Finite Group Databasebeta

← #11800020

Subgroups — #11800020

Order 1152.

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

normal subgroups only

This group has 275 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
1C3×#5189C211yesmaximal, characteristic
2#10590912C211yesmaximal, characteristic
3#10589431C211yesmaximal, characteristic, complemented
4C3×#4565C211yesmaximal, characteristic, complemented
5#10589138C211yesmaximal, characteristic, complemented
6#10589136C211yesmaximal, characteristic, complemented
7#10589109C211yesmaximal, characteristic
8#71090—31nomaximal
9#69267—31nomaximal
10C2×C32×Q16C2211yescharacteristic
11C3×#946—21no—
12C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
13D3×#208—21no—
14#63646C2211yescharacteristic
15#63639C2211yescharacteristic
16C3×#835—21no—
17C3×#819C2211yescharacteristic
18C3×#817C2211yescharacteristic
19C3×#812C2211yescharacteristic
20#63577—21no—
21C3×#470—31no—
22#5189D311yescharacteristic
23#5125—31no—
24#4893—31no—
25C3×#362—31no—
26#4565—31no—
27#4565—31no—
28#4565—61no—
29#4565D311yescharacteristic, complemented
30#4553—31no—
31#4514—31no—
32#4513—31no—
33#4494—31no—
34#4487—31no—
35#4487—31no—
36C2×C32×Q8—21no—
37C2×C3×C4×D3D411yescharacteristic
38C32×Q16—21no—
39C32×Q16—21no—
40C2×C32×C8C2311yescharacteristic
41#3748—21no—
42C3×(C2×D3)⋊C4—41no—
43C3×C4⋊Dic3—21no—
44C3×Dic3⋊C4—41no—
45C3×#208—21no—
46C2×C3×C3⋊C8D411yescharacteristic
47C3×ΓL(1,𝔽25)—21no—
48C3×ΓL(1,𝔽25)—21no—
49#3711—41no—
50D3×C3⋊C8—41no—
51C3⋊(C3⋊C16)—21no—
52C3×C3⋊C16—21no—
53#2239—91noSylow
54C2×C3×Q16—21no—
55C2×C3×Q16C2×D311yescharacteristic
56C2×C3×Q16C2×D311yescharacteristic
57C2×C3×M16—31no—
58C3×C22⋊Q8—61no—
59#946—21no—
60C2×#208—61no—
61D3×M16—61no—
62C2×ΓL(1,𝔽25)C2×D311yescharacteristic
63C3×M32—31no—
64C3×#108—31no—
65C3×#106—31no—
66C3×#103—61no—
67#835—21no—
68#822—61no—
69#819—31no—
70#819—31no—
71#819—61no—
72#819—61no—
73#819C2×D311yescharacteristic
74#817C2×D311yescharacteristic
75#812—31no—
76#812—31no—
77#812—61no—
78#812C2×D311yescharacteristic
79C22×C3×D3SD1611yescharacteristic
80C32×Q8—41no—
81C32×Q8—41no—
82C2×C32×C4—41no—
83C2×C32×C4C2×D411yescharacteristic
84C2×C3×Dic3—41no—
85C2×C3×Dic3SD1611yescharacteristic
86C3×C4×D3—21no—
87C3×C4×D3—21no—
88C32×C8#4111yescharacteristic
89C32×C8#4111yescharacteristic
90C3⋊(C3⋊C8)—41no—
91C3×C3⋊C8—21no—
92C3×C3⋊C8—41no—
93#470—31no—
94#430—91no—
95#365—91no—
96#365—91no—
97#362—31no—
98#362—91no—
99#349—91no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—41no—
103C22×C3×C4—31no—
104C2×C4×D3#6711yescharacteristic
105C3×Q16—22no—
106C3×Q16—21no—
107C3×Q16—21no—
108C3×Q16—21no—
109C3×Q16—21no—
110C3×Q16—41no—
111C3×M16—61no—
112C3×M16—61no—
113C3×M16—61no—
114C2×C3×C8—21no—
115C2×C3×C8—31no—
116C2×C3×C8C22×D311yescharacteristic
117C2×C3×C8C22×D311yescharacteristic
118C3×C4⋊C4—61no—
119C3×C4⋊C4—121no—
120C3×C22⋊C4—121no—
121(C2×D3)⋊C4—41no—
122C4⋊Dic3—21no—
123Dic3⋊C4—41no—
124#208—21no—
125#208—61no—
126#208—61no—
127#208—121no—
128#208—121no—
129C2×C3⋊C8—121no—
130C2×C3⋊C8#6711yescharacteristic
131ΓL(1,𝔽25)—21no—
132ΓL(1,𝔽25)—21no—
133ΓL(1,𝔽25)—121no—
134D3×C8—121no—
135C3×C16—61no—
136C3⋊C16—21no—
137C3⋊C16—61no—
138C3⋊C16—61no—
139C3⋊C16—121no—
140C22×C32C2×SD1611yescharacteristic
141C2×C3×D3—21no—
142C2×C3×D3—41no—
143C32×C4—81no—
144C32×C4#12311yescharacteristic
145C32×C4AGL(1,ℤ/8)11yescharacteristic
146C3×Dic3—21no—
147C3×Dic3—81no—
148C2×Q16D3211yescharacteristic
149C2×M16—31no—
150C2×M16—181no—
151C22⋊Q8—61no—
152M32—31no—
153M32—91no—
154#108—31no—
155#108—91no—
156#108—181no—
157#106—31no—
158#103—61no—
159C23×C3—31no—
160C22×D3#21411yescharacteristic
161C3×Q8—42no—
162C3×Q8—41no—
163C3×Q8—41no—
164C3×Q8—41no—
165C3×Q8—41no—
166C3×Q8—81no—
167C2×C3×C4—21no—
168C2×C3×C4—31no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—61no—
172C2×C3×C4—61no—
173C2×C3×C4—81no—
174C2×C3×C4—121no—
175C2×C3×C4D3×D411yescharacteristic
176C2×C3×C4C2×#6711yescharacteristic
177C2×Dic3—41no—
178C2×Dic3#21411yescharacteristic
179C4×D3—21no—
180C4×D3—21no—
181C3×C8—21no—
182C3×C8—21no—
183C3×C8—61no—
184C3×C8—121no—
185C3×C8#23511yescharacteristic
186C3×C8#23711yescharacteristic
187C3×C8#23711yescharacteristic
188C3×C8#23711yescharacteristic
189C3⋊C8—21no—
190C3⋊C8—41no—
191C3⋊C8—121no—
192C3⋊C8—121no—
193C3⋊C8—241no—
194C2×C32—21no—
195C2×C32#48111yescharacteristic
196C3×D3—41no—
197C2×Q8—21no—
198C22×C4—31no—
199Q16—21no—
200Q16—21no—
201M16—61no—
202M16—61no—
203M16—61no—
204M16—181no—
205M16—361no—
206C2×C8—31no—
207C2×C8—361no—
208C2×C8C2×D3211yescharacteristic
209C4⋊C4—61no—
210C4⋊C4—121no—
211C22⋊C4—121no—
212C16—61no—
213C16—181no—
214C22×C3—21no—
215C22×C3—61no—
216C22×C3—121no—
217C22×C3D3×SD1611yescharacteristic
218C22×C3C2×#21411yescharacteristic
219C2×D3—21no—
220C2×D3—41no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—61no—
224C3×C4—82no—
225C3×C4—81no—
226C3×C4—81no—
227C3×C4—241no—
228C3×C4#88311yescharacteristic
229C3×C4#91111yescharacteristic
230C3×C4#93511yescharacteristic
231C3×C4#94911yescharacteristic
232Dic3—21no—
233Dic3—81no—
234C32#223911yescharacteristic, complemented, Sylow
235C23—31no—
236Q8—41no—
237Q8—41no—
238C2×C4—31no—
239C2×C4—41no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—121no—
243C2×C4D3×#6711yescharacteristic
244C8—61no—
245C8—121no—
246C8—361no—
247C8#380411yescharacteristic
248C8#380511yescharacteristic
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—41no—
253C2×C3—121no—
254C2×C3#477711yescharacteristic
255C2×C3#503411yescharacteristic
256D3—41no—
257C22—61no—
258C22—121no—
259C22D3×#21411yescharacteristic
260C4—61no—
261C4—81no—
262C4—241no—
263C4#6392511yescharacteristic
264C4#6396911yescharacteristic
265C3—21no—
266C3#6926711yescharacteristic, complemented
267C3#7109011yescharacteristic, complemented
268C2—21no—
269C2—121no—
270C2#1059141511yescharacteristic
subgroupG/H
centerC2#10591415
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×Q16C22
socleC2×C32#481
prime radicalC22×C3×D3SD16
hypercenterC2×Q16D32
hypocenterC32#2239
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
3. #10589431C2semidirect
4. C3×#4565C2semidirect
5. #10589138C2semidirect
6. #10589136C2semidirect
29. #4565D3semidirect
234. C32#2239semidirect
266. C3#69267semidirect
267. C3#71090semidirect

Maximal subgroups

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

subgroupindex
C3×#51892
#105909122
#105894312
C3×#45652
#105891382
#105891362
#105891092
#710903
#692673