Finite Group Databasebeta

← #13166281

Subgroups — #13166281

Order 1728.

552 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C4×Dic3×C3⋊D3C211yesmaximal, characteristic
2#11723106C211yesmaximal, characteristic
3C3×#63781C211yesmaximal, characteristic
4#11722722C212yesmaximal
5#11722722C212yesmaximal
6#10593737—31nomaximal
7C22×C3⋊#157C2212yes—
8C22×C3×#157C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×Dic3×C3⋊D3C2211yescharacteristic
11C2×Dic3×C3⋊D3C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
13C32×C4×Dic3C411yescharacteristic, complemented
14Dic3×C3⋊Dic3C411yescharacteristic, complemented
15Dic3×C3⋊Dic3C411yescharacteristic, complemented
16C42×C3⋊D3—31no—
17C4×D3×Dic3—62no—
18#63781—31no—
19#63781—32no—
20#63781—32no—
21#63781D311yescharacteristic
22C22×C3×C3⋊D3C2311yescharacteristic
23C2×C3⋊#157—22no—
24C2×C3⋊#157—22no—
25C2×C3⋊#157—22no—
26C2×C3×#157—22no—
27C2×C3×#157—22no—
28C2×C3×#157—22no—
29C2×C33×C4C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
31C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C3×C4×C3⋊D3—22no—
34C2×C32×Dic3C2×C411yescharacteristic
35C2×C32×Dic3C2×C411yescharacteristic
36Dic3×C3⋊D3—22no—
37Dic3×C3⋊D3—21no—
38Dic3×C3⋊D3—21no—
39#4651—91nomaximal
40C22×#157—32no—
41C22×#157C2×D312yes—
42C2×C4×C3⋊D3—31no—
43C2×C4×C3⋊D3—31no—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×C4×D3—62no—
46C2×D3×Dic3—62no—
47C2×D3×Dic3—62no—
48C32×C42—31no—
49C4×C3⋊Dic3—31no—
50C4×C3⋊Dic3—31no—
51C4×C3⋊Dic3—31no—
52C4×C3⋊Dic3—62no—
53C4×C3⋊Dic3—122no—
54C3×C4×Dic3—22no—
55Dic32—62no—
56Dic32—62no—
57C22×C33C22×C411yescharacteristic
58C2×C3×C3⋊D3#4112yes—
59C2×C3×C3⋊D3#4111yescharacteristic
60C2×C3×C3⋊D3#4112yes—
61C2×C3×C3⋊D3#4111yescharacteristic
62C3⋊#157—42no—
63C3×#157—42no—
64C33×C4—21no—
65C3⋊(C3⋊Dic3)—21no—
66C3⋊(C3⋊Dic3)—21no—
67C3×C3⋊Dic3—21no—
68C32×Dic3—21no—
69C32×Dic3—21no—
70C2×C4×Dic3—91no—
71C42×D3—182no—
72C3×#95—91no—
73C3⋊#95—91no—
74C3⋊#95—92no—
75C3⋊#95—92no—
76C22×C3⋊D3C22×D311yescharacteristic
77C22×C3×D3—62no—
78C2×#157—22no—
79C2×#157—22no—
80C2×#157—22no—
81C2×#157—62no—
82C2×#157—62no—
83C2×#157—62no—
84C2×C32×C4—22no—
85C2×C32×C4—31no—
86C2×C32×C4—31no—
87C2×C32×C4—42no—
88C2×C32×C4C4×D311yescharacteristic
89C2×C3⋊Dic3—31no—
90C2×C3⋊Dic3—31no—
91C2×C3⋊Dic3—62no—
92C2×C3⋊Dic3—62no—
93C2×C3⋊Dic3—122no—
94C2×C3⋊Dic3—122no—
95C2×C3⋊Dic3C4×D311yescharacteristic
96C4×C3⋊D3—22no—
97C4×C3⋊D3—62no—
98C4×C3⋊D3—61no—
99C4×C3⋊D3—61no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—62no—
103C3×C4×D3—68no—
104D3×Dic3—68no—
105D3×Dic3—64no—
106D3×Dic3—64no—
107#383—271noSylow
108C2×C33#11711yescharacteristic
109C2×C33#11711yescharacteristic
110C2×C33#11711yescharacteristic
111C3×C3⋊D3#12612yes—
112C3×C3⋊D3#12612yes—
113C22×C3×C4—91no—
114C22×C3×C4—92no—
115C22×Dic3—91no—
116C22×Dic3—91no—
117C22×Dic3—92no—
118C2×C4×D3—62no—
119C2×C4×D3—182no—
120C2×C4×D3—182no—
121C3×C42—62no—
122C4×Dic3—91no—
123C4×Dic3—91no—
124C4×Dic3—91no—
125C4×Dic3—182no—
126C4×Dic3—182no—
127C4×Dic3—182no—
128C4×Dic3—362no—
129C4×Dic3#15711yescharacteristic, complemented
130C22×C32—22no—
131C22×C32—42no—
132C22×C32C2×C4×D311yescharacteristic
133C2×C3⋊D3#23512yes—
134C2×C3⋊D3#23711yescharacteristic
135C2×C3⋊D3#23712yes—
136C2×C3⋊D3#23911yescharacteristic
137C2×C3×D3—64no—
138C2×C3×D3—62no—
139C2×C3×D3—62no—
140C2×C3×D3—64no—
141#157—42no—
142#157—122no—
143C32×C4—21no—
144C32×C4—24no—
145C32×C4—44no—
146C32×C4—61no—
147C32×C4—61no—
148C3⋊Dic3—21no—
149C3⋊Dic3—61no—
150C3⋊Dic3—61no—
151C3⋊Dic3—64no—
152C3⋊Dic3—64no—
153C3⋊Dic3—124no—
154C3⋊Dic3—124no—
155C3×Dic3—24no—
156C3×Dic3—24no—
157C3×Dic3—64no—
158C2×C42—271no—
159#95—91no—
160#95—271no—
161#95—272no—
162#95—272no—
163C33#38311yescharacteristic, complemented, Sylow
164C23×C3—91no—
165C22×D3—62no—
166C2×C3×C4—22no—
167C2×C3×C4—42no—
168C2×C3×C4—62no—
169C2×C3×C4—62no—
170C2×C3×C4—91no—
171C2×C3×C4—182no—
172C2×C3×C4—182no—
173C2×C3×C4—182no—
174C2×C3×C4—182no—
175C2×C3×C4C2×#15711yescharacteristic
176C2×Dic3—62no—
177C2×Dic3—91no—
178C2×Dic3—91no—
179C2×Dic3—182no—
180C2×Dic3—182no—
181C2×Dic3—181no—
182C2×Dic3—182no—
183C2×Dic3—181no—
184C2×Dic3—182no—
185C2×Dic3—182no—
186C2×Dic3—182no—
187C2×Dic3—362no—
188C2×Dic3—362no—
189C2×Dic3C2×#15711yescharacteristic
190C2×Dic3C2×#15711yescharacteristic
191C4×D3—68no—
192C4×D3—188no—
193C4×D3—184no—
194C4×D3—184no—
195C2×C32—22no—
196C2×C32—22no—
197C2×C32—22no—
198C2×C32—42no—
199C2×C32—42no—
200C2×C32—42no—
201C2×C32#87211yescharacteristic
202C2×C32#87711yescharacteristic
203C2×C32#89211yescharacteristic
204C3⋊D3#87912yes—
205C3⋊D3#89812yes—
206C3×D3—64no—
207C3×D3—64no—
208C22×C4—91no—
209C22×C4—92no—
210C22×C4—271no—
211C22×C4—271no—
212C22×C4—272no—
213C42—31no—
214C42—271no—
215C42—271no—
216C42—271no—
217C22×C3—22no—
218C22×C3—42no—
219C22×C3—91no—
220C22×C3—92no—
221C22×C3—91no—
222C22×C3—92no—
223C22×C3C22×#15711yescharacteristic
224C2×D3—64no—
225C2×D3—64no—
226C2×D3—62no—
227C2×D3—62no—
228C3×C4—24no—
229C3×C4—21no—
230C3×C4—44no—
231C3×C4—64no—
232C3×C4—64no—
233C3×C4—181no—
234C3×C4—362no—
235Dic3—21no—
236Dic3—21no—
237Dic3—64no—
238Dic3—184no—
239Dic3—184no—
240Dic3—181no—
241Dic3—181no—
242Dic3—364no—
243Dic3—364no—
244Dic3—362no—
245C32—22no—
246C32—42no—
247C32#465111yescharacteristic, complemented
248C23—91no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—91no—
252C2×C4—182no—
253C2×C4—182no—
254C2×C4—182no—
255C2×C4—182no—
256C2×C4—271no—
257C2×C4—271no—
258C2×C4—542no—
259C2×C4—541no—
260C2×C4—541no—
261C2×C4—542no—
262C2×C4—542no—
263C2×C4—542no—
264C2×C4D3×#15711yescharacteristic
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—22no—
268C2×C3—42no—
269C2×C3—42no—
270C2×C3—42no—
271C2×C3—92no—
272C2×C3—92no—
273C2×C3#6428611yescharacteristic
274C2×C3#6428611yescharacteristic
275C2×C3#6428611yescharacteristic
276D3—64no—
277D3—64no—
278C22—92no—
279C22—91no—
280C22—91no—
281C22—92no—
282C22C2×D3×#15711yescharacteristic
283C4—21no—
284C4—61no—
285C4—61no—
286C4—181no—
287C4—362no—
288C4—541no—
289C4—541no—
290C4—1082no—
291C3—22no—
292C3—42no—
293C3#1059373711yescharacteristic, complemented
294C2—92no—
295C2—92no—
296C2#1172424611yescharacteristic
297C2#1172424711yescharacteristic
298C2#1172426111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#383
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
12. C4×C3⋊(C3⋊Dic3)C4semidirect
13. C32×C4×Dic3C4semidirect
14. Dic3×C3⋊Dic3C4semidirect
15. Dic3×C3⋊Dic3C4semidirect
129. C4×Dic3#157semidirect
163. C33#383semidirect
247. C32#4651semidirect
293. C3#10593737semidirect

Maximal subgroups

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

subgroupindex
C4×Dic3×C3⋊D32
#117231062
C3×#637812
#117227222
#105937373
#46519