Finite Group Databasebeta

← #24915921

Subgroups — #24915921

Order 10624.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#14581837C211yesmaximal, characteristic
2#19128174C212yesmaximal, complemented
3#19127673C214yesmaximal, complemented
4C2×#14581707C211yesmaximal, characteristic, complemented
5C2×#14581706C212yesmaximal, complemented
6#99×D83C214yesmaximal, complemented
7C2×#99×C83C211yesmaximal, characteristic, complemented
8#14581837—24no—
9#14581837—24no—
10C2×#13050590C412yescomplemented
11C2×#13050589C412yescomplemented
12C2×#13050589C2212yes—
13C22×C22⋊D83C412yescomplemented
14C22×C22⋊D83C2211yescharacteristic
15C22×C22⋊D83C2211yescharacteristic
16C2×D4×D83C412yescomplemented
17C2×D4×D83C2212yes—
18C4×C22⋊D83—24no—
19(C22⋊D83)⋊C4—24no—
20#14581748C2214yescomplemented
21C2×C22⋊Dic83C2212yescomplemented
22C22⋊C4×D83C2214yescomplemented
23#14581707C2214yescomplemented
24#14581706C2218yescomplemented
25C22×D4×C83C2211yescharacteristic
26C2×C22⋊C4×C83C2212yescomplemented
27#99×C83C2214yescomplemented
28C23×D83C2×C411yescharacteristic, complemented
29C23×D83C2×C412yescomplemented
30C23×D83C2311yescharacteristic
31#13050590—24no—
32#13050590—24no—
33#13050589—28no—
34#13050589—28no—
35C2×Dic166C2×C412yescomplemented
36C2×D332C2×C412yescomplemented
37C2×C22⋊D83—22no—
38C2×C22⋊D83—22no—
39C2×C22⋊D83—24no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83C2×C412yes—
42C2×C22⋊D83C2×C412yes—
43C2×C22⋊D83C2×C412yescomplemented
44C2×C22⋊D83C2×C412yescomplemented
45C2×C22⋊D83C2×C412yescomplemented
46C2×C22⋊D83D414yes—
47C2×C22⋊D83D414yes—
48C2×C22⋊D83C2312yes—
49D4×D83—28no—
50D4×D83—28no—
51C22×Dic83—22no—
52C22×Dic83C2×C412yescomplemented
53C22×Dic83C2×C411yescharacteristic, complemented
54C22×Dic83C2311yescharacteristic
55C2×C4×D83—24no—
56C2×C4×D83C2×C412yescomplemented
57C2×C4×D83C2×C412yescomplemented
58Dic83⋊C4—24no—
59C22⋊Dic83—22no—
60C22⋊Dic83C2314yes—
61(C2×D83)⋊C4—24no—
62C4×Dic83—24no—
63C24×C83C2×C411yescharacteristic, complemented
64C24×C83C2311yescharacteristic
65C2×D4×C83—22no—
66C2×D4×C83—22no—
67C2×D4×C83C2×C412yescomplemented
68C2×D4×C83C2312yes—
69C22×C4×C83—22no—
70C22×C4×C83C2×C411yescharacteristic, complemented
71C22⋊C4×C83—22no—
72C22⋊C4×C83C2314yes—
73C23×C83—21no—
74C23×C83—21no—
75C23×C83—81no—
76C23×C83—81no—
77C23×C83C22×C412yes—
78C23×C83C22×C411yescharacteristic
79C23×C83C22×C411yescharacteristic
80C23×C83C2×D412yes—
81C23×C83C2×D412yes—
82C23×C83C2411yescharacteristic
83C22×D83—22no—
84C22×D83—24no—
85C22×D83—81no—
86C22×D83—82no—
87C22×D83—81no—
88C22×D83C22×C412yes—
89C22×D83C22×C412yes—
90C22×D83C2×D414yes—
91D4×C83—24no—
92D4×C83—24no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—24no—
96C2×C4×C83—82no—
97C2×C4×C83C22×C412yes—
98C22⋊D83—22no—
99C22⋊D83—22no—
100C22⋊D83—24no—
101C22⋊D83—24no—
102C22⋊D83—24no—
103C22⋊D83C22⋊C418yes—
104C22⋊D83C22⋊C414yes—
105C22⋊D83C22⋊C414yes—
106C2×Dic83—22no—
107C2×Dic83—24no—
108C2×Dic83—24no—
109C2×Dic83—82no—
110C2×Dic83C22×C412yes—
111C2×Dic83C22×C412yes—
112C2×Dic83C2×D414yes—
113D332—24no—
114C4×D83—24no—
115C4×D83—24no—
116C4×D83—84no—
117Dic166—24no—
118C22×C83—22no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—42no—
125C22×C83—81no—
126C22×C83—81no—
127C22×C83C2×C22⋊C412yes—
128C22×C83C2×C22⋊C411yescharacteristic
129C22×C83C2×C22⋊C411yescharacteristic
130C22×C83C23×C411yescharacteristic
131C22×C83C22×D412yes—
132C2×D83—22no—
133C2×D83—44no—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—84no—
137C2×D83—82no—
138C2×D83C2×C22⋊C414yes—
139C4×C83—22no—
140C4×C83—44no—
141Dic83—22no—
142Dic83—44no—
143Dic83C2×C22⋊C414yes—
144C2×C83—22no—
145C2×C83—21no—
146C2×C83—22no—
147C2×C83—21no—
148C2×C83—21no—
149C2×C83—42no—
150C2×C83C22×C22⋊C411yescharacteristic
151D83—24no—
152D83—42no—
153#2883—831nomaximal, Sylow
154C83#288311yescharacteristic, complemented, Sylow
155C2×H(5,𝔽2)—831no—
156#518—832no—
157#410—834no—
158C2×#99—831no—
159C2×#99—832no—
160C2×#99—834no—
161C2×#99D8311yescharacteristic, complemented
162H(5,𝔽2)—1664no—
163H(5,𝔽2)—1664no—
164C2×#41—832no—
165C2×#41—832no—
166C2×#41—832no—
167C22×D4—831no—
168C22×D4—832no—
169C22×D4—831no—
170C22×D4—832no—
171C22×D4—832no—
172C22×D4C2×D8311yescharacteristic
173C4×D4—1664no—
174C4×D4—1664no—
175#117—834no—
176C2×C22⋊C4—832no—
177C2×C22⋊C4—834no—
178C2×C22⋊C4C2×D8312yescomplemented
179#99—834no—
180#99—838no—
181#99C2×D8314yescomplemented
182C24—831no—
183C24—832no—
184C24—831no—
185C24C4×D8311yescharacteristic, complemented
186C24C22×D8311yescharacteristic
187#41—1664no—
188#41—1664no—
189#41—1668no—
190#41—1668no—
191C2×Q8—832no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—832no—
195C2×D4—832no—
196C2×D4—834no—
197C2×D4—834no—
198C2×D4—832no—
199C2×D4—832no—
200C2×D4—832no—
201C2×D4—832no—
202C2×D4—832no—
203C2×D4—1668no—
204C2×D4—1668no—
205C2×D4—1664no—
206C2×D4—1664no—
207C2×D4—1662no—
208C2×D4—1662no—
209C2×D4C4×D8312yescomplemented
210C2×D4C22×D8312yes—
211C22×C4—22no—
212C22×C4—831no—
213C22×C4—832no—
214C22×C4—831no—
215C22×C4—832no—
216C22×C4—832no—
217C22×C4—1664no—
218C22×C4—1662no—
219C22×C4C4×D8311yescharacteristic, complemented
220C4⋊C4—1664no—
221C22⋊C4—22no—
222C22⋊C4—834no—
223C22⋊C4—1664no—
224C22⋊C4—1662no—
225C22⋊C4C22×D8314yes—
226C42—1664no—
227C23—21no—
228C23—21no—
229C23—81no—
230C23—81no—
231C23—832no—
232C23—834no—
233C23—832no—
234C23—6641no—
235C23—6641no—
236C23—6642no—
237C23—1664no—
238C23—1662no—
239C23C2×C4×D8312yes—
240C23C2×C4×D8311yescharacteristic
241C23C2×C4×D8311yescharacteristic
242C23D4×D8312yes—
243C23D4×D8312yes—
244C23C23×D8311yescharacteristic
245Q8—1664no—
246D4—24no—
247D4—24no—
248D4—834no—
249D4—834no—
250D4—838no—
251D4—1664no—
252D4—1664no—
253D4—1664no—
254D4—1664no—
255D4—1662no—
256D4—1662no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—82no—
261C2×C4—832no—
262C2×C4—834no—
263C2×C4—832no—
264C2×C4—6644no—
265C2×C4—6642no—
266C2×C4—1664no—
267C2×C4—1664no—
268C2×C4—1664no—
269C2×C4—1664no—
270C2×C4—1662no—
271C2×C4C2×C4×D8312yes—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—3322no—
282C22—3324no—
283C22—3322no—
284C22—834no—
285C22—6642no—
286C22—6644no—
287C22—1662no—
288C22C22⋊C4×D8311yescharacteristic
289C22C22⋊C4×D8311yescharacteristic
290C22C22⋊C4×D8312yes—
291C22C22×C4×D8311yescharacteristic
292C22C2×D4×D8312yes—
293C4—22no—
294C4—44no—
295C4—3324no—
296C4—834no—
297C4—1662no—
298C2—22no—
299C2—21no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—42no—
304C2—3322no—
305C2—1664no—
306C2C2×C22⋊C4×D8311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D83
commutatorC22×C83C23×C4
FrattiniC23C23×D83
FittingC2×#99×C83C2
socleC2×C83C22×C22⋊C4
prime radicalC2×#14581837C2
cosocleC23×C83C24
hypercenterC2×#99D83
hypocenterC83#2883
LCSG ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #19128174C2semidirect
3. #19127673C2semidirect
4. C2×#14581707C2semidirect
5. C2×#14581706C2semidirect
6. #99×D83C2semidirect
7. C2×#99×C83C2semidirect
10. C2×#13050590C4semidirect
11. C2×#13050589C4semidirect
13. C22×C22⋊D83C4semidirect
16. C2×D4×D83C4semidirect
20. #14581748C22semidirect
21. C2×C22⋊Dic83C22semidirect
22. C22⋊C4×D83C22semidirect
23. #14581707C22semidirect
24. #14581706C22semidirect
26. C2×C22⋊C4×C83C22semidirect
27. #99×C83C22semidirect
28. C23×D83C2×C4semidirect
29. C23×D83C2×C4semidirect
35. C2×Dic166C2×C4semidirect
36. C2×D332C2×C4semidirect
43. C2×C22⋊D83C2×C4semidirect
44. C2×C22⋊D83C2×C4semidirect
45. C2×C22⋊D83C2×C4semidirect
52. C22×Dic83C2×C4semidirect
53. C22×Dic83C2×C4semidirect
56. C2×C4×D83C2×C4semidirect
57. C2×C4×D83C2×C4semidirect
63. C24×C83C2×C4semidirect
67. C2×D4×C83C2×C4semidirect
70. C22×C4×C83C2×C4semidirect
154. C83#2883semidirect
161. C2×#99D83semidirect
178. C2×C22⋊C4C2×D83semidirect
181. #99C2×D83semidirect
185. C24C4×D83semidirect
209. C2×D4C4×D83semidirect
219. C22×C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818372
#191281742
#191276732
C2×#145817072
C2×#145817062
#99×D832
C2×#99×C832
#288383