Finite Group Databasebeta

← #5572⋊C9 #13147208

Subgroups — #5572⋊C9 #13147208

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×D3×Q8⋊C9C211yesmaximal, characteristic
2C3×C4×Q8⋊C9C211yesmaximal, characteristic, complemented
3Dic3×Q8⋊C9C211yesmaximal, characteristic, complemented
4C3×#5572C311yesmaximal, characteristic
5#519⋊C9—31nomaximal
6C2×C3×Q8⋊C9C2211yescharacteristic
7D3×Q8⋊C9C412yescomplemented
8C9×(C2×D3)⋊C4—41nomaximal
9C2×C3×D3×Q8C2×C311yescharacteristic
10C32×C4×Q8C2×C311yescharacteristic
11C3×Q8×Dic3C2×C311yescharacteristic
12C3×#892—31no—
13C3×#887—31no—
14C3×#872—31no—
15C3×C4×Dic6—31no—
16C22×Q8⋊C9—31no—
17C4×Q8⋊C9—21no—
18C4×Q8⋊C9—31no—
19C4×Q8⋊C9D311yescharacteristic, complemented
20C22×D3×C9—41no—
21C2×C3×C4×C9—41no—
22C2×C9×Dic3—41no—
23C3×Q8⋊C9C2×C411yescharacteristic
24C3×#519—31no—
25#5572C911yescharacteristic, complemented
26C2×C32×Q8C22×C311yescharacteristic
27C3×D3×Q8—32no—
28C3×D3×Q8C3×C412yes—
29C2×C3×C4×D3—31no—
30C2×C3×Dic6—31no—
31C32×C4⋊C4—31no—
32C32×C42—31no—
33C3×(C2×D3)⋊C4—31no—
34C3×(C2×D3)⋊C4A411yescharacteristic
35C3×C4⋊Dic3—31no—
36C3×Dic3⋊C4—31no—
37C3×Dic3⋊C4—31no—
38C3×C4×Dic3—32no—
39C3×C4×Dic3—31no—
40C2×Q8⋊C9—21no—
41C2×Q8⋊C9—32no—
42C2×Q8⋊C9C2×D311yescharacteristic
43C9×C22⋊C4—121no—
44C22×C3×C9—41no—
45C2×D3×C9—42no—
46C2×D3×C9—81no—
47C2×D3×C9—81no—
48C3×C4×C9—81no—
49C9×Dic3—81no—
50C22×C3×Q8—31no—
51C2×D3×Q8C2×C911yescharacteristic
52C3×C4×Q8—21no—
53C3×C4×Q8—31no—
54C3×C4×Q8—91no—
55C3×C4×Q8—91no—
56C3×C4×Q8C2×C911yescharacteristic, complemented
57C3×C4×Q8C3×D311yescharacteristic
58C3×#117—91no—
59C3×#117—91no—
60Q8×Dic3C2×C911yescharacteristic, complemented
61#892—31no—
62#887—31no—
63#872—31no—
64C4×Dic6—31no—
65C22×C3×D3C2×A411yescharacteristic
66C32×Q8—31no—
67C32×Q8C2×C3×C411yescharacteristic
68C2×C32×C4—31no—
69C2×C32×C4—31no—
70C2×C32×C4C2×A411yescharacteristic
71C2×C3×Dic3—31no—
72C2×C3×Dic3—31no—
73C2×C3×Dic3C2×A411yescharacteristic
74C3×C4×D3—32no—
75C3×C4×D3—32no—
76C3×Dic6—32no—
77C3×Dic6—32no—
78C23×C9—121no—
79C2×C4×C9—41no—
80C2×C4×C9—81no—
81C2×C4×C9—121no—
82Q8⋊C9—21no—
83Q8⋊C9C4×D311yescharacteristic
84#519—31noSylow
85C2×C3×C9—41no—
86C2×C3×C9—41no—
87C2×C3×C9—41no—
88D3×C9—82no—
89C2×C3×Q8—21no—
90C2×C3×Q8—32no—
91C2×C3×Q8—91no—
92C2×C3×Q8—92no—
93C2×C3×Q8C22×C911yescharacteristic
94C2×C3×Q8C2×C3×D311yescharacteristic
95C22×C3×C4—91no—
96D3×Q8—32no—
97D3×Q8C4×C912yescomplemented
98C2×C4×D3—31no—
99C2×Dic6—31no—
100C3×C4⋊C4—31no—
101C3×C4⋊C4—31no—
102C3×C4⋊C4—61no—
103C3×C4⋊C4—91no—
104C3×C4⋊C4—91no—
105C3×C4⋊C4—91no—
106C3×C22⋊C4—31no—
107C3×C22⋊C4—91no—
108C3×C42—31no—
109C3×C42—31no—
110C3×C42—61no—
111C3×C42—91no—
112C3×C42—92no—
113(C2×D3)⋊C4—31no—
114(C2×D3)⋊C4C22⋊C911yescharacteristic
115C4⋊Dic3—31no—
116Dic3⋊C4—31no—
117Dic3⋊C4—31no—
118C4×Dic3—32no—
119C4×Dic3—31no—
120C22×C32C22×A411yescharacteristic
121C2×C3×D3—21no—
122C2×C3×D3—21no—
123C2×C3×D3C4×A412yes—
124C32×C4—21no—
125C32×C4—31no—
126C32×C4—31no—
127C32×C4—61no—
128C3×Dic3—21no—
129C3×Dic3—32no—
130C3×Dic3—61no—
131C22×C9—41no—
132C22×C9—81no—
133C22×C9—122no—
134C22×C9—241no—
135C22×C9—241no—
136C4×C9—81no—
137C4×C9—82no—
138C4×C9—241no—
139C22×Q8—31no—
140C4×Q8—31no—
141C4×Q8—91no—
142C4×Q8—91no—
143C4×Q8D3×C911yescharacteristic, complemented
144#117—91no—
145#117—91no—
146C3×C9—41noSylow
147C23×C3—31no—
148C22×D3C2×C22⋊C911yescharacteristic
149C3×Q8—21no—
150C3×Q8—31no—
151C3×Q8—31no—
152C3×Q8—61no—
153C3×Q8—92no—
154C3×Q8—92no—
155C3×Q8C2×C4×C911yescharacteristic
156C3×Q8C3×C4×D311yescharacteristic
157C2×C3×C4—21no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—92no—
166C2×C3×C4—92no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×C3×C4C2×C22⋊C911yescharacteristic
170C2×C3×C4D3×A411yescharacteristic
171C2×Dic3—31no—
172C2×Dic3—31no—
173C2×Dic3C2×C22⋊C911yescharacteristic
174C4×D3—32no—
175C4×D3—32no—
176Dic6—32no—
177Dic6—32no—
178C2×C32C2×C4×A411yescharacteristic
179C2×C322−1+4⋊C311yescharacteristic
180C2×C322−1+4⋊C311yescharacteristic
181C3×D3—22no—
182C2×C9—41no—
183C2×C9—41no—
184C2×C9—41no—
185C2×C9—81no—
186C2×C9—81no—
187C2×C9—81no—
188C2×C9—242no—
189C2×Q8—32no—
190C2×Q8—91no—
191C2×Q8—92no—
192C2×Q8C2×D3×C911yescharacteristic
193C22×C4—91no—
194C4⋊C4—31no—
195C4⋊C4—91no—
196C4⋊C4—91no—
197C4⋊C4—91no—
198C22⋊C4—31no—
199C22⋊C4—91no—
200C42—31no—
201C42—91no—
202C42—92no—
203C22×C3—21no—
204C22×C3—32no—
205C22×C3—61no—
206C22×C3—61no—
207C22×C3C22×C22⋊C911yescharacteristic
208C22×C3C2×D3×A411yescharacteristic
209C2×D3—21no—
210C2×D3—21no—
211C2×D3C4×C22⋊C912yes—
212C3×C4—22no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—31no—
219C3×C4—61no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—61no—
223C3×C4—61no—
224C3×C4—92no—
225C3×C4—121no—
226C3×C4—181no—
227Dic3—21no—
228Dic3—32no—
229Dic3—61no—
230C32#519⋊C311yescharacteristic
231C9—41no—
232C9—81no—
233C23—31no—
234Q8—31no—
235Q8—92no—
236Q8—92no—
237Q8C4×D3×C911yescharacteristic
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—92no—
242C2×C4—92no—
243C2×C4—91no—
244C2×C4—91no—
245C2×C4D3×C22⋊C911yescharacteristic
246C2×C3—21no—
247C2×C3—21no—
248C2×C3—21no—
249C2×C3—62no—
250C2×C3C2×C4×C22⋊C911yescharacteristic
251C2×C32−1+4⋊C911yescharacteristic
252C2×C32−1+4⋊C911yescharacteristic
253C2×C3C4×D3×A411yescharacteristic
254C2×C3#1007⋊C311yescharacteristic
255C2×C3#1010⋊C311yescharacteristic
256D3—22no—
257C22—32no—
258C22—61no—
259C22—61no—
260C22C2×D3×C22⋊C911yescharacteristic
261C4—21no—
262C4—31no—
263C4—31no—
264C4—61no—
265C4—61no—
266C4—92no—
267C4—181no—
268C3—21no—
269C3#519⋊C911yescharacteristic, complemented
270C3#5572⋊C311yescharacteristic
271C2—62no—
272C2C4×D3×C22⋊C911yescharacteristic
273C2#1007⋊C911yescharacteristic
274C2#1010⋊C911yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutator, hypocenterC3×Q8C2×C4×C9
FittingC32×C4×Q8C2×C3
socleC22×C32C22×A4
prime radicalC22×C3×D3C2×A4
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. D3×Q8⋊C9C4semidirect
19. C4×Q8⋊C9D3semidirect
25. #5572C9semidirect
56. C3×C4×Q8C2×C9semidirect
60. Q8×Dic3C2×C9semidirect
97. D3×Q8C4×C9semidirect
143. C4×Q8D3×C9semidirect
269. C3#519⋊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
C3×C4×Q8⋊C92
Dic3×Q8⋊C92
C3×#55723
#519⋊C93
C9×(C2×D3)⋊C44