Finite Group Databasebeta

← #5574⋊C9 #13147202

Subgroups — #5574⋊C9 #13147202

Order 1728.

324 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 67 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⋊C9)C211yesmaximal, characteristic
2C3×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
3Dic3×Q8⋊C9C211yesmaximal, characteristic, complemented
4C3×#5574C311yesmaximal, characteristic
5#520⋊C9—31nomaximal
6C2×C3×Q8⋊C9C2211yescharacteristic
7C3⋊(#41⋊C9)C412yescomplemented
8C9×Dic3⋊C4—41nomaximal
9C2×C3×#239C2×C311yescharacteristic
10C32×C4×Q8C2×C311yescharacteristic
11C3×Q8×Dic3C2×C311yescharacteristic
12C3×#893—31no—
13C3×D3×C4⋊C4—31no—
14C3×C4×D12—31no—
15C3×#872—31no—
16C2×#41⋊C9—31no—
17C4×Q8⋊C9—21no—
18C4×Q8⋊C9—31no—
19C4×Q8⋊C9D311yescharacteristic, complemented
20C2×C3×C4×C9—41no—
21C2×C9×Dic3—41no—
22C2×C9×Dic3—41no—
23C3×Q8⋊C9C2×C411yescharacteristic, complemented
24C3×#520—31no—
25#5574C911yescharacteristic, complemented
26C2×C32×Q8C22×C311yescharacteristic
27C3×#239—32no—
28C3×#239C3×C412yes—
29C2×C3×D12—31no—
30C2×C3×C4×D3—31no—
31C2×C3×C4×D3—32no—
32C32×C4⋊C4—31no—
33C32×C42—31no—
34C3×(C2×D3)⋊C4—31no—
35C3×(C2×D3)⋊C4—31no—
36C3×C4⋊Dic3—31no—
37C3×Dic3⋊C4—31no—
38C3×Dic3⋊C4A411yescharacteristic
39C3×C4×Dic3—31no—
40#41⋊C9—32no—
41C2×Q8⋊C9—21no—
42C2×Q8⋊C9C2×D311yescharacteristic
43C9×C4⋊C4—121no—
44C22×C3×C9—41no—
45C3×C4×C9—81no—
46C9×Dic3—42no—
47C9×Dic3—81no—
48C2×C3×#41—31no—
49C2×#239C2×C911yescharacteristic
50C3×C4×Q8—21no—
51C3×C4×Q8—31no—
52C3×C4×Q8C2×C911yescharacteristic
53C3×C4×Q8C3×D311yescharacteristic
54C3×C4×D4—91no—
55C3×C4×D4—91no—
56C3×#117—91no—
57C2×C3×C4⋊C4—91no—
58Q8×Dic3C2×C911yescharacteristic
59#893—31no—
60D3×C4⋊C4—31no—
61C4×D12—31no—
62#872—31no—
63C22×C3×D3—31no—
64C32×Q8—31no—
65C32×Q8C2×C3×C411yescharacteristic
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4C2×A411yescharacteristic
69C2×C3×Dic3—31no—
70C2×C3×Dic3C2×A411yescharacteristic
71C2×C3×Dic3C2×A411yescharacteristic
72C3×D12—32no—
73C3×D12—32no—
74C3×C4×D3—32no—
75C3×C4×D3—32no—
76C3×C4×D3—122no—
77C2×C4×C9—41no—
78C2×C4×C9—81no—
79C2×C4×C9—121no—
80C2×C4×C9—121no—
81Q8⋊C9—21no—
82Q8⋊C9C4×D311yescharacteristic, complemented
83#520—31noSylow
84C2×C3×C9—41no—
85C2×C3×C9—41no—
86C2×C3×C9—41no—
87C3×#41—32no—
88C3×#41—92no—
89C2×C3×Q8—21no—
90C2×C3×Q8C22×C911yescharacteristic
91C2×C3×Q8C2×C3×D311yescharacteristic
92C2×C3×D4—91no—
93C22×C3×C4—91no—
94C22×C3×C4—92no—
95#239—32no—
96#239C4×C912yescomplemented
97C2×D12—31no—
98C2×C4×D3—31no—
99C2×C4×D3—32no—
100C3×C4⋊C4—31no—
101C3×C4⋊C4—31no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—61no—
104C3×C4⋊C4—91no—
105C3×C4⋊C4—91no—
106C3×C22⋊C4—91no—
107C3×C22⋊C4—91no—
108C3×C42—31no—
109C3×C42—31no—
110C3×C42—61no—
111C3×C42—91no—
112(C2×D3)⋊C4—31no—
113(C2×D3)⋊C4—31no—
114C4⋊Dic3—31no—
115Dic3⋊C4—31no—
116Dic3⋊C4C22⋊C911yescharacteristic
117C4×Dic3—31no—
118C22×C32C22×A411yescharacteristic
119C2×C3×D3—32no—
120C2×C3×D3—61no—
121C2×C3×D3—61no—
122C32×C4—21no—
123C32×C4—31no—
124C32×C4—31no—
125C32×C4—61no—
126C3×Dic3—21no—
127C3×Dic3—61no—
128C3×Dic3C4×A412yes—
129C22×C9—41no—
130C22×C9—81no—
131C4×C9—81no—
132C4×C9—82no—
133C4×C9—122no—
134C4×C9—241no—
135C2×#41—31no—
136C4×Q8—31no—
137C4×Q8D3×C911yescharacteristic
138C4×D4—91no—
139C4×D4—91no—
140#117—91no—
141C2×C4⋊C4—91no—
142C3×C9—41noSylow
143C23×C3—91no—
144C22×D3—31no—
145C3×Q8—21no—
146C3×Q8—31no—
147C3×Q8—31no—
148C3×Q8—61no—
149C3×Q8C2×C4×C911yescharacteristic
150C3×Q8C3×C4×D311yescharacteristic
151C3×D4—92no—
152C3×D4—92no—
153C2×C3×C4—21no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—61no—
161C2×C3×C4—61no—
162C2×C3×C4—92no—
163C2×C3×C4—92no—
164C2×C3×C4—91no—
165C2×C3×C4—362no—
166C2×C3×C4C2×C22⋊C911yescharacteristic
167C2×C3×C4D3×A411yescharacteristic
168C2×Dic3—31no—
169C2×Dic3C2×C22⋊C911yescharacteristic
170C2×Dic3C2×C22⋊C911yescharacteristic
171D12—32no—
172D12—32no—
173C4×D3—32no—
174C4×D3—32no—
175C4×D3—122no—
176C2×C32C2×C4×A411yescharacteristic
177C2×C32#99411yescharacteristic
178C2×C322−1+4⋊C311yescharacteristic
179C3×D3—62no—
180C2×C9—41no—
181C2×C9—41no—
182C2×C9—41no—
183C2×C9—81no—
184C2×C9—81no—
185C2×C9—81no—
186#41—32no—
187#41—92no—
188C2×Q8C2×D3×C911yescharacteristic
189C2×D4—91no—
190C22×C4—91no—
191C22×C4—92no—
192C4⋊C4—31no—
193C4⋊C4—31no—
194C4⋊C4—91no—
195C4⋊C4—91no—
196C22⋊C4—91no—
197C22⋊C4—91no—
198C42—31no—
199C42—91no—
200C22×C3—21no—
201C22×C3—92no—
202C22×C3—181no—
203C22×C3—181no—
204C22×C3C22×C22⋊C911yescharacteristic
205C22×C3C2×D3×A411yescharacteristic
206C2×D3—32no—
207C2×D3—61no—
208C2×D3—61no—
209C3×C4—22no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—31no—
213C3×C4—31no—
214C3×C4—32no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—61no—
218C3×C4—61no—
219C3×C4—61no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—121no—
223C3×C4—181no—
224Dic3—21no—
225Dic3—61no—
226Dic3C4×C22⋊C912yes—
227C32#520⋊C311yescharacteristic
228C9—41no—
229C9—81no—
230C23—91no—
231Q8—31no—
232Q8C4×D3×C911yescharacteristic
233D4—92no—
234D4—92no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—92no—
240C2×C4—92no—
241C2×C4—91no—
242C2×C4—362no—
243C2×C4D3×C22⋊C911yescharacteristic
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—21no—
247C2×C3—182no—
248C2×C3C2×C4×C22⋊C911yescharacteristic
249C2×C3#6370011yescharacteristic
250C2×C32−1+4⋊C911yescharacteristic
251C2×C3C4×D3×A411yescharacteristic
252C2×C3#1007⋊C311yescharacteristic
253C2×C3C3⋊#99411yescharacteristic
254D3—62no—
255C22—92no—
256C22—181no—
257C22—181no—
258C22C2×D3×C22⋊C911yescharacteristic
259C4—21no—
260C4—31no—
261C4—31no—
262C4—32no—
263C4—61no—
264C4—61no—
265C4—181no—
266C3—21no—
267C3#520⋊C911yescharacteristic, complemented
268C3#5574⋊C311yescharacteristic
269C2—182no—
270C2C4×D3×C22⋊C911yescharacteristic
271C2#1007⋊C911yescharacteristic
272C2C3⋊#6370011yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutator, hypocenterC3×Q8C2×C4×C9
FittingC32×C4×Q8C2×C3
socleC22×C32C22×A4
prime radicalC2×C3×#239C2×C3
cosocleC2×C32×Q8C22×C3
hypercenterC2×C3×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×Q8⋊C9C2semidirect
3. Dic3×Q8⋊C9C2semidirect
7. C3⋊(#41⋊C9)C4semidirect
19. C4×Q8⋊C9D3semidirect
23. C3×Q8⋊C9C2×C4semidirect
25. #5574C9semidirect
82. Q8⋊C9C4×D3semidirect
96. #239C4×C9semidirect
267. C3#520⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(#41⋊C9)2
C3×C4×Q8⋊C92
Dic3×Q8⋊C92
C3×#55743
#520⋊C93
C9×Dic3⋊C44