Finite Group Databasebeta

← #11800031

Subgroups — #11800031

Order 1152.

269 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 68 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#10590912C211yesmaximal, characteristic
2#10590854C211yesmaximal, characteristic
3#10589379C211yesmaximal, characteristic, complemented
4C3×#4517C211yesmaximal, characteristic, complemented
5C3×#4514C211yesmaximal, characteristic
6#10589137C211yesmaximal, characteristic, complemented
7#10589130C211yesmaximal, characteristic, complemented
8#69267—31nomaximal
9#69117—31nomaximal
10C2×C3×Dic12C2211yescharacteristic
11C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
12#63856—21no—
13D3×#208—21no—
14C32×M32C2211yescharacteristic
15#63646C2211yescharacteristic
16#63644C2211yescharacteristic
17C3×#819C2211yescharacteristic
18C3×#812C2211yescharacteristic
19#63577—21no—
20#63566—21no—
21#5189—31no—
22#5137—31no—
23#5125—31no—
24#4893—31no—
25C3×#362—31no—
26C3×#349—31no—
27#4565—31no—
28#4517—31no—
29#4517D311yescharacteristic, complemented
30#4514D311yescharacteristic
31#4513—31no—
32#4513—31no—
33#4513—31no—
34#4513—61no—
35#4494—31no—
36C2×C3×C4×D3D411yescharacteristic
37C2×C3×Dic6—21no—
38C2×C32×C8C2311yescharacteristic
39C4⋊(C3⋊Dic3)—21no—
40#3748—21no—
41C3×#208—21no—
42C2×C3×C3⋊C8D411yescharacteristic
43C3×Dic12—21no—
44C3×Dic12—21no—
45C3×ΓL(1,𝔽25)—21no—
46C3×ΓL(1,𝔽25)—21no—
47#3724—41no—
48(C2×D3)⋊Dic3—41no—
49#3711—41no—
50D3×C3⋊C8—41no—
51C32×C16—21no—
52C3×C3⋊C16—21no—
53#2239—91noSylow
54C2×C3×Q16—31no—
55C2×C3×M16—31no—
56#946—61no—
57#924—61no—
58C2×#208—61no—
59D3×M16—61no—
60C2×Dic12C2×D311yescharacteristic
61C2×ΓL(1,𝔽25)C2×D311yescharacteristic
62C3×M32—21no—
63C3×M32—31no—
64C3×M32C2×D311yescharacteristic
65C3×M32C2×D311yescharacteristic
66C3×#108—31no—
67#835—61no—
68#822—61no—
69#819—31no—
70#819—31no—
71#819—61no—
72#819—61no—
73#819C2×D311yescharacteristic
74#817—31no—
75#817—31no—
76#817—61no—
77#816—61no—
78#812C2×D311yescharacteristic
79C22×C3×D3SD1611yescharacteristic
80C2×C32×C4C2×D411yescharacteristic
81C2×C3⋊Dic3—41no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3SD1611yescharacteristic
84C3×C4×D3—21no—
85C3×C4×D3—21no—
86C3×Dic6—41no—
87C3×Dic6—41no—
88C32×C8#4111yescharacteristic
89C32×C8#4111yescharacteristic
90C3⋊(C3⋊C8)—41no—
91C3×C3⋊C8—21no—
92C3×C3⋊C8—41no—
93#470—91no—
94#430—91no—
95#365—91no—
96#365—91no—
97#362—31no—
98#362—91no—
99#349—31no—
100C2×C3×Q8—61no—
101C22×C3×C4—31no—
102C2×C4×D3D1211yescharacteristic
103C2×Dic6—21no—
104C3×Q16—61no—
105C3×Q16—61no—
106C3×M16—61no—
107C3×M16—61no—
108C3×M16—61no—
109C2×C3×C8—21no—
110C2×C3×C8—31no—
111C2×C3×C8C22×D311yescharacteristic
112C2×C3×C8C22×D311yescharacteristic
113#217—121no—
114(C2×D3)⋊C4—121no—
115C4⋊Dic3—61no—
116C4⋊Dic3—61no—
117C4⋊Dic3—121no—
118Dic3⋊C4—121no—
119Dic3⋊C4—121no—
120#208—21no—
121#208—61no—
122#208—61no—
123#208—121no—
124#208—121no—
125C2×C3⋊C8—121no—
126C2×C3⋊C8D1211yescharacteristic
127Dic12—21no—
128Dic12—21no—
129ΓL(1,𝔽25)—21no—
130ΓL(1,𝔽25)—21no—
131ΓL(1,𝔽25)—121no—
132D3×C8—121no—
133C3×C16—22no—
134C3×C16—21no—
135C3×C16—21no—
136C3×C16—61no—
137C3⋊C16—21no—
138C22×C32C2×SD1611yescharacteristic
139C2×C3×D3—21no—
140C2×C3×D3—41no—
141C32×C4#12311yescharacteristic
142C32×C4AGL(1,ℤ/8)11yescharacteristic
143C3⋊Dic3—81no—
144C3×Dic3—21no—
145C3×Dic3—81no—
146C2×Q16—31no—
147C2×M16—31no—
148C2×M16—181no—
149C22⋊Q8—181no—
150M32—31no—
151M32D3211yescharacteristic
152#108—31no—
153#108—91no—
154#108—181no—
155#106—91no—
156#103—181no—
157C23×C3—31no—
158C22×D3#20411yescharacteristic
159C3×Q8—121no—
160C3×Q8—121no—
161C2×C3×C4—21no—
162C2×C3×C4—31no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—121no—
166C2×C3×C4C2×D1211yescharacteristic
167C2×C3×C4D3×D411yescharacteristic
168C2×Dic3—41no—
169C2×Dic3—121no—
170C2×Dic3—121no—
171C2×Dic3—241no—
172C2×Dic3#20411yescharacteristic
173C4×D3—21no—
174C4×D3—21no—
175Dic6—41no—
176Dic6—41no—
177C3×C8—21no—
178C3×C8—21no—
179C3×C8—61no—
180C3×C8—121no—
181C3×C8#23511yescharacteristic
182C3×C8#23711yescharacteristic
183C3×C8#23711yescharacteristic
184C3×C8#23711yescharacteristic
185C3⋊C8—21no—
186C3⋊C8—41no—
187C3⋊C8—121no—
188C3⋊C8—121no—
189C3⋊C8—241no—
190C2×C32—21no—
191C2×C32#48111yescharacteristic
192C3×D3—41no—
193C2×Q8—61no—
194C22×C4—31no—
195Q16—61no—
196Q16—61no—
197M16—61no—
198M16—61no—
199M16—61no—
200M16—181no—
201M16—361no—
202C2×C8—31no—
203C2×C8—361no—
204C2×C8C2×D3211yescharacteristic
205C4⋊C4—181no—
206C4⋊C4—361no—
207C22⋊C4—361no—
208C16—21no—
209C16—61no—
210C22×C3—21no—
211C22×C3—61no—
212C22×C3—121no—
213C22×C3C2×#20411yescharacteristic
214C22×C3D3×SD1611yescharacteristic
215C2×D3—21no—
216C2×D3—41no—
217C3×C4—21no—
218C3×C4—21no—
219C3×C4—61no—
220C3×C4—241no—
221C3×C4#88311yescharacteristic
222C3×C4#88611yescharacteristic
223C3×C4#90811yescharacteristic
224C3×C4#91111yescharacteristic
225Dic3—21no—
226Dic3—81no—
227Dic3—241no—
228Dic3—241no—
229Dic3—481no—
230C32#223911yescharacteristic, complemented, Sylow
231C23—31no—
232Q8—121no—
233Q8—121no—
234C2×C4—31no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—121no—
238C2×C4—361no—
239C2×C4D3×D1211yescharacteristic
240C8—61no—
241C8—121no—
242C8—361no—
243C8#379511yescharacteristic
244C8#380511yescharacteristic
245C2×C3—21no—
246C2×C3—21no—
247C2×C3—21no—
248C2×C3—41no—
249C2×C3—121no—
250C2×C3#473511yescharacteristic
251C2×C3#477711yescharacteristic
252D3—41no—
253C22—61no—
254C22—121no—
255C22D3×#20411yescharacteristic
256C4—61no—
257C4—241no—
258C4—721no—
259C4#6379511yescharacteristic
260C4#6389111yescharacteristic
261C3—21no—
262C3#6911711yescharacteristic, complemented
263C3#6926711yescharacteristic, complemented
264C2—21no—
265C2—121no—
266C2#1059067611yescharacteristic
subgroupG/H
centerC2#10590676
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC32×M32C22
socleC2×C32#481
prime radicalC22×C3×D3SD16
hypercenterM32D32
hypocenterC32#2239
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

Complements

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

NH ≅ G/Ntype
3. #10589379C2semidirect
4. C3×#4517C2semidirect
6. #10589137C2semidirect
7. #10589130C2semidirect
29. #4517D3semidirect
230. C32#2239semidirect
262. C3#69117semidirect
263. C3#69267semidirect

Maximal subgroups

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

subgroupindex
#105909122
#105908542
#105893792
C3×#45172
C3×#45142
#105891372
#105891302
#692673
#691173