Finite Group Databasebeta

← C3×SD16×C22⋊C9 #13152226

Subgroups — C3×SD16×C22⋊C9 #13152226

Order 1728.

458 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C3×Q8×C22⋊C9C211yesmaximal, characteristic, complemented
2C3×D4×C22⋊C9C211yesmaximal, characteristic
3C3×C8×C22⋊C9C211yesmaximal, characteristic, complemented
4C22×C32×SD16C311yesmaximal, characteristic
5SD16×C22⋊C9C313yesmaximal, direct product factor
6C22×C3×C22⋊C9—21no—
7C3×C4×C22⋊C9—21no—
8C3×C4×C22⋊C9C2211yescharacteristic
9C3×C9×SD16—41nomaximal
10C22×C32×Q8C2×C311yescharacteristic
11C22×C32×D4C2×C311yescharacteristic
12C2×C32×SD16—31no—
13C2×C32×SD16—31no—
14C2×C32×SD16—31no—
15C2×C32×SD16—31no—
16C22×C32×C8C2×C311yescharacteristic
17Q8×C22⋊C9C2×C313yescomplemented
18D4×C22⋊C9C2×C313yes—
19C8×C22⋊C9C2×C313yescomplemented
20C2×C3×C22⋊C9—41no—
21C2×C3×C22⋊C9D411yescharacteristic
22C3×Q8×C9—41no—
23C3×D4×C9—41no—
24C3×C8×C9—41no—
25C22×C3×SD16C913yescomplemented
26C22×C3×SD16C3211yescharacteristic
27C24×C32—21no—
28C2×C32×Q8—31no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C2×C32×D4—31no—
32C2×C32×D4—31no—
33C2×C32×D4—61no—
34C22×C32×C4—21no—
35C22×C32×C4C22×C311yescharacteristic
36C22×C22⋊C9—23no—
37C32×SD16—32no—
38C32×SD16—31no—
39C32×SD16—31no—
40C32×SD16—31no—
41C32×SD16A411yescharacteristic
42C2×C32×C8—31no—
43C2×C32×C8—31no—
44C4×C22⋊C9—23no—
45C4×C22⋊C9C22×C313yes—
46C9×SD16—43no—
47C22×C3×C9—81no—
48C3×C22⋊C9SD1611yescharacteristic, direct product factor
49C3×C4×C9—41no—
50C3×C4×C9—81no—
51C22×C3×Q8C2×C913yescomplemented
52C22×C3×Q8C2×C3211yescharacteristic
53C22×C3×D4C2×C913yes—
54C22×C3×D4C2×C3211yescharacteristic
55C2×C3×SD16—33no—
56C2×C3×SD16—31no—
57C2×C3×SD16—33no—
58C2×C3×SD16—33no—
59C2×C3×SD16—33no—
60C2×C3×SD16—31no—
61C2×C3×SD16—31no—
62C2×C3×SD16—31no—
63C22×C3×C8C2×C913yescomplemented
64C22×C3×C8C2×C3211yescharacteristic
65C23×C32—41no—
66C23×C32—61no—
67C23×C32—61no—
68C23×C32—121no—
69C23×C32C3×D411yescharacteristic
70C32×Q8—31no—
71C32×Q8—61no—
72C32×Q8—61no—
73C32×Q8C2×A411yescharacteristic
74C32×D4—31no—
75C32×D4—61no—
76C32×D4—61no—
77C32×D4C2×A411yescharacteristic
78C2×C32×C4—31no—
79C2×C32×C4—31no—
80C2×C32×C4—61no—
81C2×C32×C4—61no—
82C2×C22⋊C9—43no—
83C2×C22⋊C9C3×D413yes—
84C32×C8—31no—
85C32×C8C2×A411yescharacteristic
86Q8×C9—43no—
87D4×C9—43no—
88C8×C9—43no—
89C22×SD16C3×C911yescharacteristic, complemented, Sylow
90C2×C3×C9—41no—
91C2×C3×C9—161no—
92C24×C3—23no—
93C24×C3—21no—
94C2×C3×Q8—33no—
95C2×C3×Q8—31no—
96C2×C3×Q8—33no—
97C2×C3×Q8—31no—
98C2×C3×Q8—63no—
99C2×C3×Q8—61no—
100C2×C3×D4—33no—
101C2×C3×D4—31no—
102C2×C3×D4—33no—
103C2×C3×D4—31no—
104C2×C3×D4—63no—
105C2×C3×D4—61no—
106C22×C3×C4—23no—
107C22×C3×C4—21no—
108C22×C3×C4C22×C913yes—
109C22×C3×C4C22×C3211yescharacteristic
110C3×SD16—32no—
111C3×SD16—36no—
112C3×SD16—33no—
113C3×SD16—31no—
114C3×SD16—33no—
115C3×SD16—31no—
116C3×SD16—33no—
117C3×SD16—31no—
118C3×SD16C22⋊C913yesdirect product factor
119C3×SD16C3×A411yescharacteristic
120C2×C3×C8—33no—
121C2×C3×C8—31no—
122C2×C3×C8—33no—
123C2×C3×C8—31no—
124C22×C32—21no—
125C22×C32—31no—
126C22×C32—31no—
127C22×C32—61no—
128C22×C32—121no—
129C22×C32—121no—
130C22×C32—121no—
131C22×C32—121no—
132C22×C32C3×SD1611yescharacteristic
133C32×C4—21no—
134C32×C4—31no—
135C32×C4—61no—
136C32×C4C22×A411yescharacteristic
137C22×C9—83no—
138C22⋊C9C3×SD1613yesdirect product factor
139C4×C9—43no—
140C4×C9—83no—
141C22×Q8C2×C3×C911yescharacteristic, complemented
142C22×D4C2×C3×C911yescharacteristic
143C2×SD16—31no—
144C2×SD16—31no—
145C2×SD16—31no—
146C2×SD16—31no—
147C22×C8C2×C3×C911yescharacteristic, complemented
148C3×C9—41noSylow
149C23×C3—43no—
150C23×C3—41no—
151C23×C3—61no—
152C23×C3—63no—
153C23×C3—61no—
154C23×C3—63no—
155C23×C3—123no—
156C23×C3—121no—
157C23×C3D4×C913yes—
158C23×C3C32×D411yescharacteristic
159C3×Q8—31no—
160C3×Q8—33no—
161C3×Q8—61no—
162C3×Q8—63no—
163C3×Q8—61no—
164C3×Q8—63no—
165C3×Q8C2×C22⋊C913yescomplemented
166C3×Q8C2×C3×A411yescharacteristic
167C3×D4—31no—
168C3×D4—33no—
169C3×D4—61no—
170C3×D4—63no—
171C3×D4—61no—
172C3×D4—63no—
173C3×D4C2×C22⋊C913yes—
174C3×D4C2×C3×A411yescharacteristic
175C2×C3×C4—31no—
176C2×C3×C4—33no—
177C2×C3×C4—33no—
178C2×C3×C4—31no—
179C2×C3×C4—61no—
180C2×C3×C4—63no—
181C2×C3×C4—61no—
182C2×C3×C4—63no—
183C3×C8—31no—
184C3×C8—33no—
185C3×C8C2×C22⋊C913yescomplemented
186C3×C8C2×C3×A411yescharacteristic
187C2×C32—31no—
188C2×C32—31no—
189C2×C32—41no—
190C2×C32—121no—
191C2×C32D4×A411yescharacteristic
192C2×C9—43no—
193C2×C9—163no—
194C24—21no—
195C2×Q8—31no—
196C2×Q8—31no—
197C2×Q8—61no—
198C2×D4—31no—
199C2×D4—31no—
200C2×D4—61no—
201C22×C4—21no—
202C22×C4C22×C3×C911yescharacteristic
203SD16—32no—
204SD16—31no—
205SD16—31no—
206SD16—31no—
207SD16C3×C22⋊C911yescharacteristic, direct product factor
208C2×C8—31no—
209C2×C8—31no—
210C22×C3—21no—
211C22×C3—23no—
212C22×C3—33no—
213C22×C3—31no—
214C22×C3—31no—
215C22×C3—33no—
216C22×C3—61no—
217C22×C3—63no—
218C22×C3—123no—
219C22×C3—121no—
220C22×C3—123no—
221C22×C3—121no—
222C22×C3—123no—
223C22×C3—121no—
224C22×C3—123no—
225C22×C3—121no—
226C22×C3C9×SD1613yescomplemented
227C22×C3C32×SD1611yescharacteristic
228C3×C4—21no—
229C3×C4—23no—
230C3×C4—31no—
231C3×C4—33no—
232C3×C4—61no—
233C3×C4—63no—
234C3×C4C22×C22⋊C913yes—
235C3×C4C22×C3×A411yescharacteristic
236C32A4×SD1611yescharacteristic
237C9—43no—
238C23—41no—
239C23—61no—
240C23—61no—
241C23—121no—
242C23C3×D4×C911yescharacteristic
243Q8—31no—
244Q8—61no—
245Q8—61no—
246Q8C2×C3×C22⋊C911yescharacteristic, complemented
247D4—31no—
248D4—61no—
249D4—61no—
250D4C2×C3×C22⋊C911yescharacteristic
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—61no—
254C2×C4—61no—
255C8—31no—
256C8C2×C3×C22⋊C911yescharacteristic, complemented
257C2×C3—33no—
258C2×C3—33no—
259C2×C3—31no—
260C2×C3—31no—
261C2×C3—43no—
262C2×C3—41no—
263C2×C3—123no—
264C2×C3—121no—
265C2×C3D4×C22⋊C913yes—
266C2×C3C3×D4×A411yescharacteristic
267C22—21no—
268C22—31no—
269C22—31no—
270C22—61no—
271C22—121no—
272C22—121no—
273C22—121no—
274C22—121no—
275C22C3×C9×SD1611yescharacteristic, complemented
276C4—21no—
277C4—31no—
278C4—61no—
279C4C22×C3×C22⋊C911yescharacteristic
280C3SD16×C22⋊C913yesdirect product factor
281C3C3×A4×SD1611yescharacteristic
282C2—31no—
283C2—31no—
284C2—41no—
285C2—121no—
286C2C3×D4×C22⋊C911yescharacteristic
subgroupG/H
centerC2×C32D4×A4
commutatorC22×C4C22×C3×C9
FrattiniC3×C4C22×C3×A4
FittingC22×C32×SD16C3
socleC23×C32C3×D4
prime radicalC22×C32×D4C2×C3
cosocleC22×C3×C4C22×C32
hypercenterC32×SD16A4
hypocenterC22C3×C9×SD16
LCSG ▷ C22×C4 ▷ C23 ▷ C22
UCSC1 ◁ C2×C32 ◁ C32×C4 ◁ C32×SD16

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C22⋊C9C2semidirect
3. C3×C8×C22⋊C9C2semidirect
5. SD16×C22⋊C9C3direct (N × H)
17. Q8×C22⋊C9C2×C3semidirect
19. C8×C22⋊C9C2×C3semidirect
25. C22×C3×SD16C9semidirect
48. C3×C22⋊C9SD16direct (N × H)
51. C22×C3×Q8C2×C9semidirect
63. C22×C3×C8C2×C9semidirect
89. C22×SD16C3×C9semidirect
118. C3×SD16C22⋊C9direct (N × H)
138. C22⋊C9C3×SD16direct (N × H)
141. C22×Q8C2×C3×C9semidirect
147. C22×C8C2×C3×C9semidirect
165. C3×Q8C2×C22⋊C9semidirect
185. C3×C8C2×C22⋊C9semidirect
207. SD16C3×C22⋊C9direct (N × H)
226. C22×C3C9×SD16semidirect
246. Q8C2×C3×C22⋊C9semidirect
256. C8C2×C3×C22⋊C9semidirect
275. C22C3×C9×SD16semidirect
280. C3SD16×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×C22⋊C92
C3×D4×C22⋊C92
C3×C8×C22⋊C92
C22×C32×SD163
SD16×C22⋊C93
C3×C9×SD164