Finite Group Databasebeta

← #13156537

Subgroups — #13156537

Order 1728.

582 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 98 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, complemented
2#11723429C211yesmaximal, characteristic
3#11722042C212yesmaximal, complemented
4C3×#63563C212yesmaximal
5C3×#63553C211yesmaximal, characteristic, complemented
6#10591403—31nomaximal
7#10590863—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C32×D12C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13#88815—22no—
14(C2×D3)⋊(C3⋊Dic3)—22no—
15#64145—31no—
16#63872—31no—
17#63869—32no—
18#63860—32no—
19#63860—61no—
20#63643—32no—
21C3×#832—32no—
22C3×#821—32no—
23C3×C8×Dic3—32no—
24#63567—32no—
25#63567—32no—
26#63567—62no—
27#63563D312yes—
28#63553D311yescharacteristic, complemented
29C22×C32×D3—21no—
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)—21no—
32#6308—22no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D412yes—
35C32×D12—22no—
36C32×C3⋊C8—22no—
37#4768—92no—
38#4697—91no—
39C2×C32×D4—31no—
40C2×#634—31no—
41C2×#634—32no—
42C2×#634—62no—
43C2×#634—61no—
44C2×#634—121no—
45C2×C3×D12—21no—
46C2×C3×D12C2×D312yes—
47C2×C32×C8—32no—
48#3757—62no—
49C4⋊(C3⋊Dic3)C2×D312yes—
50C4×C3⋊Dic3C2×D311yescharacteristic
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—62no—
54C3×C4×Dic3—32no—
55C3×C4×Dic3—61no—
56C2×C3×C3⋊C8—22no—
57C2×C3×C3⋊C8C2×D312yes—
58C2×C3×C3⋊C8C2×D312yes—
59#3724—62no—
60Dic3⋊Dic3—62no—
61Dic3⋊Dic3—62no—
62(C2×D3)⋊Dic3—62no—
63(C2×D3)⋊Dic3—62no—
64(C2×D3)⋊Dic3—64no—
65C22×C33C2×D411yescharacteristic
66C2×C32×D3—41no—
67C2×C32×D3—42no—
68C33×C4#4112yes—
69C3⋊(C3⋊Dic3)—41no—
70C3×C3⋊Dic3—22no—
71C3×C3⋊Dic3—42no—
72#936—92no—
73#936—181no—
74#890—92no—
75#875—91no—
76#870—91no—
77C3×#102—92no—
78C3×C4×C8—91no—
79#832—32no—
80#821—32no—
81#816—92no—
82#816—92no—
83#816—182no—
84C8×Dic3—32no—
85#810—92no—
86C23×C32—61no—
87C22×C3×D3—22no—
88C22×C3×D3—22no—
89C32×D4—62no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4C22×D311yescharacteristic
94C2×C32×C4C22×D312yes—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—61no—
97C2×C3⋊Dic3—62no—
98C2×C3⋊Dic3—62no—
99C2×C3⋊Dic3—122no—
100C2×C3⋊Dic3—122no—
101C2×C3⋊Dic3D1212yes—
102#634—62no—
103#634—64no—
104#634—62no—
105#634—62no—
106#634—122no—
107#634—122no—
108#634—124no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—122no—
115C3×D12—22no—
116C3×D12—22no—
117C3×D12—42no—
118C32×C8—62no—
119C3×C3⋊C8—22no—
120C3×C3⋊C8—22no—
121C3×C3⋊C8—42no—
122#488—271noSylow
123C2×C33#12411yescharacteristic
124C2×C33#13512yes—
125C32×D3—81no—
126C2×C3×D4—32no—
127C2×C3×D4—61no—
128C2×D12D3211yescharacteristic
129C2×Dic6—92no—
130C2×Dic6—91no—
131C2×Dic6—182no—
132C2×Dic6—181no—
133C2×Dic6—361no—
134C2×C3×C8—32no—
135C2×C3×C8—32no—
136C2×C3×C8—62no—
137C3×C4⋊C4—92no—
138C3×C42—91no—
139#217—182no—
140#217—182no—
141#217—184no—
142(C2×D3)⋊C4—182no—
143C4⋊Dic3—32no—
144C4⋊Dic3—32no—
145C4⋊Dic3—62no—
146C4⋊Dic3—182no—
147Dic3⋊C4—182no—
148Dic3⋊C4—182no—
149C4×Dic3—32no—
150C4×Dic3—61no—
151C2×C3⋊C8D3212yes—
152C22×C32—22no—
153C22×C32—21no—
154C22×C32—41no—
155C22×C32—121no—
156C22×C32—122no—
157C22×C32C2×D1211yescharacteristic
158C22×C32D3×D412yes—
159C2×C3×D3—42no—
160C2×C3×D3—44no—
161C2×C3×D3—42no—
162C2×C3×D3—42no—
163C2×C3×D3—42no—
164C32×C4—22no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—42no—
168C32×C4#23512yes—
169C32×C4#23512yes—
170C32×C4#23712yes—
171C3⋊Dic3—22no—
172C3⋊Dic3—42no—
173C3⋊Dic3—124no—
174C3⋊Dic3—122no—
175C3⋊Dic3—124no—
176C3⋊Dic3—121no—
177C3⋊Dic3—122no—
178C3×Dic3—62no—
179C3×Dic3—62no—
180C3×Dic3—122no—
181C3×Dic3—122no—
182C3×Dic3—122no—
183C3×Dic3—242no—
184#125—271no—
185#124—271no—
186#103—272no—
187#102—92no—
188C4×C8—91no—
189C33#48811yescharacteristic, complemented, Sylow
190C23×C3—62no—
191C23×C3—62no—
192C22×D3—21no—
193C3×D4—62no—
194C3×D4—62no—
195C3×D4—122no—
196C2×C3×C4—21no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—92no—
200C2×C3×C4—182no—
201C2×C3×C4C2×D3212yes—
202C2×C3×C4C2×D3211yescharacteristic
203C2×Dic3—32no—
204C2×Dic3—32no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—122no—
209C2×Dic3—182no—
210C2×Dic3—182no—
211C2×Dic3—181no—
212C2×Dic3—362no—
213C2×Dic3—362no—
214D12—22no—
215Dic6—184no—
216Dic6—182no—
217Dic6—182no—
218Dic6—182no—
219Dic6—362no—
220Dic6—362no—
221Dic6—364no—
222C3×C8—62no—
223C3×C8—62no—
224C3×C8—122no—
225C3⋊C8—22no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32—41no—
232C2×C32—42no—
233C2×C32—241no—
234C2×C32#87511yescharacteristic
235C2×C32#88512yes—
236C2×C32#90412yes—
237C2×C32#91212yes—
238C2×C32#91612yes—
239C3×D3—82no—
240C3×D3—82no—
241C2×Q8—271no—
242C2×D4—31no—
243C2×C8—32no—
244C4⋊C4—92no—
245C4⋊C4—542no—
246C22⋊C4—542no—
247C42—91no—
248C22×C3—21no—
249C22×C3—22no—
250C22×C3—41no—
251C22×C3—122no—
252C22×C3—124no—
253C22×C3—122no—
254C22×C3—122no—
255C22×C3—122no—
256C22×C3D3×D1212yes—
257C22×C3#381111yescharacteristic
258C2×D3—42no—
259C2×D3—41no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—42no—
264C3×C4—182no—
265C3×C4—362no—
266C3×C4#379812yes—
267C3×C4#380412yes—
268C3×C4#380412yes—
269Dic3—62no—
270Dic3—62no—
271Dic3—122no—
272Dic3—122no—
273Dic3—122no—
274Dic3—242no—
275Dic3—364no—
276Dic3—362no—
277Dic3—364no—
278Dic3—362no—
279Dic3—361no—
280C32—22no—
281C32—21no—
282C32—41no—
283C32#469711yescharacteristic, complemented
284C32#476812yescomplemented
285C23—61no—
286Q8—542no—
287D4—62no—
288C2×C4—92no—
289C2×C4—182no—
290C2×C4—541no—
291C2×C4D3311yescharacteristic
292C8—62no—
293C2×C3—21no—
294C2×C3—22no—
295C2×C3—22no—
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—41no—
299C2×C3—42no—
300C2×C3—242no—
301C2×C3—242no—
302C2×C3#6380612yes—
303C2×C3#6380712yes—
304C2×C3#6385212yes—
305C2×C3#6393612yes—
306C2×C3#6395311yescharacteristic
307D3—81no—
308C22—121no—
309C22—122no—
310C22#8895511yescharacteristic
311C4—182no—
312C4—362no—
313C4—1081no—
314C4#8895912yes—
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1059086312yescomplemented
319C3#1059140311yescharacteristic, complemented
320C2—241no—
321C2#1172286412yes—
322C2#1172300811yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#488
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
1. #11723680C2semidirect
3. #11722042C2semidirect
5. C3×#63553C2semidirect
28. #63553D3semidirect
189. C33#488semidirect
283. C32#4697semidirect
284. C32#4768semidirect
318. C3#10590863semidirect
319. C3#10591403semidirect

Maximal subgroups

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

subgroupindex
#117236802
#117234292
#117220422
C3×#635632
C3×#635532
#105914033
#105908633