Finite Group Databasebeta

← #11785470

Subgroups — #11785470

Order 1152.

589 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic
2C2×#63577C212yesmaximal, complemented
3#10591069C212yesmaximal
4#10589418C211yesmaximal, characteristic, complemented
5C3×#4552C211yesmaximal, characteristic, complemented
6#72000—31nomaximal
7#68940—31nomaximal
8C22×C3×C4×D3C2211yescharacteristic
9C22×C32×C8C2211yescharacteristic
10C2×#3748C2212yes—
11C2×C3×#208C2212yes—
12C22×C3×C3⋊C8C2211yescharacteristic
13C2×C3×ΓL(1,𝔽25)—24no—
14C2×C3×ΓL(1,𝔽25)C414yes—
15#63577—24no—
16(C2×D3)⋊(C3⋊C8)—24no—
17C22×C3×M16—31no—
18C22×ΓL(1,𝔽25)D311yescharacteristic
19#5207—32no—
20#5126—32no—
21C2×#822—32no—
22C2×#819—32no—
23C3×#341—31no—
24#4552—31no—
25#4552—31no—
26#4552—61no—
27#4552D311yescharacteristic, complemented
28C23×C3×D3Q811yescharacteristic
29C22×C32×C4C2311yescharacteristic
30C22×C3×Dic3Q811yescharacteristic
31C2×C3×C4×D3—22no—
32C2×C3×C4×D3—22no—
33C2×C3×C4×D3C2×C412yes—
34C2×C3×C4×D3D412yes—
35C2×C32×C8—22no—
36C2×C32×C8C2×C412yes—
37#3748—22no—
38#3748—22no—
39C2×C3⋊(C3⋊C8)—22no—
40C3×#208—22no—
41C3×#208—22no—
42C2×C3×C3⋊C8—22no—
43C2×C3×C3⋊C8C2×C412yes—
44C2×C3×C3⋊C8D414yes—
45C3×ΓL(1,𝔽25)—28no—
46#1811—91noSylow
47C23×C3×C4—31no—
48C22×C4×D3C2×D311yescharacteristic
49C2×C3×M16—34no—
50C2×C3×M16—32no—
51C2×C3×M16—64no—
52C22×C3×C8—21no—
53C22×C3×C8—31no—
54C22×C3×C8C2×D311yescharacteristic
55C22×C3×C8C2×D311yescharacteristic
56C2×#208—32no—
57C2×#208—32no—
58C2×#208—62no—
59C2×#208C2×D312yes—
60C22×C3⋊C8C2×D311yescharacteristic
61C2×ΓL(1,𝔽25)—24no—
62C2×ΓL(1,𝔽25)Dic314yes—
63#830—64no—
64#822—64no—
65(C2×D3)⋊C8—64no—
66#819—64no—
67C23×C32C2×Q811yescharacteristic
68C22×C3×D3—22no—
69C22×C3×D3—42no—
70C22×C3×D3C4⋊C412yes—
71C2×C32×C4C22×C411yescharacteristic
72C2×C32×C4C2×D411yescharacteristic
73C2×C32×C4C2×D412yes—
74C2×C32×C4#4112yes—
75C2×C3×Dic3—22no—
76C2×C3×Dic3C4⋊C412yes—
77C3×C4×D3—22no—
78C3×C4×D3—22no—
79C3×C4×D3—24no—
80C32×C8—22no—
81C3⋊(C3⋊C8)—42no—
82C3×C3⋊C8—42no—
83C3×C3⋊C8C22⋊C414yes—
84C22×M16—31no—
85C2×#108—92no—
86#407—92no—
87#341—31no—
88#341—91no—
89C24×C3—31no—
90C23×D3Dic611yescharacteristic
91C22×C3×C4—21no—
92C22×C3×C4—31no—
93C22×C3×C4—32no—
94C22×C3×C4—32no—
95C22×C3×C4—62no—
96C22×C3×C4—62no—
97C22×C3×C4C22×D311yescharacteristic
98C22×C3×C4C22×D311yescharacteristic
99C22×Dic3Dic611yescharacteristic
100C2×C4×D3—22no—
101C2×C4×D3—22no—
102C2×C4×D3D1212yes—
103C2×C4×D3C2×Dic312yes—
104C3×M16—68no—
105C3×M16—62no—
106C3×M16—62no—
107C2×C3×C8—22no—
108C2×C3×C8—22no—
109C2×C3×C8—22no—
110C2×C3×C8—32no—
111C2×C3×C8—34no—
112C2×C3×C8—42no—
113C2×C3×C8—62no—
114C2×C3×C8C4×D312yes—
115C2×C3×C8C2×Dic312yes—
116#208—22no—
117#208—22no—
118#208—64no—
119#208—62no—
120#208—62no—
121#208—62no—
122#208—62no—
123#208—122no—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—62no—
126C2×C3⋊C8—62no—
127C2×C3⋊C8—122no—
128C2×C3⋊C8C2×Dic312yes—
129C2×C3⋊C8#6714yes—
130ΓL(1,𝔽25)—28no—
131C22×C32—22no—
132C22×C32C2×C4⋊C411yescharacteristic
133C22×C32C22⋊Q812yes—
134C2×C3×D3—22no—
135C2×C3×D3—24no—
136C2×C3×D3—44no—
137C32×C4C2×C22⋊C411yescharacteristic
138C32×C4#11711yescharacteristic
139C32×C4#12112yes—
140C3×Dic3—22no—
141C23×C4—31no—
142C2×M16—34no—
143C2×M16—32no—
144C2×M16—64no—
145C2×M16—92no—
146C22×C8—31no—
147C22×C8D3211yescharacteristic
148#108—184no—
149C22⋊C8—184no—
150C23×C3—21no—
151C23×C3—32no—
152C23×C3—62no—
153C23×C3—122no—
154C23×C3C2×Dic611yescharacteristic
155C23×C3D3×Q811yescharacteristic
156C22×D3—22no—
157C22×D3—42no—
158C22×D3C4⋊Dic312yes—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—22no—
163C2×C3×C4—32no—
164C2×C3×C4—62no—
165C2×C3×C4—62no—
166C2×C3×C4—62no—
167C2×C3×C4—64no—
168C2×C3×C4C2×C4×D311yescharacteristic
169C2×C3×C4C2×D1211yescharacteristic
170C2×C3×C4#23512yes—
171C2×C3×C4#23512yes—
172C2×C3×C4D3×D411yescharacteristic
173C2×C3×C4D3×D412yes—
174C2×C3×C4C22×Dic311yescharacteristic
175C2×C3×C4C2×#6712yes—
176C2×Dic3—22no—
177C2×Dic3C4⋊Dic312yes—
178C4×D3—22no—
179C4×D3—24no—
180C4×D3—22no—
181C3×C8—22no—
182C3×C8—22no—
183C3×C8—34no—
184C3×C8—42no—
185C3×C8—122no—
186C3⋊C8—42no—
187C3⋊C8—124no—
188C3⋊C8—122no—
189C3⋊C8—122no—
190C3⋊C8#21714yes—
191C2×C32—22no—
192C2×C32#38011yescharacteristic
193C2×C32#43012yes—
194C3×D3—42no—
195C24—31no—
196C22×C4—31no—
197C22×C4—32no—
198C22×C4—32no—
199C22×C4—62no—
200C22×C4—62no—
201C22×C4C2×D3211yescharacteristic
202M16—68no—
203M16—62no—
204M16—62no—
205M16—182no—
206M16—182no—
207C2×C8—22no—
208C2×C8—32no—
209C2×C8—34no—
210C2×C8—62no—
211C2×C8—182no—
212C2×C8D3×Dic312yes—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3—21no—
217C22×C3—42no—
218C22×C3—64no—
219C22×C3—62no—
220C22×C3—124no—
221C22×C3D3×C4⋊C411yescharacteristic
222C22×C3#89612yes—
223C22×C3#92412yes—
224C22×C3C2×C4⋊Dic311yescharacteristic
225C2×D3—24no—
226C2×D3—22no—
227C2×D3—44no—
228C3×C4—21no—
229C3×C4—22no—
230C3×C4—21no—
231C3×C4—62no—
232C3×C4#87211yescharacteristic
233C3×C4D3×C22⋊C411yescharacteristic
234C3×C4#88412yes—
235C3×C4#92611yescharacteristic
236C3×C4#93012yes—
237C3×C4C2×#21711yescharacteristic
238Dic3—22no—
239C32#181111yescharacteristic, complemented, Sylow
240C23—32no—
241C23—62no—
242C23—122no—
243C23D3×Dic611yescharacteristic
244C2×C4—32no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—64no—
248C2×C4—62no—
249C2×C4#379812yes—
250C2×C4D3×D1211yescharacteristic
251C2×C4C2×D3×Dic311yescharacteristic
252C2×C4D3×#6712yes—
253C8—22no—
254C8—34no—
255C8—122no—
256C8—362no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—22no—
261C2×C3—42no—
262C2×C3—122no—
263C2×C3#466711yescharacteristic
264C2×C3#489312yes—
265C2×C3#512512yes—
266C2×C3#521211yescharacteristic
267D3—42no—
268C22—22no—
269C22—62no—
270C22—64no—
271C22—124no—
272C22#6385112yes—
273C22D3×C4⋊Dic311yescharacteristic
274C4—62no—
275C4#6385511yescharacteristic
276C4#6390612yes—
277C4D3×#21711yescharacteristic
278C3—21no—
279C3#6894011yescharacteristic, complemented
280C3#7200011yescharacteristic, complemented
281C2—22no—
282C2—122no—
283C2#1059091212yes—
284C2#1059129511yescharacteristic
subgroupG/H
centerC2×C4D3×D12
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×C8C22
socleC22×C32C2×C4⋊C4
prime radicalC23×C3×D3Q8
cosocleC22×C32×C4C23
hypercenterC22×C8D32
hypocenterC32#1811
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C2×#63577C2semidirect
4. #10589418C2semidirect
5. C3×#4552C2semidirect
27. #4552D3semidirect
239. C32#1811semidirect
279. C3#68940semidirect
280. C3#72000semidirect

Maximal subgroups

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

subgroupindex
C22×C3×ΓL(1,𝔽25)2
C2×#635772
#105910692
#105894182
C3×#45522
#720003
#689403