Finite Group Databasebeta

← #13147547

Subgroups — #13147547

Order 1728.

786 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C42×C3⋊D9C211yesmaximal, characteristic
2C9×#870C211yesmaximal, characteristic, complemented
3C3×#63432C211yesmaximal, characteristic, complemented
4#11721000C212yesmaximal
5#11720980C214yesmaximal
6#11720937C212yesmaximal, complemented
7#11720928C214yesmaximal, complemented
8#10593948—31nomaximal
9D9×#125—31nomaximal
10C2×C4×C3⋊D9C2212yes—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×(C3⋊D9)⋊C4C2214yes—
13C3×C42×C9C2211yescharacteristic
14C4×C3⋊Dic9C2212yes—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2212yes—
17C9×Dic3⋊C4C2214yes—
18C3×C4⋊Dic9C2212yes—
19C3×Dic9⋊C4C2214yes—
20Dic3⋊Dic9C2214yes—
21#88442C2214yes—
22Dic9⋊Dic3C2214yes—
23C42×C3⋊D3—31no—
24C3×#870—31no—
25C3×#870D311yescharacteristic
26#63900—32no—
27#63883—34no—
28#63872—32no—
29#63870—34no—
30C9×#125—31no—
31C4⋊C4×D9—32no—
32C4⋊C4×D9—34no—
33#63452—32no—
34#63451—34no—
35C42×D9—31no—
36C42×D9—61no—
37#63432D311yescharacteristic, complemented
38C22×C3⋊D9C2311yescharacteristic
39C2×C3×C4×C9C2312yes—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9C2312yes—
42C2×C3⋊Dic9C2311yescharacteristic
43C4×C3⋊D9—24no—
44C4×C3⋊D9Q814yes—
45C2×C9×Dic3C2314yes—
46C2×C3×Dic9C2314yes—
47(C3⋊D9)⋊C4—28no—
48D3×#125—91no—
49D3×#125—91no—
50C2×C4×C3⋊D3—31no—
51C2×C4×C3⋊D3—32no—
52C2×#624—34no—
53C32×C42C2×D311yescharacteristic
54C4×C3⋊Dic3—32no—
55C4×C3⋊Dic3—31no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3C2×D312yes—
58C3×Dic3⋊C4—34no—
59C3×Dic3⋊C4C2×D314yes—
60#3724—34no—
61Dic3⋊Dic3—34no—
62Dic3⋊Dic3—34no—
63C2×C4×D9—31no—
64C2×C4×D9—32no—
65C2×C4×D9—34no—
66C2×C4×D9—61no—
67C2×C4×D9—62no—
68C9×C4⋊C4—34no—
69C9×C4⋊C4—32no—
70C42×C9—21no—
71C42×C9C2×D311yescharacteristic
72C4⋊Dic9—34no—
73C4⋊Dic9C2×D312yes—
74Dic9⋊C4—34no—
75Dic9⋊C4—34no—
76Dic9⋊C4C2×D314yes—
77C4×Dic9—32no—
78C4×Dic9—31no—
79C4×Dic9—62no—
80C4×Dic9—61no—
81C22×C3×C9C2411yescharacteristic
82C2×C3⋊D9C2×Q812yes—
83C2×C3⋊D9#4114yes—
84C3×C4×C9—22no—
85C3×C4×C9C2×Q812yes—
86C3⋊Dic9—22no—
87C3⋊Dic9C2×Q812yes—
88C9×Dic3—24no—
89C3×Dic9—24no—
90C3×#125—31no—
91C3×#125—91no—
92D3×C4⋊C4—94no—
93D3×C4⋊C4—92no—
94D3×C4⋊C4—92no—
95D3×C4⋊C4—94no—
96#890—92no—
97#890—92no—
98#889—94no—
99#889—94no—
100C42×D3—91no—
101C42×D3—91no—
102C42×D3—181no—
103#870—31no—
104#870D911yescharacteristic, complemented
105C22×C3⋊D3—31no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—32no—
110C4×C3⋊D3—34no—
111C4×C3⋊D3—64no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3C22×D314yes—
114#624—68no—
115C22×D9—31no—
116C22×D9—61no—
117C2×C4×C9—22no—
118C2×C4×C9—21no—
119C2×C4×C9—34no—
120C2×C4×C9C22×D312yes—
121C2×C4×C9C22×D311yescharacteristic
122C2×Dic9—31no—
123C2×Dic9—32no—
124C2×Dic9—61no—
125C2×Dic9—62no—
126C2×Dic9C22×D314yes—
127C4×D9—34no—
128C4×D9—64no—
129C4×D9—68no—
130C4×D9—64no—
131C4×D9—68no—
132C2×#125—271noSylow
133C2×C3×C9C22×Q811yescharacteristic
134C2×C3×C9C2×#4112yes—
135C3⋊D9#12514yescomplemented
136C2×C4×D3—91no—
137C2×C4×D3—91no—
138C2×C4×D3—94no—
139C2×C4×D3—92no—
140C2×C4×D3—92no—
141C2×C4×D3—94no—
142C2×C4×D3—181no—
143C2×C4×D3—182no—
144C3×C4⋊C4—34no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—94no—
147C3×C4⋊C4—92no—
148C3×C42—21no—
149C3×C42C2×D911yescharacteristic
150C3×C42D3211yescharacteristic
151C4⋊Dic3—32no—
152C4⋊Dic3—94no—
153C4⋊Dic3—94no—
154C4⋊Dic3C2×D912yes—
155Dic3⋊C4—34no—
156Dic3⋊C4—94no—
157Dic3⋊C4—94no—
158Dic3⋊C4—94no—
159Dic3⋊C4—94no—
160Dic3⋊C4C2×D914yes—
161C4×Dic3—92no—
162C4×Dic3—91no—
163C4×Dic3—92no—
164C4×Dic3—91no—
165C4×Dic3—182no—
166C4×Dic3—181no—
167C22×C32C23×D311yescharacteristic
168C2×C3⋊D3—32no—
169C2×C3⋊D3—34no—
170C32×C4—22no—
171C32×C4D3×Q812yes—
172C3⋊Dic3—32no—
173C3⋊Dic3—62no—
174C3×Dic3—24no—
175C3×Dic3—64no—
176C22×C9—21no—
177C22×C9C23×D311yescharacteristic
178C2×D9—32no—
179C2×D9—34no—
180C2×D9—62no—
181C2×D9—64no—
182C4×C9—24no—
183C4×C9—22no—
184C4×C9—22no—
185C4×C9—64no—
186C4×C9D3×Q812yes—
187Dic9—24no—
188Dic9—32no—
189Dic9—62no—
190Dic9—64no—
191Dic9—62no—
192#125—31no—
193#125—91no—
194#125—274no—
195#125—272no—
196C2×C4⋊C4—274no—
197C2×C4⋊C4—272no—
198C2×C42—271no—
199C3×C9C2×#12511yescharacteristic, complemented, Sylow
200C22×D3—91no—
201C22×D3—91no—
202C22×D3—181no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—34no—
206C2×C3×C4—94no—
207C2×C3×C4C22×D912yes—
208C2×C3×C4C22×D911yescharacteristic
209C2×C3×C4C2×D3212yes—
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—34no—
212C2×Dic3—91no—
213C2×Dic3—91no—
214C2×Dic3—92no—
215C2×Dic3—92no—
216C2×Dic3—181no—
217C2×Dic3—182no—
218C2×Dic3C22×D914yes—
219C4×D3—94no—
220C4×D3—94no—
221C4×D3—184no—
222C4×D3—188no—
223C4×D3—188no—
224C4×D3—184no—
225C4×D3—184no—
226C4×D3—188no—
227C2×C32C2×D3×Q811yescharacteristic
228C2×C32D3×#4112yes—
229C3⋊D3—34no—
230C2×C9—21no—
231C2×C9—22no—
232C2×C9C2×D3×Q811yescharacteristic
233C2×C9D3×#4112yes—
234D9—34no—
235D9—64no—
236C22×C4—274no—
237C22×C4—271no—
238C22×C4—272no—
239C4⋊C4—34no—
240C4⋊C4—32no—
241C4⋊C4—94no—
242C4⋊C4—92no—
243C4⋊C4—274no—
244C4⋊C4—274no—
245C4⋊C4—274no—
246C42—272no—
247C42—271no—
248C42D3×D911yescharacteristic
249C22×C3—21no—
250C22×C3C23×D911yescharacteristic
251C22×C3C22×D3211yescharacteristic
252C2×D3—92no—
253C2×D3—94no—
254C2×D3—92no—
255C2×D3—94no—
256C2×D3—182no—
257C2×D3—184no—
258C3×C4—24no—
259C3×C4—22no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—64no—
263C3×C4—184no—
264C3×C4Q8×D912yes—
265C3×C4#379712yes—
266Dic3—24no—
267Dic3—64no—
268Dic3—92no—
269Dic3—92no—
270Dic3—182no—
271Dic3—184no—
272Dic3—182no—
273Dic3—182no—
274C32D3×#12511yescharacteristic
275C9—21no—
276C9D3×#12511yescharacteristic, complemented
277C23—271no—
278C2×C4—34no—
279C2×C4—94no—
280C2×C4—271no—
281C2×C4—274no—
282C2×C4—272no—
283C2×C4—544no—
284C2×C4—548no—
285C2×C4C2×D3×D912yes—
286C2×C4C2×D3×D911yescharacteristic
287C2×C3—21no—
288C2×C3—22no—
289C2×C3C2×Q8×D911yescharacteristic
290C2×C3#41×D912yes—
291C2×C3C2×#379711yescharacteristic
292C2×C3#6430612yes—
293D3—94no—
294D3—94no—
295D3—184no—
296C22—272no—
297C22—274no—
298C22C22×D3×D911yescharacteristic
299C4—22no—
300C4—64no—
301C4—184no—
302C4—272no—
303C4—542no—
304C4(C3⋊D9)⋊Q812yes—
305C3—21no—
306C3D9×#12511yescharacteristic, complemented
307C3#1059394811yescharacteristic
308C2—274no—
309C2C2×(C3⋊D9)⋊Q811yescharacteristic
310C2#1172247512yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC42D3×D9
hypocenterC3×C9C2×#125
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C9×#870C2semidirect
3. C3×#63432C2semidirect
6. #11720937C2semidirect
7. #11720928C2semidirect
37. #63432D3semidirect
104. #870D9semidirect
135. C3⋊D9#125semidirect
199. C3×C9C2×#125semidirect
276. C9D3×#125semidirect
306. C3D9×#125semidirect

Maximal subgroups

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

subgroupindex
C42×C3⋊D92
C9×#8702
C3×#634322
#117210002
#117209802
#117209372
#117209282
#105939483
D9×#1253