Finite Group Databasebeta

← H(3,𝔽4)×#64408 #34030880

Subgroups — H(3,𝔽4)×#64408 #34030880

Order 18688.

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#22626474C219yesmaximal, complemented
2#22626470C216yesmaximal, complemented
3#124×#64408C219yesmaximal, complemented
4#120×#64408C216yesmaximal, complemented
5H(3,𝔽4)×D73C211yesmaximal, characteristic
6C24×#64408C2212yescomplemented
7C2×Q8×#64408C2213yescomplemented
8C2×D4×#64408C2219yes—
9(C2×#64408)⋊Q8C2219yescomplemented
10#17989539C2219yes—
11#17989537C22136yescomplemented
12#17989535C22118yes—
13#17989534C22118yescomplemented
14#17989531C2219yescomplemented
15#17989528C2216yescomplemented
16C22⋊C4×#64408C22118yescomplemented
17C42×#64408C2213yescomplemented
18#17989330C416yescomplemented
19#17989329C419yescomplemented
20#124×D73C2219yes—
21#120×D73C2216yes—
22H(3,𝔽4)×C73C411yescharacteristic, complemented
23C23×#64408—412no—
24C23×#64408C2316yes—
25C23×#64408C2316yes—
26Q8×#64408—46no—
27D4×#64408—418no—
28C2×C4⋊#64408C2319yes—
29C2×C4⋊#64408C2319yes—
30C2×C22⋊#64408C23118yes—
31C2×C22⋊#64408C23118yes—
32C2×C4×#64408C2319yes—
33C2×C4×#64408C2319yes—
34(C2×#64408)⋊C4C23136yes—
35C24×D73C2312yes—
36C2×Q8×D73C2313yes—
37C2×D4×D73C2319yes—
38#14556580C2×C419yescomplemented
39#14556579C2×C419yescomplemented
40#14556578C2×C419yescomplemented
41#14556577C2×C4136yescomplemented
42#14556561C2×C4118yescomplemented
43#14556560C2×C416yescomplemented
44#14556559C2×C4118yescomplemented
45C22⋊C4×D73C23118yes—
46C42×D73C2313yes—
47#124×C73C2×C419yescomplemented
48#120×C73C2×C416yescomplemented
49C22×#64408—236no—
50C22×#64408—418no—
51C22×#64408—432no—
52C22×#64408C2416yes—
53C22×#64408C2419yes—
54C22×#64408C2411yescharacteristic
55C4⋊#64408—49no—
56C4⋊#64408—49no—
57C22⋊#64408—418no—
58C22⋊#64408—418no—
59C4×#64408—218no—
60C4×#64408—49no—
61C23×D73—46no—
62C23×D73—46no—
63C23×D73C22×C416yescomplemented
64C23×D73C2416yes—
65C2×Dic146C22×C419yescomplemented
66Q8×D73—43no—
67Q8×D73—43no—
68C2×D292C22×C419yes—
69C2×C22⋊D73C22×C4118yes—
70D4×D73—49no—
71D4×D73—49no—
72C2×C4×D73C2419yes—
73C22⋊Dic73C22×C4118yescomplemented
74#11924005C22×C4136yescomplemented
75C4×Dic73C22×C419yescomplemented
76C24×C73C22×C412yescomplemented
77C2×Q8×C73C22×C413yescomplemented
78C2×D4×C73C22×C419yes—
79C22⋊C4×C73C22×C4118yescomplemented
80C42×C73C22×C413yescomplemented
81C23×C73—46no—
82C23×C73C23×C416yes—
83C22×D73—218no—
84C22×D73—218no—
85C22×D73—418no—
86C22×D73—424no—
87C22×D73—48no—
88C22×D73C23×C416yes—
89C22×D73C2511yescharacteristic
90C2×#64408—29no—
91C2×#64408—218no—
92C2×#64408—218no—
93C2×#64408—424no—
94C2×#64408—424no—
95C2×#64408H(5,𝔽2)16yes—
96Q8×C73—43no—
97D4×C73—49no—
98C2×C4×C73C23×C419yes—
99C22⋊D73—418no—
100C2×Dic73C23×C419yes—
101D292—49no—
102C4×D73—29no—
103C4×D73—29no—
104Dic146—49no—
105C22×C73—218no—
106C22×C73—48no—
107C22×C73C24×C411yescharacteristic
108C2×D73—218no—
109C2×D73—424no—
110C2×D73—46no—
111C2×D73—418no—
112C2×D73C2×H(5,𝔽2)13yes—
113C2×D73#58513yes—
114#64408—46no—
115#64408—49no—
116#64408H(3,𝔽4)14yesdirect product factor
117C4×C73—29no—
118Dic73—29no—
119C4×H(3,𝔽4)—731nomaximal, Sylow
120C2×C73—46no—
121C2×C73C4×H(5,𝔽2)13yes—
122D73—46no—
123D73C2×H(3,𝔽4)11yescharacteristic
124D73#352913yes—
125C2×H(3,𝔽4)—731no—
126#2379—739no—
127#2375—736no—
128C4×#124—739no—
129C4×#120—736no—
130C73C4×H(3,𝔽4)11yescharacteristic, complemented, Sylow
131C24×C4—732no—
132H(3,𝔽4)—739no—
133H(3,𝔽4)—736no—
134H(3,𝔽4)#6440811yescharacteristic, direct product factor
135C2×#124—739no—
136C2×#120—736no—
137C2×C4×Q8—733no—
138C2×C4×D4—739no—
139#391—739no—
140#390—739no—
141#388—7336no—
142#386—7318no—
143#385—7318no—
144#382—739no—
145#379—736no—
146C4×C22⋊C4—7318no—
147C43—733no—
148C25—732no—
149C22×Q8—733no—
150C22×D4—739no—
151C23×C4—29212no—
152C23×C4—736no—
153C23×C4—736no—
154#124—739no—
155#124—739no—
156#124—7336no—
157#124—739no—
158#124C2×#6440819yescomplemented
159#120—736no—
160#120—7318no—
161#120—7318no—
162#120C2×#6440816yescomplemented
163C4×Q8—2926no—
164C4×D4—29218no—
165C2×C4⋊C4—739no—
166C2×C4⋊C4—739no—
167C2×C22⋊C4—7318no—
168C2×C22⋊C4—7318no—
169C2×C22⋊C4—7318no—
170C2×C42—739no—
171C2×C42—733no—
172C2×C42—739no—
173#95—7336no—
174C24—2926no—
175C24—2926no—
176C24—736no—
177C24—736no—
178C24C22×#6440812yescomplemented
179C2×Q8—2923no—
180C2×Q8—2923no—
181C2×Q8—739no—
182C2×Q8C22×#6440813yescomplemented
183C2×D4—2929no—
184C2×D4—2929no—
185C2×D4—7318no—
186C2×D4—739no—
187C2×D4C22×#6440819yes—
188C22×C4—29232no—
189C22×C4—29218no—
190C22×C4—731no—
191C22×C4—739no—
192C22×C4—736no—
193C22×C4—739no—
194C22×C4—14636no—
195C4⋊C4—2929no—
196C4⋊C4—2929no—
197C22⋊C4—29218no—
198C22⋊C4—29218no—
199C22⋊C4—7318no—
200C22⋊C4—7336no—
201C22⋊C4C22×#64408118yescomplemented
202C42—2929no—
203C42—739no—
204C42—14618no—
205C42C22×#6440813yescomplemented
206C23—46no—
207C23—29218no—
208C23—2928no—
209C23—29224no—
210C23—731no—
211C23—736no—
212C23—14618no—
213C23—14618no—
214C23C23×#6440816yes—
215Q8—43no—
216Q8—2929no—
217D4—49no—
218D4—2929no—
219D4—29218no—
220C2×C4—29224no—
221C2×C4—29224no—
222C2×C4—736no—
223C2×C4—739no—
224C2×C4—1469no—
225C2×C4—1469no—
226C2×C4—14618no—
227C2×C4—1469no—
228C2×C4—14618no—
229C2×C4C23×#6440819yes—
230C22—218no—
231C22—48no—
232C22—29224no—
233C22—2926no—
234C22—29218no—
235C22—733no—
236C22—733no—
237C22—14618no—
238C22C24×#6440811yescharacteristic
239C4—29no—
240C4—2926no—
241C4—2929no—
242C4—734no—
243C4—1469no—
244C2—46no—
245C2—2926no—
246C2—731no—
247C2—733no—
248C2H(5,𝔽2)×#6440813yes—
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
FittingH(3,𝔽4)×C73C4
prime radicalH(3,𝔽4)×D73C2
cosocleC22×D73C25
hypercenterH(3,𝔽4)#64408
hypocenterC73C4×H(3,𝔽4)
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #22626474C2semidirect
2. #22626470C2semidirect
3. #124×#64408C2semidirect
4. #120×#64408C2semidirect
6. C24×#64408C22semidirect
7. C2×Q8×#64408C22semidirect
9. (C2×#64408)⋊Q8C22semidirect
11. #17989537C22semidirect
13. #17989534C22semidirect
14. #17989531C22semidirect
15. #17989528C22semidirect
16. C22⋊C4×#64408C22semidirect
17. C42×#64408C22semidirect
18. #17989330C4semidirect
19. #17989329C4semidirect
22. H(3,𝔽4)×C73C4semidirect
38. #14556580C2×C4semidirect
39. #14556579C2×C4semidirect
40. #14556578C2×C4semidirect
41. #14556577C2×C4semidirect
42. #14556561C2×C4semidirect
43. #14556560C2×C4semidirect
44. #14556559C2×C4semidirect
47. #124×C73C2×C4semidirect
48. #120×C73C2×C4semidirect
63. C23×D73C22×C4semidirect
65. C2×Dic146C22×C4semidirect
73. C22⋊Dic73C22×C4semidirect
74. #11924005C22×C4semidirect
75. C4×Dic73C22×C4semidirect
76. C24×C73C22×C4semidirect
77. C2×Q8×C73C22×C4semidirect
79. C22⋊C4×C73C22×C4semidirect
80. C42×C73C22×C4semidirect
116. #64408H(3,𝔽4)direct (N × H)
130. C73C4×H(3,𝔽4)semidirect
134. H(3,𝔽4)#64408direct (N × H)
158. #124C2×#64408semidirect
162. #120C2×#64408semidirect
178. C24C22×#64408semidirect
182. C2×Q8C22×#64408semidirect
201. C22⋊C4C22×#64408semidirect
205. C42C22×#64408semidirect

Maximal subgroups

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

subgroupindex
#226264742
#226264702
#124×#644082
#120×#644082
H(3,𝔽4)×D732
C4×H(3,𝔽4)73