Finite Group Databasebeta

← #13161481

Subgroups — #13161481

Order 1728.

566 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 127 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×#64081C211yesmaximal, characteristic
2C3×C4⋊(C3⋊(C3⋊C8))C212yesmaximal, complemented
3#11722072C212yesmaximal
4#11722071C211yesmaximal, characteristic, complemented
5#11722070C211yesmaximal, characteristic, complemented
6#10590257—31nomaximal
7#10590253—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C33×C42C2211yescharacteristic
10C3×#3752C412yes—
11C3×#3750C412yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15#64081D311yescharacteristic
16C3×#872—32no—
17C3×#872—61no—
18C4⋊(C3⋊(C3⋊C8))D312yescomplemented
19C3×C4⋊(C3⋊C8)—32no—
20C3×C4⋊(C3⋊C8)—32no—
21C3×C4⋊(C3⋊C8)—62no—
22#63557—32no—
23(C2×D3)⋊(C3⋊C8)—32no—
24(C2×D3)⋊(C3⋊C8)—32no—
25#63555—31no—
26#63554—32no—
27#63553—31no—
28Dic3×C3⋊C8—32no—
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C2×C33×C4C2×C412yes—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3C2×C412yes—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3—22no—
35C3×C3⋊(C3⋊C8)—22no—
36C3×C3⋊(C3⋊C8)—22no—
37#5000—91no—
38#4840—92no—
39C2×C4×C3⋊D3C2×D311yescharacteristic
40C2×C3×C4×D3—32no—
41C2×C3×C4×D3—61no—
42C32×C42—22no—
43C32×C42—21no—
44C32×C42—41no—
45C32×C42C2×D312yes—
46C32×C42C2×D311yescharacteristic
47#3752Dic312yes—
48#3750Dic312yes—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C2×C3⋊(C3⋊C8)C2×D312yes—
52C3×(C2×D3)⋊C4—32no—
53C3×(C2×D3)⋊C4—32no—
54C3×(C2×D3)⋊C4—62no—
55C3×Dic3⋊C4—32no—
56C3×Dic3⋊C4—32no—
57C3×Dic3⋊C4—62no—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—62no—
65C2×C3×C3⋊C8—62no—
66C22×C33C22×C411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—22no—
69C33×C4—22no—
70C33×C4#4112yes—
71C3×C3⋊Dic3—21no—
72C3×C3⋊Dic3—22no—
73C3×#117—91no—
74#872—32no—
75#872—61no—
76C3×C4⋊C8—92no—
77#830—92no—
78(C2×D3)⋊C8—92no—
79(C2×D3)⋊C8—92no—
80#815—92no—
81C8×Dic3—92no—
82C4⋊(C3⋊C8)—32no—
83C4⋊(C3⋊C8)—32no—
84C4⋊(C3⋊C8)—62no—
85#803—91no—
86C4×C3⋊C8—91no—
87C22×C3⋊D3C2×Dic311yescharacteristic
88C22×C3×D3—32no—
89C22×C3×D3—61no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—42no—
96C2×C32×C4—41no—
97C2×C32×C4C4×D312yes—
98C2×C32×C4C4×D312yes—
99C2×C32×C4C2×Dic312yes—
100C2×C32×C4C22×D311yescharacteristic
101C2×C32×C4C22×D312yes—
102C2×C3⋊Dic3C2×Dic312yes—
103C2×C3⋊Dic3C2×Dic311yescharacteristic
104C4×C3⋊D3—22no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—32no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—62no—
110C3×C4×D3—64no—
111C3×C4×D3—62no—
112C3×C4×D3—62no—
113C3⋊(C3⋊C8)—22no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—62no—
116C3×C3⋊C8—62no—
117C3×C3⋊C8—62no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8—122no—
120C3×C3⋊C8—122no—
121#433—271noSylow
122C2×C33#11711yescharacteristic
123C2×C33#13112yes—
124C3×C3⋊D3—41no—
125C22×C3×C4—91no—
126C2×C4×D3—32no—
127C2×C4×D3—61no—
128C2×C3×C8—92no—
129C2×C3×C8—92no—
130C3×C4⋊C4—92no—
131C3×C22⋊C4—92no—
132C3×C42—22no—
133C3×C42—21no—
134C3×C42—41no—
135C3×C42—91no—
136C3×C42D3212yes—
137C3×C42D3211yescharacteristic
138(C2×D3)⋊C4—32no—
139(C2×D3)⋊C4—32no—
140(C2×D3)⋊C4—62no—
141Dic3⋊C4—32no—
142Dic3⋊C4—32no—
143Dic3⋊C4—62no—
144C4×Dic3—32no—
145C4×Dic3—61no—
146C2×C3⋊C8—32no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—62no—
151C2×C3⋊C8—62no—
152C22×C32—22no—
153C22×C32—21no—
154C22×C32—41no—
155C22×C32C2×C4×D312yes—
156C22×C32C22×Dic311yescharacteristic
157C2×C3⋊D3—21no—
158C2×C3⋊D3—22no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—62no—
163C2×C3×D3—122no—
164C32×C4—22no—
165C32×C4—24no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—42no—
172C32×C4—42no—
173C32×C4—42no—
174C32×C4—44no—
175C32×C4#23512yes—
176C32×C4#23712yes—
177C32×C4#23912yes—
178C3⋊Dic3—21no—
179C3⋊Dic3—22no—
180C3×Dic3—62no—
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—122no—
185#117—91no—
186C4⋊C8—92no—
187C22⋊C8—272no—
188#97—271no—
189C4×C8—271no—
190C33#43311yescharacteristic, complemented, Sylow
191C23×C3—91no—
192C22×D3—32no—
193C22×D3—61no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—92no—
202C2×C3×C4—91no—
203C2×C3×C4—182no—
204C2×C3×C4D3×Dic312yes—
205C2×C3×C4D3×Dic312yes—
206C2×C3×C4#62412yes—
207C2×C3×C4C2×D3212yes—
208C2×C3×C4C2×D3211yescharacteristic
209C2×Dic3—32no—
210C2×Dic3—32no—
211C2×Dic3—32no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C4×D3—64no—
215C4×D3—62no—
216C4×D3—62no—
217C3×C8—182no—
218C3×C8—182no—
219C3⋊C8—62no—
220C3⋊C8—62no—
221C3⋊C8—62no—
222C3⋊C8—62no—
223C3⋊C8—122no—
224C3⋊C8—122no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32#89212yes—
233C2×C32#90112yes—
234C2×C32#90712yes—
235C2×C32#92611yescharacteristic
236C2×C32#94812yes—
237C3⋊D3—41no—
238C3×D3—122no—
239C3×D3—122no—
240C22×C4—91no—
241C2×C8—92no—
242C2×C8—92no—
243C4⋊C4—92no—
244C22⋊C4—92no—
245C42—91no—
246C42#111311yescharacteristic
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—41no—
250C22×C3—182no—
251C22×C3—181no—
252C22×C3C2×D3×Dic312yes—
253C22×C3C2×#62411yescharacteristic
254C2×D3—62no—
255C2×D3—62no—
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—122no—
259C3×C4—24no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—42no—
267C3×C4—42no—
268C3×C4—44no—
269C3×C4—42no—
270C3×C4—182no—
271C3×C4—181no—
272C3×C4#379512yes—
273C3×C4#379612yes—
274C3×C4#379912yes—
275Dic3—62no—
276Dic3—62no—
277Dic3—62no—
278Dic3—62no—
279Dic3—122no—
280C32—22no—
281C32—21no—
282C32—41no—
283C32#484012yescomplemented
284C32#500011yescharacteristic, complemented
285C23—91no—
286C2×C4—92no—
287C2×C4—91no—
288C2×C4—182no—
289C2×C4#629412yes—
290C2×C4C2×#111311yescharacteristic
291C8—182no—
292C8—182no—
293C2×C3—22no—
294C2×C3—22no—
295C2×C3—22no—
296C2×C3—21no—
297C2×C3—22no—
298C2×C3—42no—
299C2×C3—41no—
300C2×C3—361no—
301C2×C3#6381412yes—
302C2×C3#6381512yes—
303C2×C3#6381712yes—
304C2×C3#6384112yes—
305C2×C3#6384911yescharacteristic
306D3—122no—
307D3—122no—
308C22—182no—
309C22—181no—
310C22C2×#629411yescharacteristic
311C4—22no—
312C4—182no—
313C4—181no—
314C4#8904112yes—
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1059025312yescomplemented
319C3#1059025711yescharacteristic, complemented
320C2—361no—
321C2#1172365712yes—
322C2#1172366711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×#1113
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42#1113
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. #11722071C2semidirect
5. #11722070C2semidirect
18. C4⋊(C3⋊(C3⋊C8))D3semidirect
190. C33#433semidirect
283. C32#4840semidirect
284. C32#5000semidirect
318. C3#10590253semidirect
319. C3#10590257semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C3×C4⋊(C3⋊(C3⋊C8))2
#117220722
#117220712
#117220702
#105902573
#105902533