Finite Group Databasebeta

← #24917123

Subgroups — #24917123

Order 10624.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#14581811C211yesmaximal, characteristic, complemented
2C2×#14581800C211yesmaximal, characteristic, complemented
3#19128280C212yesmaximal, complemented
4#19127960C211yesmaximal, characteristic, complemented
5#19127959C211yesmaximal, characteristic, complemented
6#19127951C211yesmaximal, characteristic, complemented
7#19127929C212yesmaximal, complemented
8#19127928C212yesmaximal, complemented
9#106×D83C211yesmaximal, characteristic, complemented
10#19127699C212yesmaximal, complemented
11#469×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×#13050589C2212yes—
14#14581811—24no—
15C2×Dic332C2211yescharacteristic
16C2×#13050578C2212yes—
17#14581800—24no—
18C2×#13050573C2211yescharacteristic
19C2×#13050573C2211yescharacteristic
20C22×C83⋊C8C2211yescharacteristic
21C2×C8×D83C2211yescharacteristic
22C2×C8×D83C2211yescharacteristic
23#14581772C2212yes—
24#14581771C2212yes—
25#14581769—22no—
26#14581765—22no—
27#14581764—22no—
28#14581763—22no—
29#14581763C2212yes—
30C4⋊C4×D83C2212yes—
31#14581724C2211yescharacteristic
32#14581723C2212yes—
33#14581717C2212yes—
34#14581715C2212yes—
35#14581713C2212yes—
36C2×D8×C83C2211yescharacteristic
37C2×M16×C83C2211yescharacteristic
38#121×C83C2212yes—
39#106×C83C2211yescharacteristic
40#102×C83C2212yes—
41#13050590—22no—
42#13050590—22no—
43#13050589—24no—
44#13050589—24no—
45C2×Dic166D411yescharacteristic
46C2×Dic166C2312yes—
47C2×D332D411yescharacteristic
48C2×C22⋊D83—22no—
49C2×C22⋊D83—22no—
50C2×C22⋊D83D411yescharacteristic
51C2×C22⋊D83D411yescharacteristic
52C22×Dic83—22no—
53C2×C4×D83—22no—
54C2×C4×D83C2311yescharacteristic
55C2×C4×D83C2311yescharacteristic
56Dic332—22no—
57#13050578—24no—
58#13050573—22no—
59#13050573—22no—
60C2×C83⋊C8—22no—
61C2×C83⋊C8C2311yescharacteristic
62C2×C83⋊C8C2311yescharacteristic
63C8×D83—22no—
64C8×D83D414yescomplemented
65C4⋊Dic83—22no—
66C4⋊Dic83C2312yes—
67Dic83⋊C4—22no—
68Dic83⋊C4—22no—
69C22⋊Dic83—22no—
70C22⋊Dic83—22no—
71(C2×D83)⋊C4—22no—
72C2×D4×C83—22no—
73C2×D4×C83C2312yes—
74C22×C4×C83C2311yescharacteristic
75D8×C83—22no—
76M16×C83—22no—
77C2×C8×C83C2311yescharacteristic
78C2×C8×C83C2311yescharacteristic
79C4⋊C4×C83C2312yes—
80C22⋊C4×C83—22no—
81C23×C83—22no—
82C23×C83C2×D411yescharacteristic
83C22×D83C2×D411yescharacteristic
84C22×D83C2×D411yescharacteristic
85D4×C83—22no—
86D4×C83—22no—
87D4×C83—82no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—22no—
94C22⋊D83—22no—
95C22⋊D83—44no—
96C22⋊D83—82no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—44no—
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102D332—22no—
103C4×D83—21no—
104C4×D83—21no—
105C4×D83—82no—
106C4×D83C2×D412yes—
107C4×D83#4112yes—
108Dic166—22no—
109Dic166—22no—
110Dic166—22no—
111C8×C83—21no—
112C8×C83C2×D412yes—
113C83⋊C8—21no—
114C83⋊C8C2×D412yes—
115C22×C83—21no—
116C22×C83—22no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83C22×D411yescharacteristic
121C2×D83—22no—
122C2×D83—21no—
123C2×D83—22no—
124C2×D83#12112yes—
125C4×C83—21no—
126C4×C83—42no—
127C4×C83C22×D411yescharacteristic
128C4×C83C2×#4111yescharacteristic
129Dic83—21no—
130Dic83—42no—
131Dic83—42no—
132Dic83#12112yes—
133C2×C83—41no—
134C2×C83—82no—
135C2×C83C2×#12111yescharacteristic
136C2×C83#57712yes—
137D83—22no—
138D83—41no—
139#3060—831nomaximal, Sylow
140C83#306011yescharacteristic, complemented, Sylow
141C2×#135—831no—
142C2×#131—831no—
143#537—832no—
144#470—831no—
145#470—831no—
146#469—831no—
147#469D8311yescharacteristic, complemented
148#465—832no—
149#465—832no—
150C2×#106—831no—
151#417—832no—
152C2×#41—831no—
153C2×#41—832no—
154#135—1664no—
155C2×Q16—831no—
156C2×SD16—832no—
157C2×D8C2×D8311yescharacteristic
158#131—1664no—
159C2×M16—831no—
160C2×M16—831no—
161C2×M16C2×D8311yescharacteristic
162C22×C8—831no—
163C22×C8—831no—
164C22×C8—831no—
165C22⋊Q8—832no—
166C22⋊Q8—832no—
167C22⋊Q8—1662no—
168#121—832no—
169#121—1662no—
170#121—1662no—
171#121—1662no—
172#121C2×D8312yes—
173C2×C4⋊C4—832no—
174#106—831no—
175#106—832no—
176#106C2×D8311yescharacteristic
177#103—832no—
178#103—832no—
179#102—832no—
180#102C2×D8312yes—
181#41—1662no—
182#41—1662no—
183#41—1664no—
184#41—1664no—
185C2×Q8—831no—
186C2×Q8—832no—
187C2×D4—22no—
188C2×D4—831no—
189C2×D4—831no—
190C2×D4—831no—
191C2×D4—1662no—
192C2×D4—1662no—
193C2×D4C22×D8312yes—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4—1662no—
197C22×C4—1662no—
198C22×C4C22×D8311yescharacteristic
199Q16—1662no—
200SD16—1664no—
201D8—22no—
202M16—22no—
203M16—1662no—
204M16—1662no—
205C2×C8—831no—
206C2×C8—834no—
207C2×C8—831no—
208C2×C8—1662no—
209C2×C8—1662no—
210C2×C8C22×D8311yescharacteristic
211C2×C8C22×D8311yescharacteristic
212C4⋊C4—832no—
213C4⋊C4—1662no—
214C4⋊C4—1662no—
215C4⋊C4—1662no—
216C4⋊C4C22×D8312yes—
217C22⋊C4—22no—
218C22⋊C4—1662no—
219C22⋊C4—1662no—
220C22⋊C4—1662no—
221C23—22no—
222C23—831no—
223C23—831no—
224C23D4×D8311yescharacteristic
225Q8—1662no—
226Q8—1662no—
227Q8—1662no—
228D4—22no—
229D4—22no—
230D4—82no—
231D4—3324no—
232D4—6642no—
233D4—1662no—
234D4—1662no—
235D4—1662no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—3324no—
240C2×C4—831no—
241C2×C4—832no—
242C2×C4—832no—
243C2×C4—831no—
244C2×C4—6642no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1661no—
248C2×C4—1661no—
249C2×C4D4×D8311yescharacteristic
250C2×C4C23×D8311yescharacteristic
251C8—21no—
252C8—832no—
253C8—1661no—
254C8D4×D8312yes—
255C22—21no—
256C22—22no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—832no—
261C22—1661no—
262C22—1662no—
263C22—1662no—
264C22C2×D4×D8311yescharacteristic
265C4—21no—
266C4—42no—
267C4—3322no—
268C4—3322no—
269C4—832no—
270C4—1661no—
271C4C2×D4×D8311yescharacteristic
272C4#41×D8311yescharacteristic
273C2—41no—
274C2—82no—
275C2—3321no—
276C2—1662no—
277C2#121×D8311yescharacteristic
278C2#1912851112yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#469×C83C2
socleC22×C83C22×D4
hypercenter#469D83
hypocenterC83#3060
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#14581811C2semidirect
2. C2×#14581800C2semidirect
3. #19128280C2semidirect
4. #19127960C2semidirect
5. #19127959C2semidirect
6. #19127951C2semidirect
7. #19127929C2semidirect
8. #19127928C2semidirect
9. #106×D83C2semidirect
10. #19127699C2semidirect
11. #469×C83C2semidirect
64. C8×D83D4semidirect
140. C83#3060semidirect
147. #469D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818112
C2×#145818002
#191282802
#191279602
#191279592
#191279512
#191279292
#191279282
#106×D832
#191276992
#469×C832
#306083