Finite Group Databasebeta

← H(3,𝔽4)×#6923 #30624909

Subgroups — H(3,𝔽4)×#6923 #30624909

Order 15616.

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#21462362C219yesmaximal, complemented
2#21462358C216yesmaximal, complemented
3#124×#6923C219yesmaximal, complemented
4#120×#6923C216yesmaximal, complemented
5H(3,𝔽4)×D61C211yesmaximal, characteristic
6C24×#6923C2212yescomplemented
7C2×Q8×#6923C2213yescomplemented
8C2×D4×#6923C2219yes—
9(C2×#6923)⋊Q8C2219yescomplemented
10#16840714C2219yes—
11#16840712C22136yescomplemented
12#16840710C22118yes—
13#16840709C22118yescomplemented
14#16840706C2219yescomplemented
15#16840703C2216yescomplemented
16C22⋊C4×#6923C22118yescomplemented
17C42×#6923C2213yescomplemented
18#16840505C416yescomplemented
19#16840504C419yescomplemented
20#124×D61C2219yes—
21#120×D61C2216yes—
22C61×H(3,𝔽4)C411yescharacteristic, complemented
23C23×#6923—412no—
24C23×#6923C2316yes—
25C23×#6923C2316yes—
26Q8×#6923—46no—
27D4×#6923—418no—
28C2×C4⋊#6923C2319yes—
29C2×C4⋊#6923C2319yes—
30C2×C22⋊#6923C23118yes—
31C2×C22⋊#6923C23118yes—
32C2×C4×#6923C2319yes—
33C2×C4×#6923C2319yes—
34(C2×#6923)⋊C4C23136yes—
35C24×D61C2312yes—
36C2×Q8×D61C2313yes—
37C2×D4×D61C2319yes—
38#14519766C2×C419yescomplemented
39#14519765C2×C419yescomplemented
40#14519764C2×C419yescomplemented
41#14519763C2×C4136yescomplemented
42#14519747C2×C4118yescomplemented
43#14519746C2×C416yescomplemented
44#14519745C2×C4118yescomplemented
45C22⋊C4×D61C23118yes—
46C42×D61C2313yes—
47#124×C61C2×C419yescomplemented
48#120×C61C2×C416yescomplemented
49C24×C61C22×C412yescomplemented
50C23×D61—46no—
51C23×D61—46no—
52C23×D61C22×C416yescomplemented
53C23×D61C2416yes—
54C22×#6923—236no—
55C22×#6923—418no—
56C22×#6923—432no—
57C22×#6923C2416yes—
58C22×#6923C2419yes—
59C22×#6923C2411yescharacteristic
60C2×Q8×C61C22×C413yescomplemented
61C2×D4×C61C22×C419yes—
62C2×C22⋊D61C22×C4118yes—
63Q8×D61—43no—
64Q8×D61—43no—
65D4×D61—49no—
66D4×D61—49no—
67C2×D244C22×C419yes—
68C2×C4×D61C2419yes—
69C2×Dic122C22×C419yescomplemented
70C22⋊#6923—418no—
71C22⋊#6923—418no—
72C4⋊#6923—49no—
73C4⋊#6923—49no—
74C4×#6923—218no—
75C4×#6923—49no—
76C22⋊C4×C61C22×C4118yescomplemented
77C42×C61C22×C413yescomplemented
78C22⋊Dic61C22×C4118yescomplemented
79#11758198C22×C4136yescomplemented
80C4×Dic61C22×C419yescomplemented
81C23×C61—46no—
82C23×C61C23×C416yes—
83C22×D61—218no—
84C22×D61—218no—
85C22×D61—418no—
86C22×D61—48no—
87C22×D61—424no—
88C22×D61C23×C416yes—
89C22×D61C2511yescharacteristic
90C2×#6923—29no—
91C2×#6923—218no—
92C2×#6923—218no—
93C2×#6923—424no—
94C2×#6923—424no—
95C2×#6923H(5,𝔽2)16yes—
96Q8×C61—43no—
97D4×C61—49no—
98C2×C4×C61C23×C419yes—
99C22⋊D61—418no—
100C2×Dic61C23×C419yes—
101D244—49no—
102C4×D61—29no—
103C4×D61—29no—
104Dic122—49no—
105C4×H(3,𝔽4)—611nomaximal, Sylow
106C22×C61—218no—
107C22×C61—48no—
108C22×C61C24×C411yescharacteristic
109C2×D61—218no—
110C2×D61—424no—
111C2×D61—46no—
112C2×D61—418no—
113C2×D61C2×H(5,𝔽2)13yes—
114C2×D61#58513yes—
115#6923—46no—
116#6923—49no—
117#6923H(3,𝔽4)14yesdirect product factor
118C4×C61—29no—
119Dic61—29no—
120C2×H(3,𝔽4)—611no—
121#2379—619no—
122#2375—616no—
123C4×#124—619no—
124C4×#120—616no—
125C2×C61—46no—
126C2×C61C4×H(5,𝔽2)13yes—
127D61—46no—
128D61C2×H(3,𝔽4)11yescharacteristic
129D61#352913yes—
130C24×C4—612no—
131H(3,𝔽4)—619no—
132H(3,𝔽4)—616no—
133H(3,𝔽4)#692311yescharacteristic, direct product factor
134C2×#124—619no—
135C2×#120—616no—
136C2×C4×Q8—613no—
137C2×C4×D4—619no—
138#391—619no—
139#390—619no—
140#388—6136no—
141#386—6118no—
142#385—6118no—
143#382—619no—
144#379—616no—
145C4×C22⋊C4—6118no—
146C43—613no—
147C61C4×H(3,𝔽4)11yescharacteristic, complemented, Sylow
148C25—612no—
149C22×Q8—613no—
150C22×D4—619no—
151C23×C4—616no—
152C23×C4—616no—
153C23×C4—24412no—
154#124—619no—
155#124—619no—
156#124—6136no—
157#124—619no—
158#124C2×#692319yescomplemented
159#120—616no—
160#120—6118no—
161#120—6118no—
162#120C2×#692316yescomplemented
163C4×Q8—2446no—
164C4×D4—24418no—
165C2×C4⋊C4—619no—
166C2×C4⋊C4—619no—
167C2×C22⋊C4—6118no—
168C2×C22⋊C4—6118no—
169C2×C22⋊C4—6118no—
170C2×C42—613no—
171C2×C42—619no—
172C2×C42—619no—
173#95—6136no—
174C24—616no—
175C24—616no—
176C24—2446no—
177C24—2446no—
178C24C22×#692312yescomplemented
179C2×Q8—619no—
180C2×Q8—2443no—
181C2×Q8—2443no—
182C2×Q8C22×#692313yescomplemented
183C2×D4—6118no—
184C2×D4—619no—
185C2×D4—2449no—
186C2×D4—2449no—
187C2×D4C22×#692319yes—
188C22×C4—611no—
189C22×C4—619no—
190C22×C4—616no—
191C22×C4—619no—
192C22×C4—12236no—
193C22×C4—24418no—
194C22×C4—24432no—
195C4⋊C4—2449no—
196C4⋊C4—2449no—
197C22⋊C4—6118no—
198C22⋊C4—6136no—
199C22⋊C4—24418no—
200C22⋊C4—24418no—
201C22⋊C4C22×#6923118yescomplemented
202C42—619no—
203C42—12218no—
204C42—2449no—
205C42C22×#692313yescomplemented
206C23—46no—
207C23—611no—
208C23—616no—
209C23—12218no—
210C23—12218no—
211C23—24418no—
212C23—2448no—
213C23—24424no—
214C23C23×#692316yes—
215Q8—43no—
216Q8—2449no—
217D4—49no—
218D4—2449no—
219D4—24418no—
220C2×C4—616no—
221C2×C4—619no—
222C2×C4—1229no—
223C2×C4—1229no—
224C2×C4—1229no—
225C2×C4—12218no—
226C2×C4—12218no—
227C2×C4—24424no—
228C2×C4—24424no—
229C2×C4C23×#692319yes—
230C22—218no—
231C22—48no—
232C22—613no—
233C22—613no—
234C22—12218no—
235C22—2446no—
236C22—24418no—
237C22—24424no—
238C22C24×#692311yescharacteristic
239C4—29no—
240C4—614no—
241C4—1229no—
242C4—2446no—
243C4—2449no—
244C2—46no—
245C2—611no—
246C2—613no—
247C2—2446no—
248C2H(5,𝔽2)×#692313yes—
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×H(3,𝔽4)C4
prime radicalH(3,𝔽4)×D61C2
cosocleC22×D61C25
hypercenterH(3,𝔽4)#6923
hypocenterC61C4×H(3,𝔽4)
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #21462362C2semidirect
2. #21462358C2semidirect
3. #124×#6923C2semidirect
4. #120×#6923C2semidirect
6. C24×#6923C22semidirect
7. C2×Q8×#6923C22semidirect
9. (C2×#6923)⋊Q8C22semidirect
11. #16840712C22semidirect
13. #16840709C22semidirect
14. #16840706C22semidirect
15. #16840703C22semidirect
16. C22⋊C4×#6923C22semidirect
17. C42×#6923C22semidirect
18. #16840505C4semidirect
19. #16840504C4semidirect
22. C61×H(3,𝔽4)C4semidirect
38. #14519766C2×C4semidirect
39. #14519765C2×C4semidirect
40. #14519764C2×C4semidirect
41. #14519763C2×C4semidirect
42. #14519747C2×C4semidirect
43. #14519746C2×C4semidirect
44. #14519745C2×C4semidirect
47. #124×C61C2×C4semidirect
48. #120×C61C2×C4semidirect
49. C24×C61C22×C4semidirect
52. C23×D61C22×C4semidirect
60. C2×Q8×C61C22×C4semidirect
69. C2×Dic122C22×C4semidirect
76. C22⋊C4×C61C22×C4semidirect
77. C42×C61C22×C4semidirect
78. C22⋊Dic61C22×C4semidirect
79. #11758198C22×C4semidirect
80. C4×Dic61C22×C4semidirect
117. #6923H(3,𝔽4)direct (N × H)
133. H(3,𝔽4)#6923direct (N × H)
147. C61C4×H(3,𝔽4)semidirect
158. #124C2×#6923semidirect
162. #120C2×#6923semidirect
178. C24C22×#6923semidirect
182. C2×Q8C22×#6923semidirect
201. C22⋊C4C22×#6923semidirect
205. C42C22×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623622
#214623582
#124×#69232
#120×#69232
H(3,𝔽4)×D612
C4×H(3,𝔽4)61