Finite Group Databasebeta

← #3027⋊C9 #11919593

Subgroups — #3027⋊C9 #11919593

Order 1152.

318 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 31 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
1#10593125C211yesmaximal, characteristic, complemented
2B(4,𝔽2)⋊C9C212yesmaximal, complemented
3C3×#3027C311yesmaximal, characteristic
4#63703C2211yescharacteristic
5D4×C22⋊C9—41nomaximal
6C3×#585C2×C311yescharacteristic
7C3×#535—31no—
8C3×B(4,𝔽2)—32no—
9C3×B(4,𝔽2)C2×C312yes—
10C3×#410—31no—
11C22×C22⋊C9—42no—
12#41⋊C9—81no—
13C4×C22⋊C9—41no—
14C4×C22⋊C9—81no—
15#3027C911yescharacteristic, complemented, Sylow
16C3×2−1+4—61no—
17C3×H(5,𝔽2)—31no—
18C3×H(5,𝔽2)—61no—
19C3×H(5,𝔽2)C22×C311yescharacteristic
20C2×C3×#41—31no—
21C2×C3×#41—61no—
22C2×C3×#41—61no—
23C22×C3×D4A411yescharacteristic
24C3×#127—61no—
25C3×#124—61no—
26C3×#121—61no—
27C3×#121—61no—
28C3×#120—32no—
29C3×#120—32no—
30C3×#117—31no—
31C3×#99—32no—
32C3×#99—32no—
33C2×C22⋊C9—41no—
34C2×C22⋊C9—81no—
35C2×C22⋊C9—82no—
36D4×C9—161no—
37Q8⋊C9—81no—
38#585C2×C911yescharacteristic, complemented
39#535—31no—
40B(4,𝔽2)—32no—
41B(4,𝔽2)C2×C912yescomplemented
42#410—31no—
43C24×C3C2×A412yes—
44C3×#41—21no—
45C3×#41—62no—
46C3×#41—62no—
47C3×#41—61no—
48C3×#41—62no—
49C3×#41—61no—
50C3×#41—62no—
51C3×#41—62no—
52C3×#41—61no—
53C2×C3×Q8—31no—
54C2×C3×Q8—61no—
55C2×C3×Q8—61no—
56C2×C3×D4—31no—
57C2×C3×D4—31no—
58C2×C3×D4—31no—
59C2×C3×D4—61no—
60C2×C3×D4—61no—
61C2×C3×D4—61no—
62C2×C3×D4—62no—
63C2×C3×D4—61no—
64C2×C3×D4—61no—
65C2×C3×D4—61no—
66C2×C3×D4—62no—
67C2×C3×D4—62no—
68C22×C3×C4—21no—
69C22×C3×C4—31no—
70C22×C3×C4—31no—
71C22×C3×C4—61no—
72C22×C3×C4C2×A411yescharacteristic
73C3×C4⋊C4—61no—
74C3×C22⋊C4—32no—
75C3×C22⋊C4—62no—
76C3×C22⋊C4—62no—
77C3×C42—61no—
78C22×C9—162no—
79C22⋊C9—41no—
80C22⋊C9—81no—
81C4×C9—161no—
822−1+4—61no—
83H(5,𝔽2)—31no—
84H(5,𝔽2)—61no—
85H(5,𝔽2)C22×C911yescharacteristic
86C2×#41—31no—
87C2×#41—61no—
88C2×#41—61no—
89C22×D4C22⋊C911yescharacteristic, complemented
90#127—61no—
91#124—61no—
92#121—61no—
93#121—61no—
94#120—32no—
95#120—32no—
96#117—31no—
97#99—32no—
98#99—32no—
99C23×C3—21no—
100C23×C3—31no—
101C23×C3—31no—
102C23×C3—62no—
103C23×C3—61no—
104C23×C3—82no—
105C23×C3C22×A411yescharacteristic
106C3×Q8—21no—
107C3×Q8—62no—
108C3×Q8—61no—
109C3×D4—41no—
110C3×D4—62no—
111C3×D4—62no—
112C3×D4—62no—
113C3×D4—61no—
114C3×D4—61no—
115C3×D4—62no—
116C3×D4—121no—
117C3×D4—242no—
118C3×D4—242no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—61no—
124C2×C3×C4—61no—
125C2×C3×C4—62no—
126C2×C3×C4—61no—
127C2×C3×C4—61no—
128C2×C3×C4—61no—
129C2×C3×C4—61no—
130C2×C3×C4—62no—
131C2×C9—161no—
132C2×C9—322no—
133C24C2×C22⋊C912yes—
134#41—21no—
135#41—62no—
136#41—62no—
137#41—61no—
138#41—62no—
139#41—61no—
140#41—62no—
141#41—62no—
142#41—61no—
143C2×Q8—31no—
144C2×Q8—61no—
145C2×Q8—61no—
146C2×D4—31no—
147C2×D4—31no—
148C2×D4—31no—
149C2×D4—61no—
150C2×D4—61no—
151C2×D4—61no—
152C2×D4—62no—
153C2×D4—61no—
154C2×D4—61no—
155C2×D4—61no—
156C2×D4—62no—
157C2×D4—62no—
158C22×C4—21no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—61no—
162C22×C4C2×C22⋊C911yescharacteristic
163C4⋊C4—61no—
164C22⋊C4—32no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C42—61no—
168C22×C3—31no—
169C22×C3—41no—
170C22×C3—42no—
171C22×C3—61no—
172C22×C3—61no—
173C22×C3—81no—
174C22×C3—122no—
175C22×C3—122no—
176C22×C3—121no—
177C22×C3—121no—
178C22×C3—241no—
179C3×C4—31no—
180C3×C4—61no—
181C3×C4—61no—
182C3×C4—122no—
183C3×C4#120⋊C311yescharacteristic
184C9—161noSylow
185C23—21no—
186C23—31no—
187C23—31no—
188C23—62no—
189C23—61no—
190C23—82no—
191C23C22×C22⋊C911yescharacteristic
192Q8—21no—
193Q8—62no—
194Q8—61no—
195D4—41no—
196D4—62no—
197D4—62no—
198D4—62no—
199D4—61no—
200D4—61no—
201D4—62no—
202D4—121no—
203D4—242no—
204D4—242no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C3—61no—
218C2×C3—82no—
219C2×C3—121no—
220C2×C3—121no—
221C2×C3C2×#120⋊C311yescharacteristic
222C22—31no—
223C22—41no—
224C22—42no—
225C22—61no—
226C22—61no—
227C22—81no—
228C22—122no—
229C22—122no—
230C22—121no—
231C22—121no—
232C22—241no—
233C4—31no—
234C4—61no—
235C4—61no—
236C4—122no—
237C4#120⋊C911yescharacteristic
238C3#3027⋊C311yescharacteristic
239C2—61no—
240C2—82no—
241C2—121no—
242C2—121no—
243C2C2×#120⋊C911yescharacteristic
subgroupG/H
center, socleC2×C3C2×#120⋊C3
commutator, hypocenterH(5,𝔽2)C22×C9
FrattiniC23×C3C22×A4
Fitting, prime radicalC3×#3027C3
cosocleC3×H(5,𝔽2)C22×C3
hypercenterC3×C4#120⋊C3
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. #10593125C2semidirect
2. B(4,𝔽2)⋊C9C2semidirect
15. #3027C9semidirect
38. #585C2×C9semidirect
41. B(4,𝔽2)C2×C9semidirect
89. C22×D4C22⋊C9semidirect

Maximal subgroups

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

subgroupindex
#105931252
B(4,𝔽2)⋊C92
C3×#30273
D4×C22⋊C94