Finite Group Databasebeta

← D9×#833 #13140385

Subgroups — D9×#833 #13140385

Order 1728.

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

normal subgroups only

This group has 604 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×D4×D9C211yesmaximal, characteristic
2D9×#208C212yesmaximal, complemented
3#11720575C211yesmaximal, characteristic, complemented
4C9×#833C211yesmaximal, characteristic, complemented
5#11720287C212yesmaximal, complemented
6D3×#833—31nomaximal
7D9×#100—31nomaximal
8C23×C3×D9C412yes—
9C2×C3×D4×C9C2211yescharacteristic
10C2×C3×#611C412yes—
11C3×D4×D9—22no—
12C3×D4×D9—22no—
13C2×C3×D36C2211yescharacteristic
14C2×C3×C4×D9C2211yescharacteristic
15#88532C2212yescomplemented
16C9×#208C2212yescomplemented
17#88413—22no—
18D9×C3⋊C8—22no—
19C2×C3×D3×D4—31no—
20D3×#208—32no—
21C2×D4×D9D311yescharacteristic
22#63660—31no—
23C3×#833D311yescharacteristic
24#63558—32no—
25M16×D9—32no—
26C9×#100—31no—
27#63391—31no—
28#63391—61no—
29#63383—32no—
30C23×C3×C9C2×C412yes—
31C22×C3×D9—22no—
32C22×C3×D9—22no—
33C22×C3×D9—44no—
34C22×C3×D9C2×C411yescharacteristic
35C22×C3×D9C2×C412yes—
36C3×D4×C9—22no—
37C2×C3×C4×C9C2311yescharacteristic
38C3×#611—24no—
39C2×C3×Dic9C2×C411yescharacteristic
40C3×D36—22no—
41C3×C4×D9D412yes—
42C3×C4×D9D412yes—
43C3⋊(C9⋊C8)—22no—
44C9×C3⋊C8—22no—
45C2×#833—91no—
46D3×#100—91no—
47C23×C3×D3—32no—
48C2×C32×D4C2×D311yescharacteristic
49C2×C3×#67—32no—
50C3×D3×D4—62no—
51C3×D3×D4—62no—
52C2×C3×D12—31no—
53C2×C3×C4×D3—31no—
54C23×D9Dic312yes—
55#3748—32no—
56C3×#208C2×D312yes—
57#3711—62no—
58D3×C3⋊C8—62no—
59C2×D4×C9—21no—
60C2×D4×C9C2×D311yescharacteristic
61C2×#611Dic312yes—
62D4×D9—22no—
63D4×D9—22no—
64C2×D36C2×D311yescharacteristic
65C2×C4×D9C2×D311yescharacteristic
66C9×M16—32no—
67#3667—32no—
68#3667—62no—
69#3663—62no—
70C8×D9—62no—
71C22×C3×C9—22no—
72C22×C3×C9—42no—
73C22×C3×C9C22×C411yescharacteristic
74C2×C3×D9—22no—
75C2×C3×D9—22no—
76C2×C3×D9—44no—
77C2×C3×D9—44no—
78C2×C3×D9—42no—
79C2×C3×D9C22⋊C411yescharacteristic
80C2×C3×D9C22⋊C411yescharacteristic
81C3×C4×C9C2×D412yes—
82C3×Dic9C22⋊C412yes—
83C22×C3×D4—91no—
84C2×D3×D4—31no—
85C2×#208—92no—
86D3×M16—92no—
87C3×#100—31no—
88#833—91no—
89#833—91no—
90#833—92no—
91#833—181no—
92#833D911yescharacteristic, direct product factor
93#823—92no—
94C23×C32C4×D312yes—
95C22×C3×D3—31no—
96C22×C3×D3—32no—
97C22×C3×D3—62no—
98C22×C3×D3—62no—
99C22×C3×D3—124no—
100C32×D4—22no—
101C2×C32×C4C22×D311yescharacteristic
102C3×#67—64no—
103C2×C3×Dic3—31no—
104C3×D12—62no—
105C3×C4×D3—32no—
106C3×C4×D3—32no—
107C23×C9—22no—
108C23×C9C2×Dic312yes—
109C22×D9—22no—
110C22×D9—22no—
111C22×D9—44no—
112C22×D9C2×Dic311yescharacteristic
113C22×D9C2×Dic312yes—
114C3⋊(C3⋊C8)—62no—
115C3×C3⋊C8—22no—
116D4×C9—22no—
117D4×C9—42no—
118C2×C4×C9—21no—
119C2×C4×C9C22×D311yescharacteristic
120#611—24no—
121C2×Dic9C2×Dic311yescharacteristic
122D36—22no—
123C4×D9#6712yes—
124C4×D9#6712yes—
125C8×C9—62no—
126C9⋊C8—62no—
127C9⋊C8—122no—
128C2×#100—271noSylow
129C2×C3×C9—21no—
130C2×C3×C9—42no—
131C2×C3×C9C2×C22⋊C411yescharacteristic
132C3×D9—21no—
133C3×D9—42no—
134C3×D9#10012yescomplemented
135C24×C3—92no—
136C23×D3—32no—
137C2×C3×D4—21no—
138C2×C3×D4—92no—
139C2×C3×D4—91no—
140C2×C3×D4—182no—
141C2×C3×D4—182no—
142C2×C3×D4C2×D911yescharacteristic
143C2×C3×D4D3211yescharacteristic
144C22×C3×C4—91no—
145C2×#67—32no—
146D3×D4—62no—
147D3×D4—62no—
148C2×D12—31no—
149C2×C4×D3—31no—
150C3×M16—32no—
151#208—92no—
152#208—92no—
153#208—182no—
154#208—182no—
155#208C2×D912yescomplemented
156C2×C3⋊C8—182no—
157ΓL(1,𝔽25)—182no—
158D3×C8—182no—
159C22×C32—22no—
160C22×C32—42no—
161C22×C32C2×C4×D311yescharacteristic
162C2×C3×D3—31no—
163C2×C3×D3—31no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C2×C3×D3—124no—
167C2×C3×D3—124no—
168C2×C3×D3—122no—
169C32×C4D3×D412yes—
170C3×Dic3—32no—
171C22×C9—22no—
172C22×C9—21no—
173C22×C9—42no—
174C22×C9—42no—
175C22×C9—82no—
176C22×C9C22×Dic311yescharacteristic
177C2×D9—22no—
178C2×D9—22no—
179C2×D9—44no—
180C2×D9—44no—
181C2×D9—42no—
182C2×D9#21711yescharacteristic
183C2×D9#21711yescharacteristic
184C4×C9—22no—
185C4×C9C2×#6712yes—
186Dic9#21712yes—
187C22×D4—91no—
188C2×M16—272no—
189#100—31no—
190#100—272no—
191#100—271no—
192C3×C9C2×#10011yescharacteristic, complemented, Sylow
193C23×C3—22no—
194C23×C3—91no—
195C23×C3—92no—
196C23×C3—182no—
197C23×C3—182no—
198C23×C3—364no—
199C23×C3C4×D912yes—
200C23×C3D3×Dic312yes—
201C22×D3—31no—
202C22×D3—32no—
203C22×D3—62no—
204C22×D3—62no—
205C22×D3—124no—
206C3×D4—22no—
207C3×D4—22no—
208C3×D4—42no—
209C3×D4—184no—
210C3×D4—182no—
211C2×C3×C4—21no—
212C2×C3×C4—91no—
213C2×C3×C4—92no—
214C2×C3×C4—92no—
215C2×C3×C4C22×D911yescharacteristic
216C2×C3×C4C2×D3211yescharacteristic
217#67—64no—
218C2×Dic3—31no—
219D12—62no—
220C4×D3—32no—
221C4×D3—32no—
222C3×C8—62no—
223C3⋊C8—22no—
224C3⋊C8—182no—
225C3⋊C8—182no—
226C3⋊C8—362no—
227C2×C32—21no—
228C2×C32—42no—
229C2×C32D3×C22⋊C411yescharacteristic
230C3×D3—32no—
231C3×D3—61no—
232C3×D3—122no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9—41no—
236C2×C9—42no—
237C2×C9—82no—
238C2×C9C2×#21711yescharacteristic
239D9—21no—
240D9—42no—
241D9#83312yesdirect product factor
242C24—92no—
243C2×D4—92no—
244C2×D4—91no—
245C2×D4—182no—
246C2×D4—182no—
247C2×D4D3×D911yescharacteristic
248C22×C4—91no—
249M16—32no—
250M16—272no—
251M16—542no—
252C2×C8—542no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—21no—
256C22×C3—42no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3—82no—
260C22×C3—91no—
261C22×C3—91no—
262C22×C3—182no—
263C22×C3—182no—
264C22×C3—362no—
265C22×C3—364no—
266C22×C3—364no—
267C22×C3C2×C4×D911yescharacteristic
268C22×C3C2×D3×Dic311yescharacteristic
269C2×D3—31no—
270C2×D3—31no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—124no—
274C2×D3—124no—
275C2×D3—122no—
276C3×C4—22no—
277C3×C4—92no—
278C3×C4D4×D912yes—
279C3×C4D3×#6712yes—
280Dic3—32no—
281C32D3×#10011yescharacteristic
282C9—21no—
283C9C2×#83311yescharacteristic, complemented
284C23—91no—
285C23—92no—
286C23—182no—
287C23—182no—
288C23—364no—
289C23Dic3×D912yes—
290D4—22no—
291D4—184no—
292D4—182no—
293C2×C4—91no—
294C2×C4—92no—
295C2×C4—92no—
296C2×C4C2×D3×D911yescharacteristic
297C8—62no—
298C8—542no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—41no—
303C2×C3—42no—
304C2×C3—42no—
305C2×C3—82no—
306C2×C3—92no—
307C2×C3—181no—
308C2×C3—362no—
309C2×C3C22⋊C4×D911yescharacteristic
310C2×C3D3×#21711yescharacteristic
311D3—32no—
312D3—61no—
313D3—122no—
314C22—22no—
315C22—42no—
316C22—91no—
317C22—91no—
318C22—182no—
319C22—182no—
320C22—362no—
321C22—364no—
322C22—364no—
323C22C2×Dic3×D911yescharacteristic
324C4—92no—
325C4D9×#6712yes—
326C3—21no—
327C3D9×#10011yescharacteristic, complemented
328C3D3×#83311yescharacteristic
329C2—21no—
330C2—42no—
331C2—92no—
332C2—181no—
333C2—362no—
334C2D9×#21711yescharacteristic
subgroupG/H
centerC2D9×#217
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32D3×C22⋊C4
prime radicalC2×C3×D4×D9C2
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×#100
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D9×#208C2semidirect
3. #11720575C2semidirect
4. C9×#833C2semidirect
5. #11720287C2semidirect
15. #88532C22semidirect
16. C9×#208C22semidirect
92. #833D9direct (N × H)
134. C3×D9#100semidirect
155. #208C2×D9semidirect
192. C3×C9C2×#100semidirect
241. D9#833direct (N × H)
283. C9C2×#833semidirect
327. C3D9×#100semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
D9×#2082
#117205752
C9×#8332
#117202872
D3×#8333
D9×#1003