋊Dic3C2×D312yes—
56C3×C4⋊Dic3C2×D312yes—
57C3×Dic3⋊C4—62no—
58C3×C4×Dic3—21no—
59C3×C4×Dic3C2×D312yes—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—61no—
62Dic3⋊Dic3—22no—
63C22×C33C2×D411yescharacteristic
64C33×C4C2×Q811yescharacteristic
65C33×C4C2×Q811yescharacteristic
66C3×C3⋊Dic3—42no—
67C32×Dic3—42no—
68C32×Dic3SD1614yes—
69C3×C4⋊Q8—92no—
70#888—32no—
71#869—32no—
72C3×#106—91no—
73#817—91no—
74C8×Dic3—91no—
75#808—32no—
76#808—92no—
77#808—92no—
78#808—92no—
79#808—92no—
80#808—182no—
81C4×C3⋊C8—92no—
82C4×C3⋊C8—181no—
83C2×C32×C4—21no—
84C2×C32×C4—22no—
85C2×C32×C4—32no—
86C2×C32×C4—41no—
87C2×C32×C4—62no—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C22×D312yes—
90C2×C3⋊Dic3—22no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—42no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—122no—
98C2×C3×Dic3#6712yes—
99C2×C3×Dic3#6712yes—
100C3×Dic6—124no—
101C3×Dic6—124no—
102#627—44no—
103C3⋊(C3⋊C8)—32no—
104C3⋊(C3⋊C8)—34no—
105C3⋊(C3⋊C8)—62no—
106C3⋊(C3⋊C8)—64no—
107C3⋊(C3⋊C8)—122no—
108C3⋊(C3⋊C8)Dic612yescomplemented
109C3×C3⋊C8—34no—
110C3×C3⋊C8—62no—
111#498—271noSylow
112C2×C33C4⋊Q811yescharacteristic
113C2×C33C2×SD1612yes—
114C2×C3×Q8—182no—
115C2×Dic6—62no—
116C2×Dic6—62no—
117C2×C3×C8—91no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—62no—
121C3×C4⋊C4—92no—
122C3×C4⋊C4—182no—
123C3×C42—32no—
124C3×C42—61no—
125C4⋊Dic3—32no—
126C4⋊Dic3—32no—
127C4⋊Dic3—62no—
128C4⋊Dic3—62no—
129C4⋊Dic3D3212yes—
130Dic3⋊C4—62no—
131C4×Dic3D3211yescharacteristic
132C2×C3⋊C8—32no—
133C2×C3⋊C8—61no—
134C2×C3⋊C8—92no—
135C2×C3⋊C8—91no—
136C2×C3⋊C8—181no—
137C2×C3⋊C8—182no—
138C2×C3⋊C8—361no—
139C22×C32—21no—
140C22×C32—22no—
141C22×C32—41no—
142C22×C32D3×D411yescharacteristic
143C22×C32C2×#6712yes—
144C32×C4—22no—
145C32×C4—21no—
146C32×C4—22no—
147C32×C4—21no—
148C32×C4—34no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—122no—
152C32×C4C2×Dic611yescharacteristic
153C32×C4D3×Q812yes—
154C32×C4D3×Q811yescharacteristic
155C32×C4D3×Q812yes—
156C3⋊Dic3—42no—
157C3×Dic3—24no—
158C3×Dic3—42no—
159C3×Dic3—42no—
160C3×Dic3—82no—
161C3×Dic3—122no—
162C3×Dic3—122no—
163C3×Dic3—242no—
164C3×Dic3#21414yes—
165C3×Dic3#21514yes—
166C4⋊Q8—92no—
167#106—91no—
168#106—272no—
169#106—271no—
170C4×C8—271no—
171C33#49811yescharacteristic, complemented, Sylow
172C3×Q8—364no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—41no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—122no—
182C2×C3×C4—182no—
183C2×C3×C4C2×D3212yes—
184C2×C3×C4C2×D3211yescharacteristic
185C2×Dic3—22no—
186C2×Dic3—62no—
187C2×Dic3—62no—
188C2×Dic3—122no—
189C2×Dic3#62512yes—
190Dic6—124no—
191Dic6—124no—
192C3×C8—92no—
193C3⋊C8—34no—
194C3⋊C8—62no—
195C3⋊C8—94no—
196C3⋊C8—92no—
197C3⋊C8—182no—
198C3⋊C8—184no—
199C3⋊C8—362no—
200C2×C32—22no—
201C2×C32—22no—
202C2×C32—21no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—41no—
206C2×C32—42no—
207C2×C32#88811yescharacteristic
208C2×C32D3×SD1612yes—
209C2×C32C2×#21412yes—
210C2×C32C2×#21512yes—
211C2×C32#94412yes—
212C2×Q8—182no—
213C2×C8—91no—
214C2×C8—271no—
215C4⋊C4—32no—
216C4⋊C4—92no—
217C4⋊C4—182no—
218C42—31no—
219C22×C3—22no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3C2×#62511yescharacteristic
223C22×C3D3×#6712yes—
224C3×C4—21no—
225C3×C4—22no—
226C3×C4—21no—
227C3×C4—22no—
228C3×C4—34no—
229C3×C4—34no—
230C3×C4—41no—
231C3×C4—41no—
232C3×C4—64no—
233C3×C4—122no—
234C3×C4—122no—
235C3×C4—242no—
236C3×C4—362no—
237C3×C4D3×Dic612yes—
238C3×C4#379711yescharacteristic
239C3×C4#379712yes—
240C3×C4#379711yescharacteristic
241Dic3—42no—
242Dic3—122no—
243Dic3—122no—
244Dic3—242no—
245Dic3#371514yes—
246C32—21no—
247C32—22no—
248C32—41no—
249C32#485511yescharacteristic, complemented
250C32#508012yescomplemented
251Q8—364no—
252C2×C4—32no—
253C2×C4—62no—
254C2×C4—182no—
255C2×C4D3311yescharacteristic
256C8—92no—
257C8—272no—
258C2×C3—21no—
259C2×C3—22no—
260C2×C3—22no—
261C2×C3—22no—
262C2×C3—22no—
263C2×C3—41no—
264C2×C3—42no—
265C2×C3C2×#371512yes—
266C2×C3#6383912yes—
267C2×C3#6387311yescharacteristic
268C2×C3D3×#21412yes—
269C2×C3D3×#21512yes—
270C22D3×#62511yescharacteristic
271C4—34no—
272C4—122no—
273C4—362no—
274C4#8895711yescharacteristic
275C4#8895811yescharacteristic
276C3—22no—
277C3—21no—
278C3—41no—
279C3#1059030911yescharacteristic, complemented
280C3#1059041212yescomplemented
281C2D3×#371512yes—
282C2#1172303211yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#498
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
34. C3⋊(C3⋊(C3⋊C8))Q8semidirect
35. C3×C3⋊(C3⋊C8)Q8semidirect
108. C3⋊(C3⋊C8)Dic6semidirect
171. C33#498semidirect
249. C32#4855semidirect
250. C32#5080semidirect
279. C3#10590309semidirect
280. C3#10590412semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
#117220792
#117220532
Dic3×C3⋊(C3⋊C8)2
C3×#635742
#105904123
#105903093