Finite Group Databasebeta

← #13156664

Subgroups — #13156664

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#11722080C211yesmaximal, characteristic
3#11722056C212yesmaximal
4Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
5C3×#63576C211yesmaximal, characteristic
6#10590411—32nomaximal
7#10590312—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#63634—32no—
19C4×C3⋊(C3⋊C8)—31no—
20C3×#807—32no—
21#63576—31no—
22#63576—32no—
23#63576—32no—
24#63576—62no—
25#63576D311yescharacteristic
26#63575—32no—
27Dic3×C3⋊C8—32no—
28Dic3×C3⋊C8—61no—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C32×Dic3—22no—
32C2×C32×Dic3D411yescharacteristic
33C2×C32×Dic3D411yescharacteristic
34C3×#627—44no—
35C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
36C3×C3⋊(C3⋊C8)Q812yescomplemented
37#5078—92no—
38#4874—91no—
39C2×C3×Dic6—62no—
40C2×C3×Dic6—62no—
41C2×#627—22no—
42C32×C4⋊C4—32no—
43C32×C42—31no—
44C4⋊(C3⋊Dic3)C2×D312yes—
45C2×C3⋊(C3⋊C8)—32no—
46C2×C3⋊(C3⋊C8)—31no—
47C2×C3⋊(C3⋊C8)—61no—
48C2×C3⋊(C3⋊C8)—62no—
49C2×C3⋊(C3⋊C8)—121no—
50C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
51C3×C4⋊Dic3—22no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—62no—
55C3×C4⋊Dic3—62no—
56C3×C4⋊Dic3C2×D312yes—
57C3×C4⋊Dic3C2×D312yes—
58C3×Dic3⋊C4—62no—
59C3×C4×Dic3—21no—
60C3×C4×Dic3C2×D312yes—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—61no—
63Dic3⋊Dic3—22no—
64C22×C33C2×D411yescharacteristic
65C33×C4C2×Q811yescharacteristic
66C33×C4C2×Q811yescharacteristic
67C3×C3⋊Dic3—42no—
68C32×Dic3—42no—
69C32×Dic3D812yes—
70C32×Dic3Q1612yes—
71C3×C4⋊Q8—92no—
72#888—32no—
73#869—32no—
74C3×#107—91no—
75#818—91no—
76C8×Dic3—91no—
77#807—32no—
78#807—92no—
79#807—92no—
80#807—92no—
81#807—92no—
82#807—182no—
83C4×C3⋊C8—92no—
84C4×C3⋊C8—181no—
85C2×C32×C4—21no—
86C2×C32×C4—22no—
87C2×C32×C4—31no—
88C2×C32×C4—31no—
89C2×C32×C4—41no—
90C2×C32×C4—62no—
91C2×C32×C4C22×D311yescharacteristic
92C2×C32×C4C22×D312yes—
93C2×C3⋊Dic3—22no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—122no—
102C2×C3×Dic3#6712yes—
103C2×C3×Dic3#6712yes—
104C3×Dic6—124no—
105C3×Dic6—124no—
106#627—44no—
107C3⋊(C3⋊C8)—32no—
108C3⋊(C3⋊C8)—34no—
109C3⋊(C3⋊C8)—62no—
110C3⋊(C3⋊C8)—64no—
111C3⋊(C3⋊C8)—122no—
112C3⋊(C3⋊C8)Dic612yescomplemented
113C3×C3⋊C8—34no—
114C3×C3⋊C8—62no—
115#500—271noSylow
116C2×C33C4⋊Q811yescharacteristic
117C2×C33C2×D811yescharacteristic
118C2×C33C2×Q1611yescharacteristic
119C2×C3×Q8—182no—
120C2×Dic6—62no—
121C2×Dic6—62no—
122C2×C3×C8—91no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—62no—
126C3×C4⋊C4—92no—
127C3×C4⋊C4—182no—
128C3×C42—32no—
129C3×C42—61no—
130C4⋊Dic3—32no—
131C4⋊Dic3—32no—
132C4⋊Dic3—62no—
133C4⋊Dic3—62no—
134C4⋊Dic3D3212yes—
135Dic3⋊C4—62no—
136C4×Dic3D3211yescharacteristic
137C2×C3⋊C8—32no—
138C2×C3⋊C8—61no—
139C2×C3⋊C8—92no—
140C2×C3⋊C8—91no—
141C2×C3⋊C8—181no—
142C2×C3⋊C8—182no—
143C2×C3⋊C8—361no—
144C22×C32—21no—
145C22×C32—22no—
146C22×C32—41no—
147C22×C32D3×D411yescharacteristic
148C22×C32C2×#6712yes—
149C32×C4—22no—
150C32×C4—21no—
151C32×C4—22no—
152C32×C4—21no—
153C32×C4—32no—
154C32×C4—32no—
155C32×C4—41no—
156C32×C4—41no—
157C32×C4—122no—
158C32×C4C2×Dic611yescharacteristic
159C32×C4D3×Q812yes—
160C32×C4D3×Q811yescharacteristic
161C32×C4D3×Q812yes—
162C3⋊Dic3—42no—
163C3×Dic3—22no—
164C3×Dic3—22no—
165C3×Dic3—42no—
166C3×Dic3—42no—
167C3×Dic3—82no—
168C3×Dic3—122no—
169C3×Dic3—122no—
170C3×Dic3—242no—
171C3×Dic3#21314yes—
172C3×Dic3#21614yes—
173C4⋊Q8—92no—
174#107—91no—
175#107—272no—
176#107—271no—
177C4×C8—271no—
178C33#50011yescharacteristic, complemented, Sylow
179C3×Q8—364no—
180C2×C3×C4—21no—
181C2×C3×C4—22no—
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—41no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—122no—
190C2×C3×C4—182no—
191C2×C3×C4C2×D3212yes—
192C2×C3×C4C2×D3211yescharacteristic
193C2×Dic3—22no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—122no—
197C2×Dic3#62511yescharacteristic
198C2×Dic3#62511yescharacteristic
199Dic6—124no—
200Dic6—124no—
201C3×C8—92no—
202C3⋊C8—34no—
203C3⋊C8—62no—
204C3⋊C8—94no—
205C3⋊C8—92no—
206C3⋊C8—182no—
207C3⋊C8—184no—
208C3⋊C8—362no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—22no—
212C2×C32—21no—
213C2×C32—22no—
214C2×C32—22no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32#88811yescharacteristic
219C2×C32D3×D811yescharacteristic
220C2×C32D3×Q1611yescharacteristic
221C2×C32C2×#21312yes—
222C2×C32C2×#21612yes—
223C2×C32#94412yes—
224C2×Q8—182no—
225C2×C8—91no—
226C2×C8—271no—
227C4⋊C4—32no—
228C4⋊C4—92no—
229C4⋊C4—182no—
230C42—31no—
231C22×C3—22no—
232C22×C3—21no—
233C22×C3—41no—
234C22×C3C2×#62511yescharacteristic
235C22×C3D3×#6712yes—
236C3×C4—21no—
237C3×C4—22no—
238C3×C4—21no—
239C3×C4—22no—
240C3×C4—34no—
241C3×C4—34no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—62no—
245C3×C4—62no—
246C3×C4—122no—
247C3×C4—122no—
248C3×C4—242no—
249C3×C4—362no—
250C3×C4D3×Dic612yes—
251C3×C4#379711yescharacteristic
252C3×C4#379712yes—
253C3×C4#379711yescharacteristic
254Dic3—42no—
255Dic3—122no—
256Dic3—122no—
257Dic3—242no—
258Dic3#371312yes—
259Dic3#371812yes—
260C32—21no—
261C32—22no—
262C32—41no—
263C32#487411yescharacteristic, complemented
264C32#507812yescomplemented
265Q8—364no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—62no—
269C2×C4—182no—
270C2×C4D3311yescharacteristic
271C8—92no—
272C8—272no—
273C2×C3—21no—
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—22no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3C2×#371311yescharacteristic
283C2×C3C2×#371811yescharacteristic
284C2×C3#6383912yes—
285C2×C3#6387311yescharacteristic
286C2×C3D3×#21312yes—
287C2×C3D3×#21612yes—
288C22D3×#62511yescharacteristic
289C4—32no—
290C4—32no—
291C4—122no—
292C4—362no—
293C4#8895711yescharacteristic
294C4#8895811yescharacteristic
295C3—21no—
296C3—22no—
297C3—41no—
298C3#1059031211yescharacteristic, complemented
299C3#1059041112yescomplemented
300C2D3×#371311yescharacteristic
301C2D3×#371811yescharacteristic
302C2#1172303211yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#500
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
35. C3⋊(C3⋊(C3⋊C8))Q8semidirect
36. C3×C3⋊(C3⋊C8)Q8semidirect
112. C3⋊(C3⋊C8)Dic6semidirect
178. C33#500semidirect
263. C32#4874semidirect
264. C32#5078semidirect
298. C3#10590312semidirect
299. C3#10590411semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
#117220802
#117220562
Dic3×C3⋊(C3⋊C8)2
C3×#635762
#105904113
#105903123