Finite Group Databasebeta

← D9×#870 #13147504

Subgroups — D9×#870 #13147504

Order 1728.

734 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 193 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#11721707C211yesmaximal, characteristic, complemented
2C3×C42×D9C211yesmaximal, characteristic
3C9×#870C211yesmaximal, characteristic, complemented
4D9×C4⋊Dic3C212yesmaximal
5D9×Dic3⋊C4C214yesmaximal
6#11720935C212yesmaximal, complemented
7#11720933C214yesmaximal, complemented
8D3×#870—31nomaximal
9D9×#125—31nomaximal
10C2×C3×C4×D9C2212yes—
11C2×C3×C4×D9C2211yescharacteristic
12C2×Dic3×D9C2214yes—
13C3×C42×C9C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2212yes—
15#88534C2214yes—
16C9×C4⋊Dic3C2212yes—
17C9×Dic3⋊C4C2214yes—
18C3×C4×Dic9C2212yes—
19C3×C4×Dic9C2211yescharacteristic
20Dic3⋊Dic9C2214yes—
21#88442C2214yes—
22Dic9⋊Dic3C2214yes—
23#64079—31no—
24C3×C42×D3—31no—
25C3×#870D311yescharacteristic
26D3×C4⋊Dic3—32no—
27D3×Dic3⋊C4—34no—
28#63869—32no—
29#63867—34no—
30C9×#125—31no—
31C4⋊C4×D9—34no—
32C4⋊C4×D9—32no—
33#63452—32no—
34#63451—34no—
35C42×D9D311yescharacteristic
36#63432—31no—
37#63432—61no—
38C22×C3×D9C2311yescharacteristic
39C2×C3×C4×C9C2312yes—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9C2314yes—
42C2×C9×Dic3C2314yes—
43C2×C3×Dic9C2312yes—
44C2×C3×Dic9C2311yescharacteristic
45C3×C4×D9—24no—
46C3×C4×D9Q814yes—
47Dic3×D9—28no—
48D3×#125—91no—
49C2×#870—91no—
50C2×C3×C4×D3—31no—
51C2×C3×C4×D3—32no—
52C2×D3×Dic3—34no—
53C32×C42C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)—32no—
55#3750—34no—
56C3×C4⋊Dic3C2×D312yes—
57C3×Dic3⋊C4C2×D314yes—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—31no—
60#3724—34no—
61Dic3⋊Dic3—34no—
62Dic3⋊Dic3—34no—
63C2×C4×D9—34no—
64C2×C4×D9C2×D312yes—
65C2×C4×D9C2×D311yescharacteristic
66C9×C4⋊C4—34no—
67C9×C4⋊C4—32no—
68C42×C9—21no—
69C42×C9C2×D311yescharacteristic
70C4⋊Dic9—32no—
71C4⋊Dic9—34no—
72C4⋊Dic9—62no—
73Dic9⋊C4—34no—
74Dic9⋊C4—34no—
75Dic9⋊C4—34no—
76Dic9⋊C4—64no—
77C4×Dic9C2×D312yes—
78C4×Dic9C2×D311yescharacteristic
79C22×C3×C9C2411yescharacteristic
80C2×C3×D9C2×Q812yes—
81C2×C3×D9#4114yes—
82C3×C4×C9—22no—
83C3×C4×C9C2×Q812yes—
84C3⋊Dic9—24no—
85C9×Dic3—24no—
86C3×Dic9—22no—
87C3×Dic9C2×Q812yes—
88C3×#125—31no—
89C2×C3×C42—91no—
90C2×C4⋊Dic3—92no—
91C2×Dic3⋊C4—94no—
92D3×C4⋊C4—94no—
93D3×C4⋊C4—92no—
94#890—92no—
95#889—94no—
96C42×D3—31no—
97#870—91no—
98#870—91no—
99#870—92no—
100#870—94no—
101#870—181no—
102#870D911yescharacteristic, direct product factor
103C22×C3×D3—31no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—34no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—32no—
109C2×C3×Dic3C22×D314yes—
110C3×C4×D3—34no—
111C3×C4×D3—64no—
112D3×Dic3—68no—
113C22×D9C22×D311yescharacteristic
114C2×C4×C9—21no—
115C2×C4×C9—22no—
116C2×C4×C9—34no—
117C2×C4×C9C22×D312yes—
118C2×C4×C9C22×D311yescharacteristic
119C2×Dic9—34no—
120C2×Dic9—64no—
121C2×Dic9C22×D312yes—
122C2×Dic9C22×D311yescharacteristic
123C4×D9—24no—
124C4×D9—68no—
125C4×D9Dic614yes—
126C2×#125—271noSylow
127C2×C3×C9C22×Q811yescharacteristic
128C2×C3×C9C2×#4112yes—
129C3×D9#12514yescomplemented
130C22×C3×C4—91no—
131C22×C3×C4—92no—
132C22×Dic3—94no—
133C2×C4×D3—31no—
134C2×C4×D3—32no—
135C2×C4×D3—94no—
136C3×C4⋊C4—34no—
137C3×C4⋊C4—32no—
138C3×C42—21no—
139C3×C42—92no—
140C3×C42—91no—
141C3×C42C2×D911yescharacteristic
142C3×C42D3211yescharacteristic
143C4⋊Dic3—94no—
144C4⋊Dic3—92no—
145C4⋊Dic3—92no—
146C4⋊Dic3—94no—
147C4⋊Dic3—182no—
148C4⋊Dic3C2×D912yes—
149Dic3⋊C4—94no—
150Dic3⋊C4—94no—
151Dic3⋊C4—94no—
152Dic3⋊C4—94no—
153Dic3⋊C4—94no—
154Dic3⋊C4—94no—
155Dic3⋊C4—184no—
156Dic3⋊C4C2×D914yes—
157C4×Dic3—32no—
158C4×Dic3—31no—
159C22×C32C23×D311yescharacteristic
160C2×C3×D3—32no—
161C2×C3×D3—34no—
162C32×C4—22no—
163C32×C4D3×Q812yes—
164C3⋊Dic3—64no—
165C3×Dic3—24no—
166C3×Dic3—32no—
167C3×Dic3—62no—
168C22×C9—21no—
169C22×C9C23×D311yescharacteristic
170C2×D9C2×Dic612yes—
171C2×D9#23514yes—
172C4×C9—22no—
173C4×C9—22no—
174C4×C9—42no—
175C4×C9—64no—
176C4×C9C2×Dic612yes—
177Dic9—22no—
178Dic9—64no—
179Dic9—124no—
180Dic9C2×Dic612yes—
181#125—31no—
182#125—274no—
183#125—272no—
184#125—271no—
185C2×C4⋊C4—274no—
186C2×C4⋊C4—272no—
187C2×C42—91no—
188C3×C9C2×#12511yescharacteristic, complemented, Sylow
189C23×C3—91no—
190C22×D3—31no—
191C2×C3×C4—21no—
192C2×C3×C4—22no—
193C2×C3×C4—34no—
194C2×C3×C4—91no—
195C2×C3×C4—94no—
196C2×C3×C4—92no—
197C2×C3×C4—184no—
198C2×C3×C4C22×D912yes—
199C2×C3×C4C22×D911yescharacteristic
200C2×C3×C4C2×D3212yes—
201C2×C3×C4C2×D3211yescharacteristic
202C2×Dic3—31no—
203C2×Dic3—32no—
204C2×Dic3—94no—
205C2×Dic3—94no—
206C2×Dic3—184no—
207C2×Dic3—188no—
208C2×Dic3C22×D914yes—
209C4×D3—34no—
210C4×D3—64no—
211C4×D3—188no—
212C2×C32C2×D3×Q811yescharacteristic
213C2×C32D3×#4112yes—
214C3×D3—34no—
215C2×C9—21no—
216C2×C9—22no—
217C2×C9C22×Dic611yescharacteristic
218C2×C9C2×#23512yes—
219D9#87014yesdirect product factor
220C22×C4—91no—
221C22×C4—92no—
222C22×C4—274no—
223C4⋊C4—34no—
224C4⋊C4—32no—
225C4⋊C4—274no—
226C4⋊C4—274no—
227C4⋊C4—274no—
228C4⋊C4—274no—
229C4⋊C4—272no—
230C42—92no—
231C42—91no—
232C42D3×D911yescharacteristic
233C22×C3—21no—
234C22×C3—92no—
235C22×C3—94no—
236C22×C3C23×D911yescharacteristic
237C22×C3C22×D3211yescharacteristic
238C2×D3—32no—
239C2×D3—34no—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—42no—
244C3×C4—64no—
245C3×C4—92no—
246C3×C4—182no—
247C3×C4Q8×D912yes—
248C3×C4D3×Dic612yes—
249Dic3—24no—
250Dic3—32no—
251Dic3—62no—
252Dic3—184no—
253Dic3—184no—
254Dic3—364no—
255C32D3×#12511yescharacteristic
256C9—21no—
257C9C2×#87011yescharacteristic, complemented
258C23—91no—
259C2×C4—34no—
260C2×C4—91no—
261C2×C4—94no—
262C2×C4—92no—
263C2×C4—184no—
264C2×C4—274no—
265C2×C4—548no—
266C2×C4C2×D3×D912yes—
267C2×C4C2×D3×D911yescharacteristic
268C2×C3—21no—
269C2×C3—22no—
270C2×C3—94no—
271C2×C3C2×Q8×D911yescharacteristic
272C2×C3#41×D912yes—
273C2×C3C2×D3×Dic611yescharacteristic
274C2×C3D3×#23512yes—
275D3—34no—
276C22—92no—
277C22—94no—
278C22C22×D3×D911yescharacteristic
279C4—22no—
280C4—64no—
281C4—92no—
282C4—182no—
283C4—544no—
284C4D9×Dic612yes—
285C3—21no—
286C3D9×#12511yescharacteristic, complemented
287C3D3×#87011yescharacteristic
288C2—94no—
289C2C2×D9×Dic611yescharacteristic
290C2D9×#23512yes—
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
1. #11721707C2semidirect
3. C9×#870C2semidirect
6. #11720935C2semidirect
7. #11720933C2semidirect
102. #870D9direct (N × H)
129. C3×D9#125semidirect
188. C3×C9C2×#125semidirect
219. D9#870direct (N × H)
257. C9C2×#870semidirect
286. C3D9×#125semidirect

Maximal subgroups

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

subgroupindex
#117217072
C3×C42×D92
C9×#8702
D9×C4⋊Dic32
D9×Dic3⋊C42
#117209352
#117209332
D3×#8703
D9×#1253