Finite Group Databasebeta

← #10591014

Subgroups — #10591014

Order 576.

358 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 58 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#64307C211yesmaximal, characteristic, complemented
2C3×#908C211yesmaximal, characteristic, complemented
3#63914C211yesmaximal, characteristic, complemented
4#63830C211yesmaximal, characteristic, complemented
5#63652C211yesmaximal, characteristic, complemented
6C3×#823C211yesmaximal, characteristic, complemented
7#63568C211yesmaximal, characteristic, complemented
8#5198—31nomaximal
9#4747—31nomaximal
10#3828C2211yescharacteristic
11C3×#235C2211yescharacteristic
12C2×C3×D12C2211yescharacteristic
13C2×C3×D12C2211yescharacteristic, complemented
14D3×#67—22no—
15C2×#625—22no—
16#3802—21no—
17D3×D12—21no—
18#3796—21no—
19#3795—21no—
20C32×M16C2211yescharacteristic, complemented
21C4×C3⋊Dic3C2211yescharacteristic, complemented
22C3×#208C2211yescharacteristic, complemented
23C3×D24—21no—
24C3×#204—21no—
25#3716—21no—
26#3714—21no—
27#1009—31no—
28#1004—31no—
29C3×AGL(1,ℤ/8)—31no—
30#940—31no—
31#940—31no—
32#932—31no—
33#908—31no—
34#908D311yescharacteristic, complemented
35C3×#100—31no—
36#837—31no—
37#825—31no—
38#825—31no—
39#825—31no—
40#825—61no—
41#823D311yescharacteristic, complemented
42C22×C3×D3—21no—
43C22×C3×D3D412yes—
44C2×D32—22no—
45C2×C32×C4C2311yescharacteristic
46#638—21no—
47C2×C3⋊Dic3—21no—
48C4⋊(C3⋊D3)D411yescharacteristic, complemented
49C4×C3⋊D3—21no—
50#634D411yescharacteristic, complemented
51C3×#67—21no—
52C3×D12—21no—
53C3×D12—21no—
54C3×D12—21no—
55C3×D12—21no—
56C3×D12D411yescharacteristic, complemented
57C3×C4×D3—21no—
58C3×Dic6D411yescharacteristic, complemented
59#626—22no—
60#625—22no—
61#625—44no—
62D3×Dic3—22no—
63C32×C8—21no—
64C3×C3⋊C8—21no—
65#453—91noSylow
66C3×#41—31no—
67C2×C3×D4—31no—
68C2×C3×D4—31no—
69C2×#67—62no—
70C2×#67—62no—
71C2×#67—62no—
72#239—61no—
73#237—61no—
74#237—61no—
75D3×D4—61no—
76D3×D4—61no—
77D3×D4—62no—
78D3×D4—61no—
79#235—31no—
80#235—31no—
81#235—61no—
82#235—61no—
83#235C2×D311yescharacteristic
84C2×D12—61no—
85C2×D12C2×D311yescharacteristic
86C2×D12C2×D311yescharacteristic, complemented
87C3×SD16—61no—
88C3×D8—61no—
89C3×M16—21no—
90C3×M16—31no—
91C3×M16C2×D311yescharacteristic, complemented
92C3×M16C2×D311yescharacteristic, complemented
93#214—61no—
94#213—61no—
95C4×Dic3—31no—
96C4×Dic3—31no—
97C4×Dic3—61no—
98#208C2×D311yescharacteristic, complemented
99D24—21no—
100D24—61no—
101#204—21no—
102#204—61no—
103C22×C32C2×D411yescharacteristic
104C2×C3⋊D3—21no—
105C2×C3×D3—22no—
106C2×C3×D3—21no—
107C2×C3×D3—42no—
108C2×C3×D3—42no—
109C2×C3×D3—41no—
110D32—82no—
111C32×C4C2×D411yescharacteristic
112C32×C4C2×D411yescharacteristic
113C3⋊Dic3—22no—
114C3⋊Dic3—21no—
115C3×Dic3—21no—
116H(5,𝔽2)—91no—
117AGL(1,ℤ/8)—31no—
118AGL(1,ℤ/8)—91no—
119#127—91no—
120#104—91no—
121#104—91no—
122#100—31no—
123C23×C3—32no—
124C23×C3—61no—
125C22×D3—21no—
126C22×D3—62no—
127C22×D3—62no—
128C22×D3D1212yes—
129C3×Q8—31no—
130C3×D4—31no—
131C3×D4—61no—
132C3×D4—61no—
133C3×D4—61no—
134C3×D4—61no—
135C3×D4—61no—
136C2×C3×C4—21no—
137C2×C3×C4—61no—
138C2×C3×C4C22×D311yescharacteristic
139C2×C3×C4C22×D311yescharacteristic
140#67—21no—
141#67—62no—
142#67—62no—
143#67—62no—
144#67—61no—
145#67—61no—
146#67—121no—
147#67—124no—
148#67—124no—
149C2×Dic3—62no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152C2×Dic3—121no—
153D12—21no—
154D12—21no—
155D12—21no—
156D12—21no—
157D12—31no—
158D12—31no—
159D12—61no—
160D12—62no—
161D12#6711yescharacteristic, complemented
162C4×D3—21no—
163C4×D3—62no—
164C4×D3—61no—
165C4×D3—61no—
166C4×D3—121no—
167Dic6—31no—
168Dic6—31no—
169Dic6—61no—
170Dic6#6711yescharacteristic, complemented
171C3×C8—21no—
172C3×C8—21no—
173C3×C8—41no—
174C3×C8—61no—
175C3⋊C8—21no—
176C2×C32—21no—
177C2×C32#12011yescharacteristic
178C3⋊D3—41no—
179C3×D3—41no—
180C3×D3—42no—
181C3×D3—81no—
182#41—31no—
183#41—91no—
184#41—181no—
185#41—181no—
186C2×D4—31no—
187C2×D4—31no—
188C2×D4—181no—
189C2×D4—181no—
190C2×D4—182no—
191C2×D4—182no—
192SD16—61no—
193SD16—181no—
194D8—61no—
195D8—181no—
196M16—31no—
197M16D3211yescharacteristic, complemented
198C42—91no—
199C22×C3—21no—
200C22×C3—61no—
201C22×C3—62no—
202C22×C3—122no—
203C22×C3—122no—
204C22×C3—121no—
205C22×C3C2×D1211yescharacteristic
206C22×C3D3×D411yescharacteristic
207C2×D3—22no—
208C2×D3—21no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—41no—
212C2×D3—61no—
213C2×D3—61no—
214C2×D3—121no—
215C2×D3—242no—
216C2×D3—242no—
217C3×C4—21no—
218C3×C4—21no—
219C3×C4—61no—
220C3×C4D3×D411yescharacteristic
221C3×C4D3×D411yescharacteristic
222C3×C4D3×D411yescharacteristic
223C3×C4C2×#6711yescharacteristic
224Dic3—21no—
225Dic3—62no—
226Dic3—62no—
227Dic3—61no—
228Dic3—61no—
229Dic3—121no—
230Dic3—122no—
231C32#45311yescharacteristic, complemented, Sylow
232C23—32no—
233C23—61no—
234C23—182no—
235Q8—31no—
236Q8—91no—
237D4—31no—
238D4—61no—
239D4—61no—
240D4—61no—
241D4—61no—
242D4—61no—
243D4—91no—
244D4—182no—
245D4—182no—
246D4—181no—
247D4—364no—
248C2×C4—61no—
249C2×C4—182no—
250C2×C4—181no—
251C2×C4—181no—
252C2×C4C2×D3211yescharacteristic
253C8—21no—
254C8—61no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—41no—
259C2×C3—122no—
260C2×C3—121no—
261C2×C3—241no—
262C2×C3#88211yescharacteristic
263C2×C3#93711yescharacteristic
264D3—41no—
265D3—42no—
266D3—81no—
267D3—121no—
268D3—121no—
269D3—241no—
270C22—61no—
271C22—62no—
272C22—122no—
273C22—122no—
274C22—121no—
275C22—181no—
276C22—722no—
277C22D3×D1211yescharacteristic
278C4—61no—
279C4—182no—
280C4—181no—
281C4#380211yescharacteristic
282C4D3×#6711yescharacteristic
283C3—21no—
284C3#474711yescharacteristic, complemented
285C3#519811yescharacteristic, complemented
286C2—21no—
287C2—122no—
288C2—121no—
289C2—241no—
290C2—361no—
291C2#6392211yescharacteristic
subgroupG/H
centerC2#63922
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×M16C22
socleC2×C32#120
hypercenterM16D32
hypocenterC32#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #64307C2semidirect
2. C3×#908C2semidirect
3. #63914C2semidirect
4. #63830C2semidirect
5. #63652C2semidirect
6. C3×#823C2semidirect
7. #63568C2semidirect
13. C2×C3×D12C22semidirect
20. C32×M16C22semidirect
21. C4×C3⋊Dic3C22semidirect
22. C3×#208C22semidirect
34. #908D3semidirect
41. #823D3semidirect
48. C4⋊(C3⋊D3)D4semidirect
50. #634D4semidirect
56. C3×D12D4semidirect
58. C3×Dic6D4semidirect
86. C2×D12C2×D3semidirect
91. C3×M16C2×D3semidirect
92. C3×M16C2×D3semidirect
98. #208C2×D3semidirect
161. D12#67semidirect
170. Dic6#67semidirect
197. M16D32semidirect
231. C32#453semidirect
284. C3#4747semidirect
285. C3#5198semidirect

Maximal subgroups

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

subgroupindex
#643072
C3×#9082
#639142
#638302
#636522
C3×#8232
#635682
#51983
#47473