Finite Group Databasebeta

← #5750⋊C9 #13147243

Subgroups — #5750⋊C9 #13147243

Order 1728.

278 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D3×#41⋊C9C211yesmaximal, characteristic
2C3×#131⋊C9C211yesmaximal, characteristic, complemented
3C3⋊(#131⋊C9)C211yesmaximal, characteristic, complemented
4C3×#5750C311yesmaximal, characteristic
5#568⋊C9—31nomaximal
6C3×#41⋊C9C2211yescharacteristic
7D3×Q8⋊C9C411yescharacteristic
8C3⋊(#41⋊C9)C411yescharacteristic
9C9×ΓL(1,𝔽25)—41nomaximal
10C3×D3×#41C2×C311yescharacteristic
11C32×#131C2×C311yescharacteristic
12C3×#948C2×C311yescharacteristic
13C3×#907—31no—
14C3×D3×M16—31no—
15C3×#901—31no—
16C2×C3×ΓL(1,𝔽25)—31no—
17C2×#41⋊C9—31no—
18#131⋊C9—21no—
19#131⋊C9—31no—
20#131⋊C9D311yescharacteristic, complemented
21C4×D3×C9—41no—
22C3×Q8⋊C9C2×C411yescharacteristic
23C3×C8×C9—41no—
24C9×C3⋊C8—41no—
25C3×#568—31no—
26#5750C911yescharacteristic, complemented
27C32×#41C22×C311yescharacteristic
28C3×#239C3×C411yescharacteristic
29C3×D3×Q8C3×C411yescharacteristic
30C3×#237—31no—
31C3×D3×D4—31no—
32C3×#235—31no—
33C2×C3×C4×D3—31no—
34C32×M16—31no—
35C2×C32×C8—31no—
36C3×#208—31no—
37C2×C3×C3⋊C8—31no—
38C3×ΓL(1,𝔽25)—31no—
39C3×ΓL(1,𝔽25)—31no—
40C3×ΓL(1,𝔽25)—31no—
41C3×ΓL(1,𝔽25)A411yescharacteristic
42C3×D3×C8—32no—
43#41⋊C9—21no—
44#41⋊C9—31no—
45#41⋊C9C2×D311yescharacteristic
46C2×Q8⋊C9—31no—
47C9×M16—121no—
48C2×D3×C9—41no—
49C3×C4×C9—41no—
50C9×Dic3—41no—
51C2×C3×#41—31no—
52D3×#41C2×C911yescharacteristic
53C3×#131—21no—
54C3×#131—31no—
55C3×#131—91no—
56C3×#131—91no—
57C3×#131C2×C911yescharacteristic, complemented
58C3×#131C3×D311yescharacteristic
59C2×C3×M16—91no—
60C2×C3×M16—91no—
61#948C2×C911yescharacteristic, complemented
62#907—31no—
63D3×M16—31no—
64#901—31no—
65C2×ΓL(1,𝔽25)—31no—
66C22×C3×D3—31no—
67C32×Q8C2×C3×C411yescharacteristic
68C32×D4—31no—
69C2×C32×C4—31no—
70C3×#67—32no—
71C2×C3×Dic3—31no—
72C3×D12—31no—
73C3×C4×D3—31no—
74C3×C4×D3—31no—
75C3×C4×D3—31no—
76C3×C4×D3C2×A411yescharacteristic
77C3×Dic6—31no—
78C32×C8—31no—
79C32×C8C2×A411yescharacteristic
80C3×C3⋊C8—31no—
81C3×C3⋊C8C2×A411yescharacteristic
82C2×C4×C9—121no—
83Q8⋊C9—21no—
84Q8⋊C9C4×D311yescharacteristic
85C8×C9—41no—
86C8×C9—81no—
87C8×C9—121no—
88#568—31noSylow
89C2×C3×C9—41no—
90D3×C9—81no—
91C3×#41—21no—
92C3×#41—31no—
93C3×#41—91no—
94C3×#41—91no—
95C3×#41C22×C911yescharacteristic
96C3×#41C2×C3×D311yescharacteristic
97C2×C3×Q8—31no—
98C2×C3×D4—91no—
99C22×C3×C4—91no—
100#239C4×C911yescharacteristic
101D3×Q8C4×C911yescharacteristic
102#237—31no—
103D3×D4—31no—
104#235—31no—
105C2×C4×D3—31no—
106C3×M16—31no—
107C3×M16—31no—
108C3×M16—31no—
109C3×M16—61no—
110C3×M16—91no—
111C3×M16—91no—
112C3×M16—91no—
113C3×M16—91no—
114C2×C3×C8—31no—
115C2×C3×C8—31no—
116C2×C3×C8—61no—
117C2×C3×C8—91no—
118C2×C3×C8—92no—
119#208—31no—
120C2×C3⋊C8—31no—
121ΓL(1,𝔽25)—31no—
122ΓL(1,𝔽25)—31no—
123ΓL(1,𝔽25)—31no—
124ΓL(1,𝔽25)C22⋊C911yescharacteristic
125D3×C8—32no—
126C22×C32—31no—
127C2×C3×D3—31no—
128C2×C3×D3—121no—
129C2×C3×D3C4×A411yescharacteristic
130C32×C4—31no—
131C32×C4C22×A411yescharacteristic
132C3×Dic3—31no—
133C3×Dic3C4×A411yescharacteristic
134C22×C9—121no—
135C4×C9—41no—
136C4×C9—81no—
137C4×C9—121no—
138C2×#41—31no—
139#131—31no—
140#131—91no—
141#131—91no—
142#131D3×C911yescharacteristic, complemented
143C2×M16—91no—
144C2×M16—91no—
145C3×C9—41noSylow
146C23×C3—91no—
147C22×D3—31no—
148C3×Q8—21no—
149C3×Q8—91no—
150C3×Q8C2×C4×C911yescharacteristic
151C3×Q8C3×C4×D311yescharacteristic
152C3×D4—31no—
153C3×D4—31no—
154C3×D4—61no—
155C3×D4—91no—
156C3×D4—92no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—61no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165#67—32no—
166C2×Dic3—31no—
167D12—31no—
168C4×D3—31no—
169C4×D3—31no—
170C4×D3—31no—
171C4×D3C2×C22⋊C911yescharacteristic
172Dic6—31no—
173C3×C8—21no—
174C3×C8—31no—
175C3×C8—31no—
176C3×C8—31no—
177C3×C8—61no—
178C3×C8—91no—
179C3×C8C2×C22⋊C911yescharacteristic
180C3×C8D3×A411yescharacteristic
181C3⋊C8—31no—
182C3⋊C8C2×C22⋊C911yescharacteristic
183C2×C32—61no—
184C2×C32C2×C4×A411yescharacteristic
185C3×D3—21no—
186C3×D3—61no—
187C2×C9—41no—
188C2×C9—81no—
189C2×C9—241no—
190#41—31no—
191#41—91no—
192#41—91no—
193#41C2×D3×C911yescharacteristic
194C2×Q8—31no—
195C2×D4—91no—
196C22×C4—91no—
197M16—31no—
198M16—31no—
199M16—91no—
200M16—91no—
201M16—91no—
202M16—91no—
203C2×C8—31no—
204C2×C8—91no—
205C2×C8—92no—
206C22×C3—31no—
207C22×C3—31no—
208C22×C3—31no—
209C22×C3—61no—
210C22×C3—91no—
211C22×C3—361no—
212C2×D3—31no—
213C2×D3—121no—
214C2×D3C4×C22⋊C911yescharacteristic
215C3×C4—21no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—31no—
219C3×C4—61no—
220C3×C4—91no—
221C3×C4C22×C22⋊C911yescharacteristic
222C3×C4C2×D3×A411yescharacteristic
223Dic3—31no—
224Dic3C4×C22⋊C911yescharacteristic
225C32#568⋊C311yescharacteristic
226C9—41no—
227C9—81no—
228C23—91no—
229Q8—91no—
230Q8C4×D3×C911yescharacteristic
231D4—31no—
232D4—91no—
233D4—92no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C8—31no—
241C8—31no—
242C8—91no—
243C8D3×C22⋊C911yescharacteristic
244C2×C3—21no—
245C2×C3—61no—
246C2×C3—61no—
247C2×C3—61no—
248C2×C3—121no—
249C2×C3—181no—
250C2×C3C2×C4×C22⋊C911yescharacteristic
251C2×C3C4×D3×A411yescharacteristic
252D3—21no—
253D3—61no—
254C22—31no—
255C22—31no—
256C22—91no—
257C22—361no—
258C4—31no—
259C4—31no—
260C4—91no—
261C4C2×D3×C22⋊C911yescharacteristic
262C3—21no—
263C3#568⋊C911yescharacteristic, complemented
264C3#5750⋊C311yescharacteristic
265C2—61no—
266C2—61no—
267C2—181no—
268C2C4×D3×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC3×C4C2×D3×A4
commutator, hypocenterC3×Q8C2×C4×C9
FittingC32×#131C2×C3
socleC2×C32C2×C4×A4
prime radicalC3×D3×#41C2×C3
cosocleC32×#41C22×C3
hypercenterC3×C8D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#131⋊C9C2semidirect
3. C3⋊(#131⋊C9)C2semidirect
20. #131⋊C9D3semidirect
26. #5750C9semidirect
57. C3×#131C2×C9semidirect
61. #948C2×C9semidirect
142. #131D3×C9semidirect
263. C3#568⋊C9semidirect

Maximal subgroups

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

subgroupindex
D3×#41⋊C92
C3×#131⋊C92
C3⋊(#131⋊C9)2
C3×#57503
#568⋊C93
C9×ΓL(1,𝔽25)4