Finite Group Databasebeta

← #24917116

Subgroups — #24917116

Order 10624.

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

normal subgroups only

This group has 460 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×Q16×D83C211yesmaximal, characteristic, complemented
2C2×#14581800C211yesmaximal, characteristic
3#19128274C212yesmaximal
4#19127960C211yesmaximal, characteristic, complemented
5#19127959C211yesmaximal, characteristic, complemented
6#19127956C211yesmaximal, characteristic, complemented
7#19127941C212yesmaximal, complemented
8#19127940C212yesmaximal, complemented
9#19127736C211yesmaximal, characteristic, complemented
10#19127708C212yesmaximal, complemented
11#470×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C2×Q8×D83C2212yes—
14C2×Dic332C2211yescharacteristic
15C2×#13050580C2212yes—
16Q16×D83—24no—
17#14581800—24no—
18C2×#13050573C2211yescharacteristic
19C2×#13050573C2211yescharacteristic
20C22×C83⋊C8C2211yescharacteristic
21C2×C8×D83C2211yescharacteristic
22C2×C8×D83C2211yescharacteristic
23#14581791—22no—
24#14581778—22no—
25#14581777—22no—
26#14581774—22no—
27#14581772C2212yes—
28#14581771C2212yes—
29#14581768C2212yes—
30#14581750C2212yes—
31#14581725C2212yes—
32#14581724C2211yescharacteristic
33#14581717C2212yes—
34#14581716C2212yes—
35#14581715C2212yes—
36C2×Q16×C83C2211yescharacteristic
37C2×M16×C83C2211yescharacteristic
38C22⋊Q8×C83C2212yes—
39#106×C83C2211yescharacteristic
40#103×C83C2212yes—
41#13050590—22no—
42#13050590—22no—
43C2×Dic166—22no—
44C2×Dic166D411yescharacteristic
45C2×Dic166C2312yes—
46Q8×D83—24no—
47Q8×D83—24no—
48C2×D332D411yescharacteristic
49C2×C22⋊D83D411yescharacteristic
50C2×C22⋊D83D411yescharacteristic
51C2×C4×D83—22no—
52C2×C4×D83C2311yescharacteristic
53C2×C4×D83C2311yescharacteristic
54Dic332—22no—
55#13050580—24no—
56#13050573—22no—
57#13050573—22no—
58C2×C83⋊C8—22no—
59C2×C83⋊C8C2311yescharacteristic
60C2×C83⋊C8C2311yescharacteristic
61C8×D83—22no—
62C8×D83D414yes—
63C4⋊Dic83—22no—
64C4⋊Dic83C2312yes—
65Dic83⋊C4—22no—
66Dic83⋊C4—22no—
67C22⋊Dic83—22no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—22no—
71C4×Dic83—22no—
72C2×Q8×C83C2312yes—
73C22×C4×C83C2311yescharacteristic
74Q16×C83—22no—
75M16×C83—22no—
76C2×C8×C83C2311yescharacteristic
77C2×C8×C83C2311yescharacteristic
78C4⋊C4×C83—22no—
79C4⋊C4×C83C2312yes—
80C22⋊C4×C83—22no—
81C23×C83C2×D411yescharacteristic
82C22×D83C2×D411yescharacteristic
83C22×D83C2×D411yescharacteristic
84Q8×C83—22no—
85Q8×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2411yescharacteristic
92C22⋊D83—22no—
93C22⋊D83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83C2×D411yescharacteristic
97C2×Dic83C2×D411yescharacteristic
98D332—22no—
99C4×D83—21no—
100C4×D83—21no—
101C4×D83—44no—
102C4×D83C2×D412yes—
103C4×D83#4112yes—
104Dic166—22no—
105Dic166—22no—
106Dic166—22no—
107Dic166—44no—
108C8×C83—21no—
109C8×C83C2×D412yes—
110C83⋊C8—21no—
111C83⋊C8C2×D412yes—
112C22×C83—21no—
113C22×C83—22no—
114C22×C83C22×D411yescharacteristic
115C2×D83—22no—
116C2×D83—21no—
117C2×D83—22no—
118C2×D83#12112yes—
119C4×C83—21no—
120C4×C83—42no—
121C4×C83—42no—
122C4×C83C22×D411yescharacteristic
123C4×C83C2×#4111yescharacteristic
124Dic83—21no—
125Dic83—42no—
126Dic83—42no—
127Dic83#12112yes—
128C2×C83—41no—
129C2×C83C2×#12111yescharacteristic
130C2×C83#57812yes—
131D83—22no—
132D83—41no—
133#3059—831nomaximal, Sylow
134C83#305911yescharacteristic, complemented, Sylow
135C22×Q16—831no—
136C2×#131—831no—
137#536—832no—
138#470—831no—
139#470—831no—
140#470—831no—
141#470D8311yescharacteristic, complemented
142#467—832no—
143#467—832no—
144#427—831no—
145#419—832no—
146C2×#41—831no—
147C22×Q8—832no—
148C2×Q16—832no—
149C2×Q16—831no—
150C2×Q16—1664no—
151C2×Q16C2×D8311yescharacteristic
152#131—1664no—
153C2×M16—831no—
154C2×M16—831no—
155C2×M16C2×D8311yescharacteristic
156C22×C8—831no—
157C22×C8—831no—
158C22×C8—831no—
159C4⋊Q8—1662no—
160#124—1662no—
161#123—1662no—
162#123—1662no—
163C22⋊Q8—832no—
164C22⋊Q8—832no—
165C22⋊Q8—832no—
166C22⋊Q8C2×D8312yes—
167#117—832no—
168#107—832no—
169#106—831no—
170#106C2×D8311yescharacteristic
171#103—832no—
172#103—832no—
173#103—832no—
174#103C2×D8312yes—
175#41—1662no—
176#41—1662no—
177C2×Q8—831no—
178C2×Q8—832no—
179C2×Q8—1662no—
180C2×Q8—1664no—
181C2×Q8—1664no—
182C2×Q8C22×D8312yes—
183C2×D4—831no—
184C2×D4—831no—
185C2×D4—831no—
186C22×C4—831no—
187C22×C4—831no—
188C22×C4—1662no—
189C22×C4C22×D8311yescharacteristic
190Q16—22no—
191Q16—1664no—
192Q16—1662no—
193M16—22no—
194M16—1662no—
195M16—1662no—
196C2×C8—831no—
197C2×C8—834no—
198C2×C8—831no—
199C2×C8—1662no—
200C2×C8—1662no—
201C2×C8C22×D8311yescharacteristic
202C2×C8C22×D8311yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—832no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C4⋊C4—1662no—
208C4⋊C4C22×D8312yes—
209C22⋊C4—22no—
210C22⋊C4—1662no—
211C22⋊C4—1662no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C42—1662no—
215C23—831no—
216C23—831no—
217C23D4×D8311yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—3324no—
221Q8—1662no—
222Q8—1662no—
223Q8—1662no—
224D4—1662no—
225D4—1662no—
226D4—1662no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—3324no—
232C2×C4—831no—
233C2×C4—832no—
234C2×C4—832no—
235C2×C4—831no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1661no—
239C2×C4—1661no—
240C2×C4D4×D8311yescharacteristic
241C2×C4C23×D8311yescharacteristic
242C8—21no—
243C8—832no—
244C8—1661no—
245C8D4×D8312yes—
246C22—21no—
247C22—22no—
248C22—832no—
249C22—1661no—
250C22—1662no—
251C22—1662no—
252C22C2×D4×D8311yescharacteristic
253C4—21no—
254C4—42no—
255C4—42no—
256C4—3322no—
257C4—3322no—
258C4—832no—
259C4—1661no—
260C4C2×D4×D8311yescharacteristic
261C4#41×D8311yescharacteristic
262C2—41no—
263C2—3321no—
264C2—1662no—
265C2#121×D8311yescharacteristic
266C2#1912851712yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#470×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050590C22
hypercenter#470D83
hypocenterC83#3059
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D83C2semidirect
4. #19127960C2semidirect
5. #19127959C2semidirect
6. #19127956C2semidirect
7. #19127941C2semidirect
8. #19127940C2semidirect
9. #19127736C2semidirect
10. #19127708C2semidirect
11. #470×C83C2semidirect
134. C83#3059semidirect
141. #470D83semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D832
C2×#145818002
#191282742
#191279602
#191279592
#191279562
#191279412
#191279402
#191277362
#191277082
#470×C832
#305983