Finite Group Databasebeta

← #24918648

Subgroups — #24918648

Order 10624.

368 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 111 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#19128249C211yesmaximal, characteristic
2#125×D83C211yesmaximal, characteristic, complemented
3#19128111C211yesmaximal, characteristic, complemented
4#19128110C211yesmaximal, characteristic, complemented
5#19128065C211yesmaximal, characteristic, complemented
6#19128018C212yesmaximal, complemented
7#19128013C212yesmaximal, complemented
8#19128003C212yesmaximal, complemented
9#19127999C212yesmaximal, complemented
10#19127756C211yesmaximal, characteristic, complemented
11#490×C83C211yesmaximal, characteristic, complemented
12C2×#13050589—21no—
13#14581792C2211yescharacteristic
14#14581784C2211yescharacteristic
15#14581783C2211yescharacteristic
16#14581783C2211yescharacteristic
17#14581782—21no—
18#14581782—21no—
19#14581779—21no—
20#14581763—21no—
21#14581763C2212yes—
22D4×Dic83—21no—
23D4×Dic83C2212yes—
24C4⋊C4×D83—21no—
25C4⋊C4×D83—21no—
26C4⋊C4×D83C2211yescharacteristic
27C4⋊C4×D83C2211yescharacteristic
28C42×D83C2211yescharacteristic
29#14581726C2212yes—
30#14581725C2212yes—
31#14581724C2212yes—
32#14581723C2212yes—
33Dic83⋊C8C2212yes—
34#14581713C2212yes—
35#14581713C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581703C2211yescharacteristic
38#127×C83C2211yescharacteristic
39#125×C83C2211yescharacteristic
40#102×C83C2212yes—
41#102×C83C2212yes—
42#97×C83C2211yescharacteristic
43#13050589—44no—
44C2×Dic166—21no—
45C2×C22⋊D83—22no—
46C22×Dic83—22no—
47C2×C4×D83—21no—
48C2×C4×D83—21no—
49C2×C4×D83D411yescharacteristic
50C2×C4×D83D411yescharacteristic
51C2×C4×D83C2311yescharacteristic
52C2×C83⋊C8C2312yes—
53C4⋊Dic83—21no—
54C4⋊Dic83—21no—
55C4⋊Dic83—21no—
56C4⋊Dic83C2312yes—
57C4⋊Dic83C2311yescharacteristic
58C4⋊Dic83C2311yescharacteristic
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62Dic83⋊C4—21no—
63Dic83⋊C4—21no—
64Dic83⋊C4—21no—
65C22⋊Dic83—22no—
66C22⋊Dic83—22no—
67C4×Dic83D411yescharacteristic
68C4×Dic83D411yescharacteristic
69C4×Dic83C2311yescharacteristic
70C2×D4×C83—21no—
71C2×D4×C83—21no—
72C2×D4×C83C2312yes—
73C2×C8×C83C2312yes—
74C4⋊C4×C83—21no—
75C4⋊C4×C83—21no—
76C4⋊C4×C83C2311yescharacteristic
77C4⋊C4×C83C2311yescharacteristic
78C42×C83C2311yescharacteristic
79C23×C83—22no—
80C22×D83C2×D411yescharacteristic
81D4×C83—24no—
82D4×C83—42no—
83D4×C83—42no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2411yescharacteristic
89C22⋊D83—44no—
90C2×Dic83—22no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—44no—
94C2×Dic83C2×D411yescharacteristic
95C2×Dic83C2×D411yescharacteristic
96C2×Dic83C2×D411yescharacteristic
97C4×D83—22no—
98C4×D83—22no—
99C4×D83—41no—
100C4×D83—42no—
101C4×D83—42no—
102Dic166—42no—
103C8×C83—22no—
104C83⋊C8—22no—
105C22×C83—42no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83C22×D411yescharacteristic
109C2×D83—21no—
110C2×D83—21no—
111C2×D83AGL(1,ℤ/8)12yes—
112C4×C83—21no—
113C4×C83—21no—
114C4×C83—41no—
115C4×C83—41no—
116C4×C832−1+412yes—
117Dic83—21no—
118Dic83—21no—
119Dic83—21no—
120Dic83—42no—
121Dic83—41no—
122Dic83—41no—
123C2×C83—82no—
124C2×C83#53611yescharacteristic
125C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C83#57511yescharacteristic
127D83—22no—
128#3234—831nomaximal, Sylow
129C83#323411yescharacteristic, complemented, Sylow
130#532—831no—
131C2×#125—831no—
132#501—831no—
133#501—831no—
134#490—831no—
135#490D8311yescharacteristic, complemented
136#481—832no—
137#480—832no—
138#478—832no—
139#478—832no—
140#432—831no—
141C2×#41—1661no—
142C4⋊Q8—831no—
143#127C2×D8311yescharacteristic
144#125—831no—
145#125—831no—
146#125—831no—
147#125—1661no—
148#125—1661no—
149#125C2×D8311yescharacteristic
150#124—1661no—
151#121—832no—
152#121—1661no—
153C4×D4—832no—
154C4×D4—1661no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—1661no—
158C2×C4⋊C4—1661no—
159C2×C42—831no—
160#107—832no—
161#107—832no—
162#106—832no—
163#106—832no—
164C4⋊C8—832no—
165#102—832no—
166#102—832no—
167#102C2×D8312yes—
168#102C2×D8312yes—
169C22⋊C8—832no—
170#97—831no—
171#97C2×D8311yescharacteristic
172#41—3324no—
173C2×Q8—1661no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—1662no—
177C2×D4C22×D8312yes—
178C22×C4—831no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—1661no—
182C22×C4—1661no—
183C22×C4—1662no—
184C2×C8—832no—
185C2×C8C22×D8312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—831no—
189C4⋊C4—831no—
190C4⋊C4—832no—
191C4⋊C4—1661no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C4⋊C4—1661no—
195C4⋊C4—1661no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C4⋊C4—1661no—
200C4⋊C4C22×D8311yescharacteristic
201C4⋊C4C22×D8311yescharacteristic
202C22⋊C4—1662no—
203C22⋊C4—1662no—
204C42—831no—
205C42—831no—
206C42—831no—
207C42C22×D8311yescharacteristic
208C23—22no—
209C23—831no—
210Q8—3322no—
211D4—24no—
212D4—42no—
213D4—42no—
214D4—3324no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—3321no—
218C2×C4—3322no—
219C2×C4—3322no—
220C2×C4—3324no—
221C2×C4—831no—
222C2×C4—831no—
223C2×C4—831no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1661no—
227C2×C4—1661no—
228C2×C4—1662no—
229C2×C4D4×D8311yescharacteristic
230C2×C4D4×D8311yescharacteristic
231C2×C4C23×D8311yescharacteristic
232C8—22no—
233C8—1662no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—832no—
238C22—1661no—
239C22—1661no—
240C22C2×D4×D8311yescharacteristic
241C4—21no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—3322no—
246C4—3321no—
247C4—3321no—
248C4—1661no—
249C4—1661no—
250C4—1661no—
251C4#1458183912yes—
252C2—82no—
253C2—1662no—
254C2#1912827111yescharacteristic
255C2AGL(1,ℤ/8)×D8311yescharacteristic
256C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#490×C83C2
socleC22×C83C22×D4
prime radical#19128249C2
hypercenter#490D83
hypocenterC83#3234
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #125×D83C2semidirect
3. #19128111C2semidirect
4. #19128110C2semidirect
5. #19128065C2semidirect
6. #19128018C2semidirect
7. #19128013C2semidirect
8. #19128003C2semidirect
9. #19127999C2semidirect
10. #19127756C2semidirect
11. #490×C83C2semidirect
129. C83#3234semidirect
135. #490D83semidirect

Maximal subgroups

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

subgroupindex
#191282492
#125×D832
#191281112
#191281102
#191280652
#191280182
#191280132
#191280032
#191279992
#191277562
#490×C832
#323483