Finite Group Databasebeta

← #261×H(3,𝔽4) #16807815

Subgroups — #261×H(3,𝔽4) #16807815

Order 3328.

2664 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

normal subgroups only

This group has 2664 conjugacy classes of subgroups — too many to draw. Only its 776 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#13131379C219yesmaximal, complemented
2#13131375C216yesmaximal, complemented
3#124×#261C219yesmaximal, complemented
4#120×#261C216yesmaximal, complemented
5D13×H(3,𝔽4)C211yesmaximal, characteristic
6C24×#261C2212yescomplemented
7C2×Q8×#261C2213yescomplemented
8C2×D4×#261C2219yes—
9C13×H(3,𝔽4)C411yescharacteristic, complemented
10#11719388C419yescomplemented
11D13×#124C2219yes—
12#11719299C416yescomplemented
13D13×#120C2216yes—
14#11719171C2216yescomplemented
15(C2×#261)⋊Q8C2219yescomplemented
16#11719151C2219yes—
17#11719150C22118yes—
18#11719071C22136yescomplemented
19#11719070C22118yescomplemented
20C22⋊C4×#261C22118yescomplemented
21#11719060C2219yescomplemented
22C42×#261C2213yescomplemented
23C24×D13C2312yes—
24C23×#261—412no—
25C23×#261C2316yes—
26C23×#261C2316yes—
27C2×Q8×D13C2313yes—
28C2×D4×D13C2319yes—
29C2×C22⋊#261C23118yes—
30C2×C22⋊#261C23118yes—
31Q8×#261—46no—
32D4×#261—418no—
33C2×C4⋊#261C2319yes—
34C2×C4⋊#261C2319yes—
35C2×C4×#261C2319yes—
36C2×C4×#261C2319yes—
37C13×#124C2×C419yescomplemented
38C13×#120C2×C416yescomplemented
39#88207C2×C416yescomplemented
40#88201C2×C419yescomplemented
41#88191C2×C4118yescomplemented
42#88190C2×C419yescomplemented
43#88139C2×C4136yescomplemented
44#88136C2×C4118yescomplemented
45C22⋊C4×D13C23118yes—
46#88129C2×C419yescomplemented
47C42×D13C2313yes—
48(C2×#261)⋊C4C23136yes—
49C4×H(3,𝔽4)—131nomaximal, Sylow
50C24×C13C22×C412yescomplemented
51C23×D13—46no—
52C23×D13—46no—
53C23×D13C22×C416yescomplemented
54C23×D13C2416yes—
55C22×#261—236no—
56C22×#261—418no—
57C22×#261—432no—
58C22×#261C2416yes—
59C22×#261C2419yes—
60C22×#261C2411yescharacteristic
61C2×Q8×C13C22×C413yescomplemented
62C2×D4×C13C22×C419yes—
63C2×C22⋊D13C22×C4118yes—
64Q8×D13—43no—
65Q8×D13—43no—
66D4×D13—49no—
67D4×D13—49no—
68C2×D52C22×C419yes—
69C2×C4×D13C2419yes—
70C2×Dic26C22×C419yescomplemented
71C22⋊#261—418no—
72C22⋊#261—418no—
73C4⋊#261—49no—
74C4⋊#261—49no—
75C4×#261—218no—
76C4×#261—49no—
77C13×C22⋊C4C22×C4118yescomplemented
78C42×C13C22×C413yescomplemented
79C22⋊Dic13C22×C4118yescomplemented
80#6103C22×C4136yescomplemented
81C4×Dic13C22×C419yescomplemented
82C2×H(3,𝔽4)—131no—
83#2379—139no—
84#2375—136no—
85C4×#124—139no—
86C4×#120—136no—
87C23×C13—46no—
88C23×C13C23×C416yes—
89C22×D13—218no—
90C22×D13—218no—
91C22×D13—418no—
92C22×D13—48no—
93C22×D13—424no—
94C22×D13C23×C416yes—
95C22×D13C2511yescharacteristic
96C2×#261—29no—
97C2×#261—218no—
98C2×#261—218no—
99C2×#261—424no—
100C2×#261—424no—
101C2×#261H(5,𝔽2)16yes—
102Q8×C13—43no—
103D4×C13—49no—
104C2×C4×C13C23×C419yes—
105C22⋊D13—418no—
106C2×Dic13C23×C419yes—
107D52—49no—
108C4×D13—29no—
109C4×D13—29no—
110Dic26—49no—
111C24×C4—132no—
112H(3,𝔽4)—139no—
113H(3,𝔽4)—136no—
114H(3,𝔽4)#26111yescharacteristic, direct product factor
115C2×#124—139no—
116C2×#120—136no—
117C2×C4×Q8—133no—
118C2×C4×D4—139no—
119#391—139no—
120#390—139no—
121#388—1336no—
122#386—1318no—
123#385—1318no—
124#382—139no—
125#379—136no—
126C4×C22⋊C4—1318no—
127C43—133no—
128C22×C13—218no—
129C22×C13—48no—
130C22×C13C24×C411yescharacteristic
131C2×D13—218no—
132C2×D13—424no—
133C2×D13—46no—
134C2×D13—418no—
135C2×D13C2×H(5,𝔽2)13yes—
136C2×D13#58513yes—
137#261—46no—
138#261—49no—
139#261H(3,𝔽4)14yesdirect product factor
140C4×C13—29no—
141Dic13—29no—
142C25—132no—
143C22×Q8—133no—
144C22×D4—139no—
145C23×C4—136no—
146C23×C4—136no—
147C23×C4—5212no—
148#124—139no—
149#124—139no—
150#124—1336no—
151#124—139no—
152#124C2×#26119yescomplemented
153#120—136no—
154#120—1318no—
155#120—1318no—
156#120C2×#26116yescomplemented
157C4×Q8—526no—
158C4×D4—5218no—
159C2×C4⋊C4—139no—
160C2×C4⋊C4—139no—
161C2×C22⋊C4—1318no—
162C2×C22⋊C4—1318no—
163C2×C22⋊C4—1318no—
164C2×C42—133no—
165C2×C42—139no—
166C2×C42—139no—
167#95—1336no—
168C2×C13—46no—
169C2×C13C4×H(5,𝔽2)13yes—
170D13—46no—
171D13C2×H(3,𝔽4)11yescharacteristic
172D13#352913yes—
173C24—136no—
174C24—136no—
175C24—526no—
176C24—526no—
177C24C22×#26112yescomplemented
178C2×Q8—139no—
179C2×Q8—523no—
180C2×Q8—523no—
181C2×Q8C22×#26113yescomplemented
182C2×D4—1318no—
183C2×D4—139no—
184C2×D4—529no—
185C2×D4—529no—
186C2×D4C22×#26119yes—
187C22×C4—131no—
188C22×C4—139no—
189C22×C4—136no—
190C22×C4—139no—
191C22×C4—2636no—
192C22×C4—5218no—
193C22×C4—5232no—
194C4⋊C4—529no—
195C4⋊C4—529no—
196C22⋊C4—1318no—
197C22⋊C4—1336no—
198C22⋊C4—5218no—
199C22⋊C4—5218no—
200C22⋊C4C22×#261118yescomplemented
201C42—139no—
202C42—2618no—
203C42—529no—
204C42C22×#26113yescomplemented
205C13C4×H(3,𝔽4)11yescharacteristic, complemented, Sylow
206C23—46no—
207C23—131no—
208C23—136no—
209C23—2618no—
210C23—2618no—
211C23—5218no—
212C23—528no—
213C23—5224no—
214C23C23×#26116yes—
215Q8—43no—
216Q8—529no—
217D4—49no—
218D4—529no—
219D4—5218no—
220C2×C4—136no—
221C2×C4—139no—
222C2×C4—269no—
223C2×C4—269no—
224C2×C4—269no—
225C2×C4—2618no—
226C2×C4—2618no—
227C2×C4—5224no—
228C2×C4—5224no—
229C2×C4C23×#26119yes—
230C22—218no—
231C22—48no—
232C22—133no—
233C22—133no—
234C22—2618no—
235C22—526no—
236C22—5218no—
237C22—5224no—
238C22C24×#26111yescharacteristic
239C4—29no—
240C4—134no—
241C4—269no—
242C4—529no—
243C4—526no—
244C2—46no—
245C2—131no—
246C2—133no—
247C2—526no—
248C2H(5,𝔽2)×#26113yes—
subgroupG/H
center, FrattiniC22C24×#261
commutator, socleC22×C13C24×C4
FittingC13×H(3,𝔽4)C4
prime radicalD13×H(3,𝔽4)C2
cosocleC22×D13C25
hypercenterH(3,𝔽4)#261
hypocenterC13C4×H(3,𝔽4)
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #13131379C2semidirect
2. #13131375C2semidirect
3. #124×#261C2semidirect
4. #120×#261C2semidirect
6. C24×#261C22semidirect
7. C2×Q8×#261C22semidirect
9. C13×H(3,𝔽4)C4semidirect
10. #11719388C4semidirect
12. #11719299C4semidirect
14. #11719171C22semidirect
15. (C2×#261)⋊Q8C22semidirect
18. #11719071C22semidirect
19. #11719070C22semidirect
20. C22⋊C4×#261C22semidirect
21. #11719060C22semidirect
22. C42×#261C22semidirect
37. C13×#124C2×C4semidirect
38. C13×#120C2×C4semidirect
39. #88207C2×C4semidirect
40. #88201C2×C4semidirect
41. #88191C2×C4semidirect
42. #88190C2×C4semidirect
43. #88139C2×C4semidirect
44. #88136C2×C4semidirect
46. #88129C2×C4semidirect
50. C24×C13C22×C4semidirect
53. C23×D13C22×C4semidirect
61. C2×Q8×C13C22×C4semidirect
70. C2×Dic26C22×C4semidirect
77. C13×C22⋊C4C22×C4semidirect
78. C42×C13C22×C4semidirect
79. C22⋊Dic13C22×C4semidirect
80. #6103C22×C4semidirect
81. C4×Dic13C22×C4semidirect
114. H(3,𝔽4)#261direct (N × H)
139. #261H(3,𝔽4)direct (N × H)
152. #124C2×#261semidirect
156. #120C2×#261semidirect
177. C24C22×#261semidirect
181. C2×Q8C22×#261semidirect
200. C22⋊C4C22×#261semidirect
204. C42C22×#261semidirect
205. C13C4×H(3,𝔽4)semidirect

Maximal subgroups

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

subgroupindex
#131313792
#131313752
#124×#2612
#120×#2612
D13×H(3,𝔽4)2
C4×H(3,𝔽4)13