Finite Group Databasebeta

← #24919767

Subgroups — #24919767

Order 10624.

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

normal subgroups only

This group has 420 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#19128354C211yesmaximal, characteristic, complemented
2#19128354C211yesmaximal, characteristic, complemented
3#19128035C212yesmaximal, complemented
4#19127978C211yesmaximal, characteristic, complemented
5#19127950C212yesmaximal, complemented
6#19127903C211yesmaximal, characteristic, complemented
7#19127849C212yesmaximal, complemented
8C4×D664C211yesmaximal, characteristic, complemented
9#19127780C211yesmaximal, characteristic, complemented
10#19127701C212yesmaximal, complemented
11#487×C83C211yesmaximal, characteristic, complemented
12C2×#13050591—21no—
13C2×#13050591C2212yes—
14C2×#13050577C2212yes—
15C2×D664C2211yescharacteristic
16C2×#13050573C2212yes—
17#14581777—22no—
18#14581777—22no—
19#14581768—21no—
20#14581768C2212yes—
21#14581767—21no—
22#14581767C2212yes—
23C4×D332C2211yescharacteristic
24C4×D332C2211yescharacteristic
25C4×D332C2211yescharacteristic
26C4⋊C4×D83—21no—
27C4⋊C4×D83C2212yes—
28#14581726C2211yescharacteristic
29#14581725C2212yes—
30C4⋊(C83⋊C8)C2211yescharacteristic
31#14581716C2212yes—
32#14581714C2211yescharacteristic
33#14581714C2211yescharacteristic
34#14581712C2212yes—
35(C2×D83)⋊C8C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4⋊Q8×C83C2211yescharacteristic
38#103×C83C2212yes—
39#103×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41#13050591—24no—
42#13050591—42no—
43#13050591—44no—
44C2×D332—22no—
45C2×D332—22no—
46C2×D332D411yescharacteristic
47C2×D332C2311yescharacteristic
48C2×D332C2311yescharacteristic
49C2×C4×D83—22no—
50C2×C4×D83—21no—
51C2×C4×D83—21no—
52C2×C4×D83—22no—
53C2×C4×D83C2312yes—
54#13050577—42no—
55D664—22no—
56#13050573—42no—
57C2×C83⋊C8C2312yes—
58C4⋊Dic83D411yescharacteristic
59C4⋊Dic83C2311yescharacteristic
60C4⋊Dic83C2311yescharacteristic
61Dic83⋊C4—22no—
62Dic83⋊C4—22no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4D412yes—
68C2×Q8×C83—21no—
69C2×Q8×C83C2312yes—
70C2×C8×C83C2311yescharacteristic
71C2×C8×C83C2311yescharacteristic
72C4⋊C4×C83—21no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2312yes—
76C42×C83C2311yescharacteristic
77C22×D83—21no—
78C22×D83—21no—
79C22×D83C2×D412yes—
80Q8×C83—22no—
81Q8×C83—22no—
82Q8×C83—42no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2411yescharacteristic
88C2×Dic83—21no—
89C2×Dic83—21no—
90C2×Dic83C2×D412yes—
91D332—22no—
92D332—21no—
93D332—21no—
94D332—21no—
95D332—44no—
96D332—44no—
97D332#4112yescomplemented
98C4×D83—22no—
99C4×D83—22no—
100C4×D83—44no—
101C4×D83—44no—
102C4×D83—42no—
103C4×D83—42no—
104C8×C83—21no—
105C8×C83—21no—
106C83⋊C8—22no—
107C22×C83C22×D411yescharacteristic
108C2×D83—22no—
109C2×D83—22no—
110C2×D83—22no—
111C2×D83—22no—
112C2×D83—41no—
113C2×D83—41no—
114C2×D83—41no—
115C2×D83—41no—
116C2×D83—41no—
117C4×C83—21no—
118C4×C83—42no—
119C4×C83—42no—
120C4×C83AGL(1,ℤ/8)12yes—
121C4×C83C2×#4111yescharacteristic
122C4×C83H(5,𝔽2)11yescharacteristic
123Dic83—22no—
124Dic83—41no—
125Dic83—41no—
126C2×C83#54611yescharacteristic
127C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C83#57811yescharacteristic
129D83—42no—
130D83—42no—
131D83—81no—
132#3330—831nomaximal, Sylow
133C83#333011yescharacteristic, complemented, Sylow
134#547—831no—
135#547—831no—
136#487D8311yescharacteristic, complemented
137#484—832no—
138#474—831no—
139#468—832no—
140#461—831no—
141#449—832no—
142C4×D8—831no—
143#436—831no—
144#417—832no—
145C2×#41—832no—
146C2×#41—1661no—
147C2×SD16—832no—
148C2×D8—831no—
149C2×M16—832no—
150C4⋊Q8C2×D8311yescharacteristic
151C4⋊Q8C2×D8311yescharacteristic
152#123—1662no—
153#123—1662no—
154C22⋊Q8—832no—
155C22⋊Q8—1661no—
156#121—832no—
157#121—1661no—
158C4×D4—831no—
159C4×D4—831no—
160C4×D4—831no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4—1661no—
163#107—831no—
164#107—832no—
165C4⋊C8—831no—
166#103—832no—
167#103C2×D8312yes—
168#103C2×D8312yes—
169#102—831no—
170#102—831no—
171#102—832no—
172C22⋊C8—832no—
173C4×C8C2×D8311yescharacteristic
174#41—3324no—
175#41—3322no—
176#41—1664no—
177C2×Q8—21no—
178C2×Q8C22×D8312yes—
179C2×D4—831no—
180C2×D4—831no—
181C2×D4—831no—
182C2×D4—1662no—
183C2×D4—1662no—
184C22×C4—832no—
185C22×C4—1662no—
186C22×C4—1662no—
187C22×C4—1661no—
188C22×C4—1661no—
189SD16—3322no—
190D8—1662no—
191M16—3322no—
192C2×C8—832no—
193C2×C8C22×D8311yescharacteristic
194C2×C8C22×D8311yescharacteristic
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—831no—
199C4⋊C4—831no—
200C4⋊C4—831no—
201C4⋊C4—1662no—
202C4⋊C4—1662no—
203C4⋊C4C22×D8312yes—
204C22⋊C4—832no—
205C22⋊C4—1662no—
206C22⋊C4—1662no—
207C22⋊C4—1661no—
208C22⋊C4—1661no—
209C42C22×D8311yescharacteristic
210C23—832no—
211C23—1661no—
212C23—1661no—
213Q8—22no—
214Q8—22no—
215Q8—42no—
216D4—3324no—
217D4—3324no—
218D4—832no—
219D4—1662no—
220D4—1661no—
221D4—1661no—
222D4—1661no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—3324no—
226C2×C4—3324no—
227C2×C4—3322no—
228C2×C4—3322no—
229C2×C4—832no—
230C2×C4—1661no—
231C2×C4—1661no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4D4×D8311yescharacteristic
235C2×C4D4×D8311yescharacteristic
236C2×C4C23×D8311yescharacteristic
237C8—21no—
238C8—21no—
239C8—1662no—
240C22—3321no—
241C22—3321no—
242C22—3321no—
243C22—3321no—
244C22—3321no—
245C22—1662no—
246C22—1662no—
247C22—1662no—
248C22—1662no—
249C22C2×D4×D8311yescharacteristic
250C4—21no—
251C4—42no—
252C4—42no—
253C4—3321no—
254C4—3321no—
255C4—1662no—
256C4#1458181912yes—
257C4#41×D8311yescharacteristic
258C4#1458183811yescharacteristic
259C2—3322no—
260C2—3322no—
261C2—6641no—
262C2#1912834411yescharacteristic
263C2C2×#1458181911yescharacteristic
264C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#487×C83C2
socleC22×C83C22×D4
hypercenter#487D83
hypocenterC83#3330
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
1. #19128354C2semidirect
2. #19128354C2semidirect
3. #19128035C2semidirect
4. #19127978C2semidirect
5. #19127950C2semidirect
6. #19127903C2semidirect
7. #19127849C2semidirect
8. C4×D664C2semidirect
9. #19127780C2semidirect
10. #19127701C2semidirect
11. #487×C83C2semidirect
97. D332#41semidirect
133. C83#3330semidirect
136. #487D83semidirect

Maximal subgroups

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

subgroupindex
#191283542
#191280352
#191279782
#191279502
#191279032
#191278492
C4×D6642
#191277802
#191277012
#487×C832
#333083