Finite Group Databasebeta

← D9×#836 #13140759

Subgroups — D9×#836 #13140759

Order 1728.

476 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×C3×Q8×D9C211yesmaximal, characteristic
2D9×#208C212yesmaximal
3#11720581C211yesmaximal, characteristic, complemented
4C9×#836C211yesmaximal, characteristic, complemented
5#11720289C212yesmaximal, complemented
6D3×#836—31nomaximal
7D9×#101—31nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9C3×Q8×D9—22no—
10C3×Q8×D9—22no—
11C2×C3×C4×D9C412yes—
12C2×C3×C4×D9C2211yescharacteristic
13C2×C3×Dic18C412yes—
14C2×C3×Dic18C2211yescharacteristic
15#88532C2212yes—
16C9×#208C2212yes—
17#88413—22no—
18D9×C3⋊C8—22no—
19C2×C3×D3×Q8—31no—
20D3×#208—32no—
21C2×Q8×D9D311yescharacteristic
22#63663—31no—
23C3×#836D311yescharacteristic
24#63560—32no—
25M16×D9—32no—
26C9×#101—31no—
27#63394—31no—
28#63394—61no—
29#63382—32no—
30C22×C3×D9C2×C411yescharacteristic
31C3×Q8×C9—22no—
32C2×C3×C4×C9C2×C412yes—
33C2×C3×C4×C9C2311yescharacteristic
34C2×C3×Dic9C2×C412yes—
35C2×C3×Dic9C2×C411yescharacteristic
36C3×C4×D9—22no—
37C3×C4×D9—22no—
38C3×C4×D9D412yes—
39C3×C4×D9D412yes—
40C3×Dic18—22no—
41C3×Dic18—24no—
42C3⋊(C9⋊C8)—22no—
43C9×C3⋊C8—22no—
44C2×#836—91no—
45D3×#101—91no—
46C2×C32×Q8C2×D311yescharacteristic
47C3×D3×Q8—62no—
48C3×D3×Q8—62no—
49C2×C3×C4×D3—31no—
50C2×C3×C4×D3—32no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—31no—
53#3748—32no—
54C3×#208C2×D312yes—
55#3711—62no—
56D3×C3⋊C8—62no—
57C2×Q8×C9—21no—
58C2×Q8×C9C2×D311yescharacteristic
59Q8×D9—22no—
60Q8×D9—22no—
61C2×C4×D9Dic312yes—
62C2×C4×D9C2×D311yescharacteristic
63C2×Dic18Dic312yes—
64C2×Dic18C2×D311yescharacteristic
65C9×M16—32no—
66#3667—32no—
67#3667—62no—
68#3663—62no—
69C8×D9—62no—
70C22×C3×C9C22×C411yescharacteristic
71C2×C3×D9—22no—
72C2×C3×D9C22⋊C411yescharacteristic
73C2×C3×D9C22⋊C411yescharacteristic
74C3×C4×C9—22no—
75C3×C4×C9C2×D412yes—
76C3×Dic9—22no—
77C3×Dic9C22⋊C412yes—
78C22×C3×Q8—91no—
79C2×D3×Q8—31no—
80C2×#208—92no—
81D3×M16—92no—
82C3×#101—31no—
83#836—91no—
84#836—91no—
85#836—92no—
86#836—181no—
87#836D911yescharacteristic, direct product factor
88#824—92no—
89C22×C3×D3—31no—
90C32×Q8—22no—
91C2×C32×C4C4×D312yes—
92C2×C32×C4C22×D311yescharacteristic
93C2×C3×Dic3—31no—
94C2×C3×Dic3—32no—
95C3×C4×D3—32no—
96C3×C4×D3—32no—
97C3×C4×D3—62no—
98C3×C4×D3—62no—
99C3×Dic6—64no—
100C3×Dic6—62no—
101C22×D9C2×Dic311yescharacteristic
102C3⋊(C3⋊C8)—62no—
103C3×C3⋊C8—22no—
104Q8×C9—22no—
105Q8×C9—42no—
106C2×C4×C9—22no—
107C2×C4×C9—21no—
108C2×C4×C9C2×Dic312yes—
109C2×C4×C9C22×D311yescharacteristic
110C2×Dic9C2×Dic312yes—
111C2×Dic9C2×Dic311yescharacteristic
112C4×D9—22no—
113C4×D9—22no—
114C4×D9#6712yes—
115C4×D9#6712yes—
116Dic18—22no—
117Dic18—24no—
118C8×C9—62no—
119C9⋊C8—62no—
120C9⋊C8—122no—
121C2×#101—271noSylow
122C2×C3×C9—21no—
123C2×C3×C9C2×C22⋊C411yescharacteristic
124C3×D9—21no—
125C3×D9#10112yescomplemented
126C2×C3×Q8—21no—
127C2×C3×Q8—92no—
128C2×C3×Q8—91no—
129C2×C3×Q8—182no—
130C2×C3×Q8—182no—
131C2×C3×Q8C2×D911yescharacteristic
132C2×C3×Q8D3211yescharacteristic
133C22×C3×C4—91no—
134C22×C3×C4—92no—
135D3×Q8—62no—
136D3×Q8—62no—
137C2×C4×D3—31no—
138C2×C4×D3—32no—
139C2×Dic6—32no—
140C2×Dic6—31no—
141C3×M16—32no—
142#208—92no—
143#208—92no—
144#208—182no—
145#208—182no—
146#208C2×D912yes—
147C2×C3⋊C8—182no—
148ΓL(1,𝔽25)—182no—
149D3×C8—182no—
150C22×C32C2×C4×D311yescharacteristic
151C2×C3×D3—31no—
152C2×C3×D3—31no—
153C2×C3×D3—62no—
154C32×C4—22no—
155C32×C4D3×D412yes—
156C3×Dic3—32no—
157C3×Dic3—62no—
158C22×C9—21no—
159C22×C9C22×Dic311yescharacteristic
160C2×D9—22no—
161C2×D9#21711yescharacteristic
162C2×D9#21711yescharacteristic
163C4×C9—22no—
164C4×C9—22no—
165C4×C9—42no—
166C4×C9C2×#6712yes—
167Dic9—22no—
168Dic9#21712yes—
169C22×Q8—91no—
170C2×M16—272no—
171#101—31no—
172#101—272no—
173#101—271no—
174C3×C9C2×#10111yescharacteristic, complemented, Sylow
175C23×C3—91no—
176C22×D3—31no—
177C3×Q8—22no—
178C3×Q8—22no—
179C3×Q8—42no—
180C3×Q8—184no—
181C3×Q8—182no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—91no—
185C2×C3×C4—92no—
186C2×C3×C4—92no—
187C2×C3×C4—92no—
188C2×C3×C4—182no—
189C2×C3×C4—182no—
190C2×C3×C4C4×D912yes—
191C2×C3×C4C22×D911yescharacteristic
192C2×C3×C4D3×Dic312yes—
193C2×C3×C4C2×D3211yescharacteristic
194C2×Dic3—31no—
195C2×Dic3—32no—
196C4×D3—32no—
197C4×D3—32no—
198C4×D3—62no—
199C4×D3—62no—
200Dic6—64no—
201Dic6—62no—
202C3×C8—62no—
203C3⋊C8—22no—
204C3⋊C8—182no—
205C3⋊C8—182no—
206C3⋊C8—362no—
207C2×C32—21no—
208C2×C32D3×C22⋊C411yescharacteristic
209C3×D3—32no—
210C3×D3—61no—
211C2×C9—21no—
212C2×C9—21no—
213C2×C9—41no—
214C2×C9C2×#21711yescharacteristic
215D9—21no—
216D9#83612yesdirect product factor
217C2×Q8—92no—
218C2×Q8—91no—
219C2×Q8—182no—
220C2×Q8—182no—
221C2×Q8D3×D911yescharacteristic
222C22×C4—91no—
223C22×C4—92no—
224M16—32no—
225M16—272no—
226M16—542no—
227C2×C8—542no—
228C22×C3—21no—
229C22×C3—91no—
230C22×C3—91no—
231C22×C3—182no—
232C22×C3C2×C4×D911yescharacteristic
233C22×C3C2×D3×Dic311yescharacteristic
234C2×D3—31no—
235C2×D3—31no—
236C2×D3—62no—
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—42no—
241C3×C4—92no—
242C3×C4—182no—
243C3×C4D4×D912yes—
244C3×C4D3×#6712yes—
245Dic3—32no—
246Dic3—62no—
247C32D3×#10111yescharacteristic
248C9—21no—
249C9C2×#83611yescharacteristic, complemented
250C23—91no—
251Q8—22no—
252Q8—184no—
253Q8—182no—
254C2×C4—91no—
255C2×C4—92no—
256C2×C4—92no—
257C2×C4—92no—
258C2×C4—182no—
259C2×C4—182no—
260C2×C4Dic3×D912yes—
261C2×C4C2×D3×D911yescharacteristic
262C8—62no—
263C8—542no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—41no—
268C2×C3—92no—
269C2×C3—181no—
270C2×C3C22⋊C4×D911yescharacteristic
271C2×C3D3×#21711yescharacteristic
272D3—32no—
273D3—61no—
274C22—91no—
275C22—91no—
276C22—182no—
277C22C2×Dic3×D911yescharacteristic
278C4—22no—
279C4—92no—
280C4—182no—
281C4D9×#6712yes—
282C3—21no—
283C3D9×#10111yescharacteristic, complemented
284C3D3×#83611yescharacteristic
285C2—21no—
286C2—92no—
287C2—181no—
288C2D9×#21711yescharacteristic
subgroupG/H
centerC2D9×#217
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC2×C32D3×C22⋊C4
prime radicalC22×C3×D9C2×C4
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9C2×#101
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #11720581C2semidirect
4. C9×#836C2semidirect
5. #11720289C2semidirect
87. #836D9direct (N × H)
125. C3×D9#101semidirect
174. C3×C9C2×#101semidirect
216. D9#836direct (N × H)
249. C9C2×#836semidirect
283. C3D9×#101semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D92
D9×#2082
#117205812
C9×#8362
#117202892
D3×#8363
D9×#1013