Finite Group Databasebeta

← #24919867

Subgroups — #24919867

Order 10624.

366 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128401C212yesmaximal, complemented
2#19128103C211yesmaximal, characteristic, complemented
3#19128024C212yesmaximal, complemented
4#19128018C212yesmaximal, complemented
5#19128001C212yesmaximal, complemented
6#19127934C211yesmaximal, characteristic, complemented
7#19127934C211yesmaximal, characteristic, complemented
8C8×D332C211yesmaximal, characteristic, complemented
9#19127738C211yesmaximal, characteristic, complemented
10#107×D83C211yesmaximal, characteristic, complemented
11C4×D8×C83C211yesmaximal, characteristic, complemented
12C2×C8×D83C2211yescharacteristic
13C2×C8×D83C2211yescharacteristic
14#14581785—22no—
15#14581784C2212yes—
16#14581775—22no—
17#14581765—22no—
18#14581763C2212yes—
19#14581763C2212yes—
20C4×D332C2211yescharacteristic
21C4×C22⋊D83—22no—
22#14581750C2212yes—
23C4⋊C4×D83C2212yes—
24#14581726C2211yescharacteristic
25#14581726C2211yescharacteristic
26#14581725C2212yes—
27#14581724C2212yes—
28#14581723C2212yes—
29C4⋊(C83⋊C8)C2211yescharacteristic
30#14581713C2212yes—
31#14581713C2212yes—
32(C2×D83)⋊C8C2211yescharacteristic
33(C2×D83)⋊C8C2211yescharacteristic
34C2×D8×C83C2211yescharacteristic
35C4×D4×C83C2212yes—
36#107×C83C2211yescharacteristic
37#102×C83C2212yes—
38C4×C8×C83C2211yescharacteristic
39C2×D332D411yescharacteristic
40C2×C22⋊D83—22no—
41C2×C22⋊D83—22no—
42C2×C4×D83—22no—
43C2×C4×D83C2311yescharacteristic
44C2×C4×D83C2311yescharacteristic
45C2×C83⋊C8C2311yescharacteristic
46C2×C83⋊C8C2311yescharacteristic
47C8×D83—22no—
48C8×D83—22no—
49C4⋊Dic83D411yescharacteristic
50C4⋊Dic83C2312yes—
51C4⋊Dic83C2312yes—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55C22⋊Dic83—22no—
56C22⋊Dic83—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4D411yescharacteristic
59(C2×D83)⋊C4D411yescharacteristic
60C4×Dic83—22no—
61C2×D4×C83C2312yes—
62C22×C4×C83—22no—
63D8×C83—22no—
64C2×C8×C83C2311yescharacteristic
65C2×C8×C83C2311yescharacteristic
66C4⋊C4×C83C2312yes—
67C22⋊C4×C83—22no—
68C42×C83C2311yescharacteristic
69C23×C83—22no—
70C22×D83C2×D411yescharacteristic
71C22×D83C2×D411yescharacteristic
72D4×C83—22no—
73D4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—82no—
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2411yescharacteristic
79C22⋊D83—44no—
80C22⋊D83—82no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83C2×D411yescharacteristic
84C2×Dic83C2×D411yescharacteristic
85D332—22no—
86C4×D83—21no—
87C4×D83—21no—
88C4×D83—21no—
89C4×D83—21no—
90C4×D83—82no—
91C8×C83—21no—
92C8×C83#4112yes—
93C83⋊C8—21no—
94C83⋊C8—21no—
95C22×C83—42no—
96C22×C83—42no—
97C22×C83—42no—
98C22×C83C22×D411yescharacteristic
99C2×D83—21no—
100C2×D83—21no—
101C2×D83—21no—
102C2×D83—21no—
103C2×D83—21no—
104C2×D83#13512yes—
105C4×C83—21no—
106C4×C83—21no—
107C4×C83—42no—
108C4×C83C2×#4111yescharacteristic
109C4×C832−1+411yescharacteristic
110Dic83—21no—
111Dic83—21no—
112Dic83—42no—
113Dic83—42no—
114C2×C83—82no—
115C2×C83#54711yescharacteristic
116C2×C83C2×#13511yescharacteristic
117C2×C83#57611yescharacteristic
118D83—22no—
119D83—41no—
120#3339—831nomaximal, Sylow
121C83#333911yescharacteristic, complemented, Sylow
122#553—832no—
123#499—831no—
124#482—832no—
125#481—832no—
126#478—832no—
127#466—831no—
128#466—831no—
129C4×D8D8311yescharacteristic, complemented
130C8×D4—831no—
131#427—831no—
132C2×#107—831no—
133C2×D8C2×D8311yescharacteristic
134C22×C8—831no—
135C22×C8—831no—
136#126—1662no—
137#125—832no—
138#123—1662no—
139#121—832no—
140#121—832no—
141#121—1662no—
142C4×D4—831no—
143C4×D4—1662no—
144C4×D4C2×D8312yes—
145#117—832no—
146C2×C4⋊C4—832no—
147#107—831no—
148#107—831no—
149#107—832no—
150#107C2×D8311yescharacteristic
151#106—832no—
152#106—832no—
153C4⋊C8—831no—
154#102—832no—
155#102—832no—
156#102C2×D8312yes—
157C22⋊C8—831no—
158C22⋊C8—831no—
159C4×C8C2×D8311yescharacteristic
160C2×D4—831no—
161C2×D4—1662no—
162C2×D4—1662no—
163C2×D4C22×D8312yes—
164C22×C4—22no—
165C22×C4—831no—
166C22×C4—831no—
167C22×C4—1662no—
168D8—22no—
169C2×C8—831no—
170C2×C8—831no—
171C2×C8—1662no—
172C2×C8—1662no—
173C2×C8C22×D8311yescharacteristic
174C2×C8C22×D8311yescharacteristic
175C4⋊C4—831no—
176C4⋊C4—832no—
177C4⋊C4—832no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C4⋊C4—1662no—
181C4⋊C4C22×D8312yes—
182C22⋊C4—22no—
183C22⋊C4—831no—
184C22⋊C4—831no—
185C22⋊C4—1662no—
186C22⋊C4—1662no—
187C22⋊C4—1662no—
188C42—1662no—
189C42C22×D8311yescharacteristic
190C23—22no—
191C23—831no—
192C23—831no—
193D4—22no—
194D4—22no—
195D4—3324no—
196D4—6642no—
197D4—1662no—
198C2×C4—22no—
199C2×C4—82no—
200C2×C4—831no—
201C2×C4—831no—
202C2×C4—6642no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1661no—
206C2×C4—1661no—
207C2×C4—1661no—
208C2×C4—1661no—
209C2×C4D4×D8311yescharacteristic
210C2×C4D4×D8311yescharacteristic
211C2×C4C23×D8311yescharacteristic
212C8—21no—
213C8—1661no—
214C8—1661no—
215C8#1305059012yes—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—832no—
220C22—1661no—
221C22—1661no—
222C22—1661no—
223C22—1661no—
224C22—1661no—
225C22C2×D4×D8311yescharacteristic
226C4—21no—
227C4—21no—
228C4—42no—
229C4—3322no—
230C4—3322no—
231C4—1661no—
232C4—1661no—
233C4C2×#1305059011yescharacteristic
234C4#1458183911yescharacteristic
235C2—82no—
236C2—3321no—
237C2—1662no—
238C2#1912835611yescharacteristic
239C2#135×D8311yescharacteristic
240C2#1912850211yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×D8×C83C2
socleC22×C83C22×D4
hypercenterC4×D8D83
hypocenterC83#3339
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #19128401C2semidirect
2. #19128103C2semidirect
3. #19128024C2semidirect
4. #19128018C2semidirect
5. #19128001C2semidirect
6. #19127934C2semidirect
7. #19127934C2semidirect
8. C8×D332C2semidirect
9. #19127738C2semidirect
10. #107×D83C2semidirect
11. C4×D8×C83C2semidirect
121. C83#3339semidirect
129. C4×D8D83semidirect

Maximal subgroups

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

subgroupindex
#191284012
#191281032
#191280242
#191280182
#191280012
#191279342
C8×D3322
#191277382
#107×D832
C4×D8×C832
#333983