Finite Group Databasebeta

← #5836⋊C9 #13147296

Subgroups — #5836⋊C9 #13147296

Order 1728.

302 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 49 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×#63700C211yesmaximal, characteristic, complemented
2#1010⋊C9C211yesmaximal, characteristic, complemented
3C3⋊(#131⋊C9)C211yesmaximal, characteristic, complemented
4C3×#5836C311yesmaximal, characteristic
5#577⋊C9—31nomaximal
6C3×#41⋊C9—21no—
7C3×#41⋊C9C2211yescharacteristic
8D3×Q8⋊C9—21no—
9C9×#215—41nomaximal
10C32×H(5,𝔽2)C2×C311yescharacteristic
11C3×#1010C2×C311yescharacteristic
12C3×#951—31no—
13C3×#950—31no—
14C3×#948C2×C311yescharacteristic
15C2×C3×#214—31no—
16C3×#932—31no—
172−1+4⋊C9—31no—
18#63700—21no—
19#63700D311yescharacteristic, complemented
20#131⋊C9—31no—
21C3×Q8×C9—41no—
22C9×D12—41no—
23C3×Q8⋊C9D411yescharacteristic
24C9×C3⋊C8—41no—
25C3×#577—31no—
26#5836C911yescharacteristic, complemented
27C32×#41—21no—
28C32×#41—31no—
29C32×#41C22×C311yescharacteristic
30C2×C32×D4—31no—
31C2×C32×D4—61no—
32C3×D3×Q8—21no—
33C3×#237—61no—
34C3×#235—31no—
35C2×C3×Dic6—31no—
36C3×#216—31no—
37C3×#215A411yescharacteristic
38C3×#214—32no—
39C3×#214—31no—
40C3×#213—31no—
41C3×#208—31no—
42C2×C3×C3⋊C8—31no—
43#41⋊C9—22no—
44#41⋊C9—21no—
45#41⋊C9—21no—
46#41⋊C9C2×D311yescharacteristic
47C2×Q8⋊C9—61no—
48C9×SD16—121no—
49C2×D3×C9—81no—
50C3×C4×C9—41no—
51C3×C4×C9—81no—
52C3×2−1+4—31no—
53C3×H(5,𝔽2)—21no—
54C3×H(5,𝔽2)C2×C911yescharacteristic, complemented
55C3×H(5,𝔽2)C3×D311yescharacteristic
56#1010C2×C911yescharacteristic
57C3×#137—91no—
58C3×AGL(1,ℤ/8)—91no—
59C3×#135—91no—
60C2×C3×SD16—91no—
61C3×#131—31no—
62#951—31no—
63#950—31no—
64#948C2×C911yescharacteristic, complemented
65C2×#214—31no—
66#932—31no—
67C23×C32—61no—
68C32×Q8C3×D411yescharacteristic
69C32×Q8C2×A411yescharacteristic
70C32×D4—31no—
71C32×D4—31no—
72C32×D4—61no—
73C32×D4—61no—
74C2×C32×C4—31no—
75C2×C32×C4—61no—
76C3×#67—61no—
77C2×C3×Dic3—61no—
78C3×D12C2×A411yescharacteristic
79C3×C4×D3—61no—
80C3×Dic6—31no—
81C3×Dic6—61no—
82C3×C3⋊C8—31no—
83C3×C3⋊C8C2×A411yescharacteristic
84Q8×C9—41no—
85Q8×C9—81no—
86D4×C9—121no—
87Q8⋊C9—21no—
88Q8⋊C9#6711yescharacteristic
89C8×C9—121no—
90#577—31noSylow
91C2×C3×C9—41no—
92D3×C9—161no—
93C3×#41—22no—
94C3×#41—21no—
95C3×#41—21no—
96C3×#41—21no—
97C3×#41—31no—
98C3×#41—31no—
99C3×#41—61no—
100C3×#41—91no—
101C3×#41—181no—
102C3×#41C22×C911yescharacteristic
103C3×#41C2×C3×D311yescharacteristic
104C2×C3×Q8—61no—
105C2×C3×Q8—91no—
106C2×C3×D4—31no—
107C2×C3×D4—31no—
108C2×C3×D4—61no—
109C2×C3×D4—61no—
110C2×C3×D4—61no—
111C2×C3×D4—62no—
112D3×Q8—21no—
113#237—61no—
114#235—31no—
115C2×Dic6—31no—
116C3×Q16—91no—
117C3×SD16—31no—
118C3×SD16—91no—
119C3×SD16—92no—
120C3×D8—91no—
121C3×M16—91no—
122C2×C3×C8—91no—
123#216—31no—
124#215C22⋊C911yescharacteristic
125#214—32no—
126#214—31no—
127#213—31no—
128#208—31no—
129C2×C3⋊C8—31no—
130C22×C32—31no—
131C22×C32—61no—
132C22×C32—241no—
133C2×C3×D3—21no—
134C32×C4—21no—
135C32×C4—31no—
136C32×C4C22×A411yescharacteristic
137C3×Dic3—61no—
138C22×C9—241no—
139C4×C9—41no—
140C4×C9—81no—
141C4×C9—82no—
142C4×C9—81no—
1432−1+4—31no—
144H(5,𝔽2)D3×C911yescharacteristic, complemented
145#137—91no—
146AGL(1,ℤ/8)—91no—
147#135—91no—
148C2×SD16—91no—
149#131—31no—
150C3×C9—41noSylow
151C23×C3—61no—
152C23×C3—61no—
153C23×C3—62no—
154C3×Q8—21no—
155C3×Q8—21no—
156C3×Q8—91no—
157C3×Q8—181no—
158C3×Q8D4×C911yescharacteristic
159C3×Q8C2×C22⋊C911yescharacteristic
160C3×Q8C3×#6711yescharacteristic
161C3×Q8D3×A411yescharacteristic
162C3×D4—31no—
163C3×D4—31no—
164C3×D4—31no—
165C3×D4—31no—
166C3×D4—31no—
167C3×D4—61no—
168C3×D4—61no—
169C3×D4—61no—
170C3×D4—61no—
171C3×D4—62no—
172C3×D4—61no—
173C3×D4—61no—
174C3×D4—62no—
175C3×D4—181no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—62no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184#67—61no—
185C2×Dic3—61no—
186D12C2×C22⋊C911yescharacteristic
187C4×D3—61no—
188Dic6—31no—
189Dic6—61no—
190C3×C8—31no—
191C3×C8—91no—
192C3⋊C8—31no—
193C3⋊C8C2×C22⋊C911yescharacteristic
194C2×C32—61no—
195C2×C32—121no—
196C2×C32D4×A411yescharacteristic
197C3×D3—41no—
198C2×C9—41no—
199C2×C9—81no—
200C2×C9—481no—
201#41—21no—
202#41—31no—
203#41—91no—
204#41—181no—
205#41C2×D3×C911yescharacteristic
206C2×Q8—61no—
207C2×Q8—91no—
208C2×D4—31no—
209C2×D4—61no—
210Q16—91no—
211SD16—31no—
212SD16—91no—
213SD16—92no—
214D8—91no—
215M16—91no—
216C2×C8—91no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—61no—
220C22×C3—62no—
221C22×C3—61no—
222C22×C3—61no—
223C22×C3—61no—
224C22×C3—241no—
225C22×C3—241no—
226C22×C3—242no—
227C2×D3—21no—
228C3×C4—22no—
229C3×C4—21no—
230C3×C4—21no—
231C3×C4—21no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—61no—
235C3×C4—181no—
236C3×C4C22×C22⋊C911yescharacteristic
237C3×C4C2×D3×A411yescharacteristic
238Dic3—61no—
239C32#577⋊C311yescharacteristic
240C9—41no—
241C9—81no—
242C23—61no—
243Q8—91no—
244Q8—181no—
245Q8C9×#6711yescharacteristic
246Q8D3×C22⋊C911yescharacteristic
247D4—31no—
248D4—31no—
249D4—31no—
250D4—61no—
251D4—61no—
252D4—181no—
253C2×C4—31no—
254C2×C4—61no—
255C2×C4—181no—
256C2×C4—181no—
257C8—31no—
258C8—91no—
259C2×C3—21no—
260C2×C3—61no—
261C2×C3—61no—
262C2×C3—121no—
263C2×C3—122no—
264C2×C3—121no—
265C2×C3—121no—
266C2×C3—121no—
267C2×C3D4×C22⋊C911yescharacteristic
268C2×C3A4×#6711yescharacteristic
269D3—41no—
270C22—31no—
271C22—61no—
272C22—61no—
273C22—241no—
274C4—21no—
275C4—31no—
276C4—181no—
277C4C2×D3×C22⋊C911yescharacteristic
278C3—21no—
279C3#577⋊C911yescharacteristic, complemented
280C3#5836⋊C311yescharacteristic
281C2—61no—
282C2—121no—
283C2—121no—
284C2#67×C22⋊C911yescharacteristic
subgroupG/H
centerC2×C3A4×#67
commutatorC3×#41C22×C9
FrattiniC3×C4C2×D3×A4
FittingC32×H(5,𝔽2)C2×C3
socleC2×C32D4×A4
prime radicalC3×#5836C3
cosocleC32×#41C22×C3
hypercenterC3×Q8D3×A4
hypocenterC3×Q8D4×C9
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#63700C2semidirect
2. #1010⋊C9C2semidirect
3. C3⋊(#131⋊C9)C2semidirect
19. #63700D3semidirect
26. #5836C9semidirect
54. C3×H(5,𝔽2)C2×C9semidirect
64. #948C2×C9semidirect
144. H(5,𝔽2)D3×C9semidirect
279. C3#577⋊C9semidirect

Maximal subgroups

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

subgroupindex
C3×#637002
#1010⋊C92
C3⋊(#131⋊C9)2
C3×#58363
#577⋊C93
C9×#2154