Finite Group Databasebeta

← #24917601

Subgroups — #24917601

Order 10624.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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#19128235C211yesmaximal, characteristic, complemented
2#19128031C211yesmaximal, characteristic, complemented
3#19128014C211yesmaximal, characteristic, complemented
4#19128001C212yesmaximal, complemented
5#19127916C212yesmaximal, complemented
6#19127852C211yesmaximal, characteristic, complemented
7#19127835C211yesmaximal, characteristic
8#19127811C212yesmaximal, complemented
9#19127801C212yesmaximal, complemented
10#19127758C211yesmaximal, characteristic, complemented
11C2×C4×D4×C83C211yesmaximal, characteristic, complemented
12C2×#13050578C2212yes—
13C2×#13050575C2212yes—
14#14581786—21no—
15#14581784C2211yescharacteristic, complemented
16#14581780C2211yescharacteristic, complemented
17#14581775—21no—
18#14581771C2211yescharacteristic
19#14581766C2211yescharacteristic
20C4×Dic166C2211yescharacteristic, complemented
21C4×D332C2211yescharacteristic, complemented
22C4×C22⋊D83—21no—
23#14581749—21no—
24#14581725C2212yescomplemented
25#14581723C2212yescomplemented
26C4⋊(C83⋊C8)C2212yescomplemented
27#14581715C2212yescomplemented
28#14581713C2212yes—
29#14581713C2212yes—
30#14581712C2212yescomplemented
31C22⋊(C83⋊C8)C2211yescharacteristic
32C22⋊(C83⋊C8)C2211yescharacteristic
33#14581703C2212yescomplemented
34C22×D4×C83C2211yescharacteristic
35C23×C4×C83—21no—
36C4×D4×C83—21no—
37C4×D4×C83—21no—
38C4×D4×C83C2212yes—
39C4×D4×C83C2212yes—
40C2×C4⋊C4×C83C2211yescharacteristic
41C2×C22⋊C4×C83—21no—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83—21no—
47#13050578—24no—
48#13050575—24no—
49C2×C83⋊C8C2312yes—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83C2311yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53Dic83⋊C4—21no—
54Dic83⋊C4—21no—
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57C22⋊Dic83—21no—
58(C2×D83)⋊C4—21no—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62C4×Dic83—21no—
63C24×C83—21no—
64C2×D4×C83—24no—
65C2×D4×C83—21no—
66C2×D4×C83C2312yes—
67C22×C4×C83—21no—
68C22×C4×C83—22no—
69C22×C4×C83—22no—
70C22×C4×C83—44no—
71C22×C4×C83D411yescharacteristic
72C22×C4×C83D411yescharacteristic
73C22×C4×C83C2311yescharacteristic
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2312yes—
76C22⋊C4×C83—22no—
77C22⋊C4×C83—22no—
78C42×C83D411yescharacteristic
79C42×C83D411yescharacteristic
80C42×C83C2311yescharacteristic
81C42×C83C2311yescharacteristic
82C23×C83—24no—
83C23×C83—22no—
84C23×C83—42no—
85C23×C83—42no—
86C23×C83C2×D411yescharacteristic
87C22×D83—21no—
88D4×C83—24no—
89D4×C83—22no—
90D4×C83#4114yes—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—41no—
95C2×C4×C83—44no—
96C2×C4×C83—44no—
97C2×C4×C83—42no—
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2×D411yescharacteristic
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—42no—
105C2×Dic83—21no—
106C2×Dic83—21no—
107C2×Dic83—21no—
108D332—22no—
109C4×D83—42no—
110Dic166—22no—
111C83⋊C8—22no—
112C83⋊C8—22no—
113C22×C83—21no—
114C22×C83—21no—
115C22×C83—24no—
116C22×C83—48no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83AGL(1,ℤ/8)12yes—
120C22×C83C22×D411yescharacteristic
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—41no—
124C4×C83—21no—
125C4×C83—21no—
126C4×C83—21no—
127C4×C83—21no—
128C4×C83—21no—
129C4×C83—42no—
130C4×C83C2×#4112yes—
131Dic83—41no—
132Dic83—41no—
133Dic83—41no—
134C2×C83—22no—
135C2×C83—44no—
136C2×C83#52511yescharacteristic
137C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C83#57511yescharacteristic
139D83—81no—
140#3114—831nomaximal, Sylow
141C83#311411yescharacteristic, complemented, Sylow
142#529—831no—
143C2×C4×D4D8311yescharacteristic, complemented
144#483—831no—
145#480—831no—
146#478—832no—
147#463—832no—
148#450—831no—
149#447—831no—
150#442—832no—
151#440—832no—
152#432—831no—
153C22×D4C2×D8311yescharacteristic
154C23×C4—21no—
155C2×SD16—832no—
156C2×D8—832no—
157#126—1661no—
158#125—831no—
159#124—831no—
160#123—1661no—
161C22⋊Q8—831no—
162#121—831no—
163C4×Q8—831no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—831no—
167C4×D4—1661no—
168C4×D4C2×D8312yes—
169C4×D4C2×D8312yes—
170#117—1661no—
171C2×C4⋊C4C2×D8311yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D8311yescharacteristic
174#107—832no—
175#106—832no—
176C4⋊C8—832no—
177#103—832no—
178#102—832no—
179#102—832no—
180#102—832no—
181C22⋊C8—831no—
182C22⋊C8—831no—
183#97—832no—
184C24—21no—
185C2×Q8—831no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—831no—
189C2×D4—1661no—
190C2×D4C22×D8312yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—1661no—
196C22×C4C22⋊D8311yescharacteristic
197C22×C4C22⋊D8311yescharacteristic
198C22×C4C22×D8311yescharacteristic
199SD16—1664no—
200D8—1664no—
201C2×C8—832no—
202C2×C8—832no—
203C4⋊C4—21no—
204C4⋊C4—831no—
205C4⋊C4—831no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4C22×D8312yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—1661no—
214C22⋊C4—1661no—
215C22⋊C4—1661no—
216C22⋊C4—1661no—
217C22⋊C4—1661no—
218C42—1661no—
219C42C22⋊D8311yescharacteristic
220C42C22⋊D8311yescharacteristic
221C42C22×D8311yescharacteristic
222C42C22×D8311yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—1661no—
228C23C2×C22⋊D8311yescharacteristic
229Q8—1662no—
230D4—24no—
231D4—22no—
232D4—3322no—
233D4—1662no—
234D4#1305059014yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—3322no—
243C2×C4—1661no—
244C2×C4—1661no—
245C2×C4—1661no—
246C2×C4C2×C22⋊D8311yescharacteristic
247C2×C4C2×C22⋊D8311yescharacteristic
248C2×C4C2×C22⋊D8311yescharacteristic
249C2×C4C2×C22⋊D8311yescharacteristic
250C2×C4C2×C22⋊D8311yescharacteristic
251C2×C4C23×D8311yescharacteristic
252C8—1662no—
253C8—1662no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—3321no—
261C22—3321no—
262C22—3321no—
263C22#1458181812yes—
264C22C22×C22⋊D8311yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—3321no—
272C4—3321no—
273C4—3321no—
274C4C2×#1305059012yes—
275C2—22no—
276C2—44no—
277C2—6641no—
278C2#1912821211yescharacteristic
279C2C2×#1458181811yescharacteristic
280C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C4×D4×C83C2
socleC22×C83C22×D4
prime radical#19127835C2
hypercenterC2×C4×D4D83
hypocenterC83#3114
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #19128235C2semidirect
2. #19128031C2semidirect
3. #19128014C2semidirect
4. #19128001C2semidirect
5. #19127916C2semidirect
6. #19127852C2semidirect
8. #19127811C2semidirect
9. #19127801C2semidirect
10. #19127758C2semidirect
11. C2×C4×D4×C83C2semidirect
15. #14581784C22semidirect
16. #14581780C22semidirect
20. C4×Dic166C22semidirect
21. C4×D332C22semidirect
24. #14581725C22semidirect
25. #14581723C22semidirect
26. C4⋊(C83⋊C8)C22semidirect
27. #14581715C22semidirect
30. #14581712C22semidirect
33. #14581703C22semidirect
141. C83#3114semidirect
143. C2×C4×D4D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191280312
#191280142
#191280012
#191279162
#191278522
#191278352
#191278112
#191278012
#191277582
C2×C4×D4×C832
#311483