Finite Group Databasebeta

← #5576⋊C9 #13147209

Subgroups — #5576⋊C9 #13147209

Order 1728.

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

normal subgroups only

This group has 340 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×D3×Q8⋊C9C211yesmaximal, characteristic, complemented
2C2×C3⋊(#41⋊C9)C211yesmaximal, characteristic, complemented
3C3×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
4C3×#5576C311yesmaximal, characteristic
5#548⋊C9—31nomaximal
6C2×C3×Q8⋊C9C2211yescharacteristic
7D3×Q8⋊C9—21no—
8C3⋊(#41⋊C9)—21no—
9C9×(C2×D3)⋊C4—41nomaximal
10C2×C3×#239C2×C311yescharacteristic
11C2×C3×D3×Q8C2×C311yescharacteristic
12C32×C4×Q8C2×C311yescharacteristic
13C3×#897—31no—
14C3×#895—31no—
15C3×#875—31no—
16C3×C4×D12—31no—
17C2×#41⋊C9—31no—
18C22×Q8⋊C9—31no—
19C4×Q8⋊C9—21no—
20C4×Q8⋊C9D311yescharacteristic, complemented
21C22×D3×C9—41no—
22C2×C3×C4×C9—41no—
23C2×C9×Dic3—41no—
24C3×Q8⋊C9D411yescharacteristic, complemented
25C3×#548—31no—
26#5576C911yescharacteristic, complemented
27C2×C32×Q8C22×C311yescharacteristic
28C3×#239—21no—
29C3×#239—61no—
30C3×D3×Q8—21no—
31C3×D3×Q8—61no—
32C2×C3×D12—31no—
33C2×C3×D12—31no—
34C2×C3×C4×D3—31no—
35C2×C3×C4×D3—31no—
36C2×C3×Dic6—31no—
37C32×C4⋊C4—31no—
38C32×C42—31no—
39C3×(C2×D3)⋊C4—32no—
40C3×(C2×D3)⋊C4—31no—
41C3×(C2×D3)⋊C4A411yescharacteristic
42C3×C4⋊Dic3—31no—
43#41⋊C9—61no—
44C2×Q8⋊C9—21no—
45C2×Q8⋊C9—61no—
46C2×Q8⋊C9C2×D311yescharacteristic
47C9×C22⋊C4—121no—
48C22×C3×C9—41no—
49C2×D3×C9—42no—
50C2×D3×C9—81no—
51C2×D3×C9—81no—
52C3×C4×C9—81no—
53C9×Dic3—81no—
54C2×C3×#41—31no—
55C22×C3×Q8—31no—
56C2×#239C2×C911yescharacteristic, complemented
57C2×D3×Q8C2×C911yescharacteristic
58C3×#124—91no—
59C3×C22⋊Q8—91no—
60C3×#121—91no—
61C3×C4×Q8—21no—
62C3×C4×Q8C2×C911yescharacteristic, complemented
63C3×C4×Q8C3×D311yescharacteristic
64C3×C4×D4—91no—
65#897—31no—
66#895—31no—
67#875—31no—
68C4×D12—31no—
69C22×C3×D3—31no—
70C22×C3×D3C2×A411yescharacteristic
71C32×Q8—31no—
72C32×Q8C3×D411yescharacteristic
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4C2×A411yescharacteristic
76C2×C3×Dic3—31no—
77C2×C3×Dic3C2×A411yescharacteristic
78C3×D12—61no—
79C3×D12—61no—
80C3×D12—62no—
81C3×C4×D3—61no—
82C3×C4×D3—61no—
83C3×C4×D3—61no—
84C3×C4×D3—61no—
85C3×Dic6—61no—
86C3×Dic6—61no—
87C23×C9—121no—
88C2×C4×C9—41no—
89C2×C4×C9—81no—
90C2×C4×C9—121no—
91Q8⋊C9—21no—
92Q8⋊C9D1211yescharacteristic, complemented
93#548—31noSylow
94C2×C3×C9—41no—
95C2×C3×C9—41no—
96C2×C3×C9—41no—
97D3×C9—82no—
98C3×#41—61no—
99C3×#41—181no—
100C2×C3×Q8—21no—
101C2×C3×Q8—61no—
102C2×C3×Q8—91no—
103C2×C3×Q8—181no—
104C2×C3×Q8C22×C911yescharacteristic
105C2×C3×Q8C2×C3×D311yescharacteristic
106C2×C3×D4—91no—
107C2×C3×D4—91no—
108C22×C3×C4—91no—
109C22×C3×C4—91no—
110#239—21no—
111#239—61no—
112D3×Q8—21no—
113D3×Q8—61no—
114C2×D12—31no—
115C2×D12—31no—
116C2×C4×D3—31no—
117C2×C4×D3—31no—
118C2×Dic6—31no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—91no—
123C3×C22⋊C4—31no—
124C3×C22⋊C4—92no—
125C3×C22⋊C4—91no—
126C3×C42—31no—
127C3×C42—31no—
128C3×C42—61no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—31no—
131(C2×D3)⋊C4C22⋊C911yescharacteristic
132C4⋊Dic3—31no—
133C22×C32C22×A411yescharacteristic
134C2×C3×D3—21no—
135C2×C3×D3—21no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3#41⋊C312yes—
140C32×C4—21no—
141C32×C4—31no—
142C32×C4—31no—
143C32×C4—61no—
144C3×Dic3—21no—
145C3×Dic3—61no—
146C22×C9—41no—
147C22×C9—81no—
148C22×C9—122no—
149C22×C9—241no—
150C22×C9—241no—
151C4×C9—81no—
152C4×C9—82no—
153C4×C9—241no—
154C2×#41—31no—
155C22×Q8—31no—
156#124—91no—
157C22⋊Q8—91no—
158#121—91no—
159C4×Q8D3×C911yescharacteristic, complemented
160C4×D4—91no—
161C3×C9—41noSylow
162C23×C3—31no—
163C23×C3—91no—
164C22×D3—31no—
165C22×D3C2×C22⋊C911yescharacteristic
166C3×Q8—21no—
167C3×Q8—31no—
168C3×Q8—31no—
169C3×Q8—61no—
170C3×Q8—181no—
171C3×Q8—181no—
172C3×Q8D4×C911yescharacteristic
173C3×Q8C3×D1211yescharacteristic
174C3×D4—181no—
175C3×D4—181no—
176C3×D4—182no—
177C2×C3×C4—21no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—91no—
186C2×C3×C4—181no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4—181no—
190C2×C3×C4C2×C22⋊C911yescharacteristic
191C2×C3×C4D3×A411yescharacteristic
192C2×Dic3—31no—
193C2×Dic3C2×C22⋊C911yescharacteristic
194D12—61no—
195D12—61no—
196D12—62no—
197C4×D3—61no—
198C4×D3—61no—
199C4×D3—61no—
200C4×D3—61no—
201Dic6—61no—
202Dic6—61no—
203C2×C32D4×A411yescharacteristic
204C2×C32C2×#41⋊C311yescharacteristic
205C2×C322−1+4⋊C311yescharacteristic
206C3×D3—22no—
207C3×D3—121no—
208C2×C9—41no—
209C2×C9—41no—
210C2×C9—41no—
211C2×C9—81no—
212C2×C9—81no—
213C2×C9—81no—
214C2×C9—242no—
215#41—61no—
216#41—181no—
217C2×Q8—61no—
218C2×Q8—91no—
219C2×Q8—181no—
220C2×Q8C2×D3×C911yescharacteristic
221C2×D4—91no—
222C2×D4—91no—
223C22×C4—91no—
224C22×C4—91no—
225C4⋊C4—31no—
226C4⋊C4—91no—
227C22⋊C4—31no—
228C22⋊C4—92no—
229C22⋊C4—91no—
230C42—31no—
231C22×C3—21no—
232C22×C3—32no—
233C22×C3—61no—
234C22×C3—61no—
235C22×C3—181no—
236C22×C3—181no—
237C22×C3—181no—
238C22×C3C22×C22⋊C911yescharacteristic
239C22×C3C2×D3×A411yescharacteristic
240C2×D3—21no—
241C2×D3—21no—
242C2×D3—61no—
243C2×D3—61no—
244C2×D3—61no—
245C2×D3#41⋊C912yescomplemented
246C3×C4—22no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—61no—
258C3×C4—121no—
259C3×C4—181no—
260Dic3—21no—
261Dic3—61no—
262C32#548⋊C311yescharacteristic
263C9—41no—
264C9—81no—
265C23—31no—
266C23—91no—
267Q8—31no—
268Q8—181no—
269Q8—181no—
270Q8C9×D1211yescharacteristic
271D4—181no—
272D4—181no—
273D4—182no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—91no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4D3×C22⋊C911yescharacteristic
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—62no—
287C2×C3—361no—
288C2×C3D4×C22⋊C911yescharacteristic
289C2×C3C2×#41⋊C911yescharacteristic
290C2×C32−1+4⋊C911yescharacteristic
291C2×C3A4×D1211yescharacteristic
292C2×C3#1007⋊C311yescharacteristic
293C2×C3D3×#41⋊C311yescharacteristic
294D3—22no—
295D3—121no—
296C22—32no—
297C22—61no—
298C22—61no—
299C22—181no—
300C22—181no—
301C22—181no—
302C22C2×D3×C22⋊C911yescharacteristic
303C4—21no—
304C4—31no—
305C4—31no—
306C4—61no—
307C4—61no—
308C4—181no—
309C3—21no—
310C3#548⋊C911yescharacteristic, complemented
311C3#5576⋊C311yescharacteristic
312C2—62no—
313C2—361no—
314C2D12×C22⋊C911yescharacteristic
315C2#1007⋊C911yescharacteristic
316C2D3×#41⋊C911yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutatorC2×C3×Q8C22×C9
FittingC32×C4×Q8C2×C3
socleC22×C32C22×A4
prime radicalC3×#5576C3
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×D3×Q8⋊C9C2semidirect
2. C2×C3⋊(#41⋊C9)C2semidirect
3. C3×C4×Q8⋊C9C2semidirect
20. C4×Q8⋊C9D3semidirect
24. C3×Q8⋊C9D4semidirect
26. #5576C9semidirect
56. C2×#239C2×C9semidirect
62. C3×C4×Q8C2×C9semidirect
92. Q8⋊C9D12semidirect
159. C4×Q8D3×C9semidirect
245. C2×D3#41⋊C9semidirect
310. C3#548⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Q8⋊C92
C2×C3⋊(#41⋊C9)2
C3×C4×Q8⋊C92
C3×#55763
#548⋊C93
C9×(C2×D3)⋊C44