Finite Group Databasebeta

← #24918596

Subgroups — #24918596

Order 10624.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128256C211yesmaximal, characteristic, complemented
2#19128249C211yesmaximal, characteristic
3#19128104C211yesmaximal, characteristic, complemented
4#19128099C211yesmaximal, characteristic, complemented
5#19128047C211yesmaximal, characteristic, complemented
6#19128024C214yesmaximal, complemented
7#19128003C214yesmaximal, complemented
8#19127752C211yesmaximal, characteristic, complemented
9#486×C83C211yesmaximal, characteristic, complemented
10C2×#13050589—21no—
11#14581792C2211yescharacteristic
12#14581792C2211yescharacteristic
13#14581783C2212yes—
14#14581779—21no—
15#14581772—21no—
16#14581768—21no—
17#14581763—21no—
18#14581763C2212yes—
19Q8×Dic83—21no—
20#14581757—21no—
21D4×Dic83—21no—
22D4×Dic83C2212yes—
23#14581750C2212yes—
24C42×D83C2211yescharacteristic
25#14581725C2214yes—
26#14581724C2214yes—
27Dic83⋊C8C2212yes—
28#14581713C2214yes—
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic
31C4⋊Q8×C83C2211yescharacteristic
32#127×C83C2211yescharacteristic
33#102×C83C2214yes—
34C4×C8×C83C2211yescharacteristic
35#13050589—44no—
36C2×Dic166—21no—
37C2×Dic166—21no—
38C2×C22⋊D83—22no—
39C22×Dic83—22no—
40C2×C4×D83D411yescharacteristic
41C2×C4×D83D411yescharacteristic
42C2×C4×D83C2311yescharacteristic
43C2×C83⋊C8C2312yes—
44C4⋊Dic83—21no—
45C4⋊Dic83—21no—
46C4⋊Dic83—22no—
47C4⋊Dic83C2312yes—
48C4⋊Dic83C2312yes—
49Dic83⋊C4—22no—
50C22⋊Dic83—22no—
51C22⋊Dic83—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C4×Dic83D411yescharacteristic
55C4×Dic83D411yescharacteristic
56C4×Dic83C2311yescharacteristic
57C2×Q8×C83—21no—
58C2×D4×C83—21no—
59C2×D4×C83—21no—
60C2×D4×C83C2312yes—
61C2×C8×C83C2312yes—
62C4⋊C4×C83—21no—
63C4⋊C4×C83C2312yes—
64C42×C83C2311yescharacteristic
65C23×C83—22no—
66C22×D83C2×D411yescharacteristic
67Q8×C83—42no—
68D4×C83—24no—
69D4×C83—42no—
70D4×C83—42no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2411yescharacteristic
75C22⋊D83—44no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—44no—
79C2×Dic83C2×D411yescharacteristic
80C2×Dic83C2×D411yescharacteristic
81C2×Dic83C2×D411yescharacteristic
82C4×D83—22no—
83C4×D83—22no—
84C4×D83—22no—
85Dic166—42no—
86Dic166—42no—
87C8×C83—22no—
88C83⋊C8—22no—
89C22×C83—42no—
90C22×C83—42no—
91C22×C83—42no—
92C22×C83C22×D411yescharacteristic
93C2×D83—21no—
94C2×D83—21no—
95C2×D83—21no—
96C4×C83—42no—
97C4×C83#13512yes—
98C4×C83AGL(1,ℤ/8)12yes—
99C4×C832−1+412yes—
100Dic83—21no—
101Dic83—21no—
102Dic83—21no—
103Dic83—42no—
104Dic83—42no—
105C2×C83—82no—
106C2×C83#53611yescharacteristic
107C2×C83C2×#13511yescharacteristic
108C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
109D83—41no—
110#3228—831nomaximal, Sylow
111C83#322811yescharacteristic, complemented, Sylow
112#533—831no—
113#532—831no—
114#500—831no—
115#498—831no—
116#486—831no—
117#486D8311yescharacteristic, complemented
118#482—834no—
119#478—834no—
120#431—831no—
121C2×#41—1661no—
122C4⋊Q8—831no—
123C4⋊Q8—831no—
124C4⋊Q8C2×D8311yescharacteristic
125#127C2×D8311yescharacteristic
126#125—832no—
127#124—1661no—
128C22⋊Q8—1661no—
129C22⋊Q8—1661no—
130#121—832no—
131#121—1661no—
132C4×Q8—1661no—
133C4×Q8—1661no—
134C4×D4—832no—
135C4×D4—1661no—
136#117—832no—
137C2×C42—831no—
138#107—834no—
139#106—834no—
140C4⋊C8—832no—
141#102—834no—
142#102C2×D8314yes—
143C22⋊C8—832no—
144C4×C8—831no—
145C4×C8C2×D8311yescharacteristic
146#41—3324no—
147C2×Q8—21no—
148C2×Q8—1661no—
149C2×Q8—1661no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—1662no—
153C2×D4C22×D8312yes—
154C22×C4—831no—
155C22×C4—831no—
156C22×C4—831no—
157C22×C4—1662no—
158C2×C8—832no—
159C2×C8C22×D8312yes—
160C4⋊C4—21no—
161C4⋊C4—832no—
162C4⋊C4—832no—
163C4⋊C4—1662no—
164C4⋊C4—1662no—
165C4⋊C4—1661no—
166C4⋊C4—1661no—
167C4⋊C4C22×D8312yes—
168C22⋊C4—1662no—
169C22⋊C4—1662no—
170C22⋊C4—1662no—
171C42—831no—
172C42—831no—
173C42—831no—
174C42—1662no—
175C42C22×D8311yescharacteristic
176C23—22no—
177C23—831no—
178Q8—42no—
179Q8—3322no—
180Q8—3322no—
181D4—24no—
182D4—42no—
183D4—42no—
184D4—3324no—
185C2×C4—22no—
186C2×C4—3324no—
187C2×C4—831no—
188C2×C4—831no—
189C2×C4—831no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4D4×D8311yescharacteristic
196C2×C4D4×D8311yescharacteristic
197C2×C4C23×D8311yescharacteristic
198C8—22no—
199C8—1662no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—1661no—
204C22—1661no—
205C22—1661no—
206C22C2×D4×D8311yescharacteristic
207C4—42no—
208C4—3322no—
209C4—3322no—
210C4—1661no—
211C4—1661no—
212C4—1661no—
213C4#1458181312yes—
214C4#1458181612yes—
215C4#1458183912yes—
216C2—82no—
217C2—3321no—
218C2#1912827111yescharacteristic
219C2C2×#1458181311yescharacteristic
220C2C2×#1458181611yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#486×C83C2
socleC22×C83C22×D4
prime radical#19128249C2
hypercenter#486D83
hypocenterC83#3228
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128104C2semidirect
4. #19128099C2semidirect
5. #19128047C2semidirect
6. #19128024C2semidirect
7. #19128003C2semidirect
8. #19127752C2semidirect
9. #486×C83C2semidirect
111. C83#3228semidirect
117. #486D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282492
#191281042
#191280992
#191280472
#191280242
#191280032
#191277522
#486×C832
#322883