Finite Group Databasebeta

← #24917858

Subgroups — #24917858

Order 10624.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128252C211yesmaximal, characteristic
2C4⋊Q8×D83C211yesmaximal, characteristic, complemented
3#19128062C214yesmaximal, complemented
4#19128051C211yesmaximal, characteristic, complemented
5#19128049C211yesmaximal, characteristic, complemented
6#19128048C211yesmaximal, characteristic, complemented
7#19127701C214yesmaximal, complemented
8C4×#13050573C211yesmaximal, characteristic, complemented
9#487×C83C211yesmaximal, characteristic, complemented
10C2×#13050591C2212yes—
11C2×Q8×D83—21no—
12C2×#13050573C2212yes—
13#14581792C2211yescharacteristic
14#14581791—21no—
15#14581791C2212yes—
16#14581790—21no—
17#14581789C2211yescharacteristic
18#14581781C2212yes—
19#14581767—21no—
20#14581767—21no—
21#14581754—21no—
22#14581754—21no—
23C4⋊C4×D83—21no—
24C4⋊C4×D83C2212yes—
25C42×D83C2211yescharacteristic
26#14581716C2214yes—
27#14581714C2214yes—
28#14581712C2214yes—
29#14581702C2212yes—
30C4×C83⋊C8C2211yescharacteristic
31C4⋊Q8×C83C2211yescharacteristic
32C4⋊Q8×C83C2211yescharacteristic
33#103×C83C2214yes—
34C4×C8×C83C2211yescharacteristic
35#13050591—28no—
36C2×Dic166—21no—
37C2×Dic166—22no—
38Q8×D83—44no—
39C2×D332—22no—
40C2×D332—21no—
41C2×D332—21no—
42C2×D332C2312yes—
43C2×C4×D83—22no—
44C2×C4×D83—22no—
45C2×C4×D83D411yescharacteristic
46C2×C4×D83D411yescharacteristic
47C2×C4×D83C2311yescharacteristic
48#13050573—24no—
49C2×C83⋊C8C2312yes—
50C4⋊Dic83—21no—
51C4⋊Dic83C2312yes—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4—22no—
56C4×Dic83D411yescharacteristic
57C4×Dic83D411yescharacteristic
58C4×Dic83C2311yescharacteristic
59C2×Q8×C83—21no—
60C2×Q8×C83C2312yes—
61C2×C8×C83C2312yes—
62C4⋊C4×C83—21no—
63C4⋊C4×C83—21no—
64C4⋊C4×C83—21no—
65C4⋊C4×C83C2312yes—
66C42×C83C2311yescharacteristic
67C22×D83—22no—
68C22×D83C2×D411yescharacteristic
69Q8×C83—24no—
70Q8×C83—42no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2411yescharacteristic
76C2×Dic83—22no—
77C2×Dic83C2×D411yescharacteristic
78C2×Dic83C2×D411yescharacteristic
79C2×Dic83C2×D411yescharacteristic
80D332—24no—
81D332—44no—
82D332—42no—
83D332—42no—
84C4×D83—22no—
85C4×D83—22no—
86C4×D83—44no—
87C4×D83—44no—
88C4×D83#4114yes—
89Dic166—44no—
90Dic166—42no—
91C8×C83—22no—
92C83⋊C8—22no—
93C22×C83C22×D411yescharacteristic
94C2×D83—21no—
95C2×D83—21no—
96C2×D83—42no—
97C2×D83—42no—
98C2×D83—42no—
99C2×D83#12412yes—
100C4×C83—42no—
101C4×C83—42no—
102C4×C83AGL(1,ℤ/8)12yes—
103C4×C83#13712yes—
104C4×C83C2×#4112yes—
105Dic83—21no—
106Dic83—21no—
107Dic83—42no—
108Dic83#12412yes—
109C2×C83C2×#12411yescharacteristic
110C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C83C2×#13711yescharacteristic
112D83—22no—
113D83—82no—
114#3141—831nomaximal, Sylow
115C83#314111yescharacteristic, complemented, Sylow
116#532—831no—
117C2×C4⋊Q8—831no—
118#489—834no—
119#487—831no—
120#487D8311yescharacteristic, complemented
121#486—831no—
122#486—831no—
123#417—834no—
124C4×M16—831no—
125C2×#41—832no—
126C22×Q8—1661no—
127C2×M16—832no—
128C4⋊Q8—832no—
129C4⋊Q8—831no—
130C4⋊Q8—1661no—
131C4⋊Q8—1661no—
132C4⋊Q8C2×D8311yescharacteristic
133C4⋊Q8C2×D8311yescharacteristic
134#127—831no—
135#124—832no—
136#121—1661no—
137#121—1661no—
138C4×D4—1661no—
139C4×D4—1661no—
140C2×C4⋊C4—832no—
141C2×C4⋊C4—1661no—
142C2×C42—831no—
143#103—834no—
144#103C2×D8314yes—
145#102—834no—
146#102—834no—
147#97—832no—
148C4×C8—831no—
149C4×C8C2×D8311yescharacteristic
150#41—1668no—
151C2×Q8—21no—
152C2×Q8—3324no—
153C2×Q8—1662no—
154C2×Q8—1661no—
155C2×Q8C22×D8312yes—
156C2×D4—832no—
157C2×D4—1662no—
158C2×D4—1661no—
159C2×D4—1661no—
160C22×C4—831no—
161C22×C4—831no—
162C22×C4—831no—
163C22×C4—1662no—
164C22×C4—1662no—
165M16—1664no—
166C2×C8—832no—
167C2×C8C22×D8312yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—832no—
172C4⋊C4—1662no—
173C4⋊C4—1662no—
174C4⋊C4—1661no—
175C4⋊C4C22×D8312yes—
176C22⋊C4—1662no—
177C22⋊C4—1662no—
178C42—831no—
179C42—831no—
180C42—831no—
181C42C22×D8311yescharacteristic
182C23—831no—
183C23—1662no—
184Q8—24no—
185Q8—42no—
186Q8—3324no—
187Q8—3322no—
188D4—3324no—
189D4—3322no—
190D4—3322no—
191D4—1664no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—3324no—
195C2×C4—3324no—
196C2×C4—831no—
197C2×C4—834no—
198C2×C4—831no—
199C2×C4—831no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4D4×D8311yescharacteristic
204C2×C4D4×D8311yescharacteristic
205C2×C4C23×D8311yescharacteristic
206C8—22no—
207C8—1662no—
208C22—3322no—
209C22—3322no—
210C22—3322no—
211C22—832no—
212C22—1661no—
213C22—1661no—
214C22C2×D4×D8311yescharacteristic
215C4—42no—
216C4—42no—
217C4—3322no—
218C4—832no—
219C4—1661no—
220C4—1661no—
221C4#1458181912yes—
222C4#1458182412yes—
223C4#41×D8312yes—
224C2—6642no—
225C2—1662no—
226C2#124×D8311yescharacteristic
227C2C2×#1458181911yescharacteristic
228C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#487×C83C2
socleC22×C83C22×D4
prime radical#19128252C2
hypercenter#487D83
hypocenterC83#3141
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D83C2semidirect
3. #19128062C2semidirect
4. #19128051C2semidirect
5. #19128049C2semidirect
6. #19128048C2semidirect
7. #19127701C2semidirect
8. C4×#13050573C2semidirect
9. #487×C83C2semidirect
115. C83#3141semidirect
120. #487D83semidirect

Maximal subgroups

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

subgroupindex
#191282522
C4⋊Q8×D832
#191280622
#191280512
#191280492
#191280482
#191277012
C4×#130505732
#487×C832
#314183