Finite Group Databasebeta

← #5834⋊C9 #13147287

Subgroups — #5834⋊C9 #13147287

Order 1728.

355 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 61 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×#41⋊C9C211yesmaximal, characteristic, complemented
2C2×C3⋊(#41⋊C9)C211yesmaximal, characteristic, complemented
3Dic3×Q8⋊C9C211yesmaximal, characteristic, complemented
4C3×#5834C311yesmaximal, characteristic
5C4⋊(#41⋊C9)—31nomaximal
6C3×#41⋊C9—21no—
7C2×C3×Q8⋊C9C2211yescharacteristic
8C3⋊(#41⋊C9)—21no—
9C9×Dic3⋊C4—41nomaximal
10C2×C32×#41C2×C311yescharacteristic
11C2×C3×#239C2×C311yescharacteristic
12C3×Q8×Dic3C2×C311yescharacteristic
13C3×#940—31no—
14C3×#938—31no—
15C3×#930—31no—
16C3×C4×#67—31no—
17C2×#41⋊C9—21no—
18C2×#41⋊C9—31no—
19C2×#41⋊C9D311yescharacteristic, complemented
20C4×Q8⋊C9—31no—
21C2×C3×C4×C9—41no—
22C2×C9×Dic3—41no—
23C2×C9×Dic3—41no—
24C3×Q8⋊C9D411yescharacteristic, complemented
25C3×#550—31no—
26#5834C911yescharacteristic, complemented
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8C22×C311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C3×#67—32no—
33C2×C3×#67—31no—
34C3×#239—21no—
35C3×#239—61no—
36C2×C3×D12—31no—
37C2×C3×C4×D3—31no—
38C3×#217—31no—
39C3×(C2×D3)⋊C4—31no—
40C3×C4⋊Dic3—31no—
41C3×Dic3⋊C4A411yescharacteristic
42C3×C4×Dic3—31no—
43#41⋊C9—22no—
44#41⋊C9—21no—
45#41⋊C9—61no—
46C2×Q8⋊C9—21no—
47C2×Q8⋊C9C2×D311yescharacteristic
48C9×C4⋊C4—121no—
49C22×C3×C9—41no—
50C3×C4×C9—81no—
51C9×Dic3—42no—
52C9×Dic3—81no—
53C2×C3×#41—21no—
54C2×C3×#41—31no—
55C2×C3×#41C2×C911yescharacteristic
56C2×C3×#41C3×D311yescharacteristic
57C2×#239C2×C911yescharacteristic
58C3×#127—91no—
59C3×#121—91no—
60C3×#121—91no—
61C3×C4×Q8—31no—
62C3×C4×D4—91no—
63Q8×Dic3C2×C911yescharacteristic
64#940—31no—
65#938—31no—
66#930—31no—
67C4×#67—31no—
68C23×C32—31no—
69C22×C3×D3—31no—
70C32×Q8—31no—
71C32×Q8C3×D411yescharacteristic
72C32×D4—61no—
73C32×D4—61no—
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C2×C32×C4—61no—
77C2×C32×C4C2×A411yescharacteristic
78C3×#67—62no—
79C3×#67—122no—
80C2×C3×Dic3—31no—
81C2×C3×Dic3C2×A411yescharacteristic
82C2×C3×Dic3C2×A411yescharacteristic
83C3×D12—61no—
84C3×D12—61no—
85C3×C4×D3—61no—
86C3×C4×D3—61no—
87C2×C4×C9—41no—
88C2×C4×C9—81no—
89C2×C4×C9—121no—
90C2×C4×C9—121no—
91Q8⋊C9—21no—
92Q8⋊C9#6711yescharacteristic, complemented
93#550—31noSylow
94C2×C3×C9—41no—
95C2×C3×C9—41no—
96C2×C3×C9—41no—
97C3×#41—22no—
98C3×#41—21no—
99C3×#41—21no—
100C3×#41—62no—
101C3×#41—61no—
102C3×#41—61no—
103C3×#41—61no—
104C3×#41—181no—
105C2×C3×Q8—21no—
106C2×C3×Q8C22×C911yescharacteristic
107C2×C3×Q8C2×C3×D311yescharacteristic
108C2×C3×D4—31no—
109C2×C3×D4—31no—
110C2×C3×D4—61no—
111C2×C3×D4—91no—
112C2×C3×D4—91no—
113C2×C3×D4—92no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—61no—
117C22×C3×C4—91no—
118C2×#67—32no—
119C2×#67—31no—
120#239—21no—
121#239—61no—
122C2×D12—31no—
123C2×C4×D3—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—91no—
126C3×C22⋊C4—91no—
127C3×C22⋊C4—91no—
128C3×C42—91no—
129#217—31no—
130(C2×D3)⋊C4—31no—
131C4⋊Dic3—31no—
132Dic3⋊C4C22⋊C911yescharacteristic
133C4×Dic3—31no—
134C22×C32—61no—
135C22×C32—61no—
136C22×C32—61no—
137C22×C32C22×A411yescharacteristic
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—61no—
141C32×C4—21no—
142C32×C4—31no—
143C32×C4—31no—
144C3×Dic3—21no—
145C3×Dic3—61no—
146C3×Dic3#41⋊C312yes—
147C22×C9—41no—
148C22×C9—81no—
149C4×C9—81no—
150C4×C9—82no—
151C4×C9—122no—
152C4×C9—241no—
153C2×#41—31no—
154C2×#41D3×C911yescharacteristic
155#127—91no—
156#121—91no—
157#121—91no—
158C4×Q8—31no—
159C4×D4—91no—
160C3×C9—41noSylow
161C23×C3—31no—
162C23×C3—31no—
163C23×C3—61no—
164C23×C3—91no—
165C22×D3—31no—
166C3×Q8—21no—
167C3×Q8—31no—
168C3×Q8—31no—
169C3×Q8—61no—
170C3×Q8D4×C911yescharacteristic
171C3×Q8C3×#6711yescharacteristic
172C3×D4—61no—
173C3×D4—61no—
174C3×D4—61no—
175C3×D4—61no—
176C3×D4—62no—
177C3×D4—62no—
178C3×D4—182no—
179C3×D4—181no—
180C3×D4—181no—
181C3×D4—362no—
182C2×C3×C4—21no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—91no—
195C2×C3×C4—181no—
196C2×C3×C4—181no—
197C2×C3×C4C2×C22⋊C911yescharacteristic
198C2×C3×C4D3×A411yescharacteristic
199#67—62no—
200#67—122no—
201C2×Dic3—31no—
202C2×Dic3C2×C22⋊C911yescharacteristic
203C2×Dic3C2×C22⋊C911yescharacteristic
204D12—61no—
205D12—61no—
206C4×D3—61no—
207C4×D3—61no—
208C2×C32—121no—
209C2×C32D4×A411yescharacteristic
210C2×C32C2×#41⋊C311yescharacteristic
211C2×C32#99411yescharacteristic
212C3×D3—121no—
213C2×C9—41no—
214C2×C9—41no—
215C2×C9—41no—
216C2×C9—81no—
217C2×C9—81no—
218C2×C9—81no—
219#41—21no—
220#41—61no—
221#41—61no—
222#41—181no—
223C2×Q8C2×D3×C911yescharacteristic
224C2×D4—31no—
225C2×D4—91no—
226C2×D4—91no—
227C2×D4—92no—
228C22×C4—31no—
229C22×C4—91no—
230C4⋊C4—31no—
231C4⋊C4—91no—
232C22⋊C4—91no—
233C22⋊C4—91no—
234C42—91no—
235C22×C3—21no—
236C22×C3—62no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—121no—
244C22×C3—121no—
245C22×C3—181no—
246C22×C3—181no—
247C22×C3—181no—
248C22×C3C22×C22⋊C911yescharacteristic
249C22×C3C2×D3×A411yescharacteristic
250C2×D3—61no—
251C2×D3—61no—
252C2×D3—61no—
253C3×C4—22no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—32no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—181no—
265Dic3—21no—
266Dic3—61no—
267Dic3#41⋊C912yescomplemented
268C32C4⋊(#41⋊C3)11yescharacteristic
269C9—41no—
270C9—81no—
271C23—31no—
272C23—91no—
273Q8—31no—
274Q8C9×#6711yescharacteristic
275D4—61no—
276D4—61no—
277D4—182no—
278D4—181no—
279D4—181no—
280D4—362no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—91no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4D3×C22⋊C911yescharacteristic
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—122no—
294C2×C3—121no—
295C2×C3—121no—
296C2×C3—361no—
297C2×C3D4×C22⋊C911yescharacteristic
298C2×C3C2×#41⋊C911yescharacteristic
299C2×C3#6370011yescharacteristic
300C2×C3C3⋊#99411yescharacteristic
301C2×C3D3×#41⋊C311yescharacteristic
302C2×C3A4×#6711yescharacteristic
303D3—121no—
304C22—61no—
305C22—61no—
306C22—61no—
307C22—181no—
308C22—181no—
309C22—181no—
310C22C2×D3×C22⋊C911yescharacteristic
311C4—21no—
312C4—32no—
313C4—31no—
314C4—31no—
315C4—61no—
316C4—181no—
317C3—21no—
318C3C4⋊(#41⋊C9)11yescharacteristic, complemented
319C3#5834⋊C311yescharacteristic
320C2—121no—
321C2—361no—
322C2C3⋊#6370011yescharacteristic
323C2D3×#41⋊C911yescharacteristic
324C2#67×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutatorC2×C3×Q8C22×C9
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
prime radicalC3×#5834C3
cosocleC2×C32×Q8C22×C3
hypercenterC2×C3×C4D3×A4
hypocenterC3×Q8D4×C9
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. C2×C3⋊(#41⋊C9)C2semidirect
3. Dic3×Q8⋊C9C2semidirect
19. C2×#41⋊C9D3semidirect
24. C3×Q8⋊C9D4semidirect
26. #5834C9semidirect
92. Q8⋊C9#67semidirect
267. Dic3#41⋊C9semidirect
318. C3C4⋊(#41⋊C9)semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
C2×C3⋊(#41⋊C9)2
Dic3×Q8⋊C92
C3×#58343
C4⋊(#41⋊C9)3
C9×Dic3⋊C44