Finite Group Databasebeta

← #24912352

Subgroups — #24912352

Order 10624.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14581760C211yesmaximal, characteristic
2C2×D4×Dic83C211yesmaximal, characteristic, complemented
3C22×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
4#19127623C212yesmaximal, complemented
5#19127616C212yesmaximal, complemented
6#19127591C212yesmaximal, complemented
7#19127578C212yesmaximal, complemented
8#19127564C211yesmaximal, characteristic, complemented
9#19127559C211yesmaximal, characteristic, complemented
10#19127556C211yesmaximal, characteristic, complemented
11C2×#123×C83C211yesmaximal, characteristic, complemented
12C24×D83C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C23×Dic83C2211yescharacteristic
15C23×Dic83C2211yescharacteristic, complemented
16#14581760—24no—
17D4×Dic83—24no—
18C2×C4⋊Dic83C2211yescharacteristic, complemented
19C2×Dic83⋊C4C2212yescomplemented
20C2×C22⋊Dic83C2211yescharacteristic
21C2×C22⋊Dic83C2211yescharacteristic
22C2×C22⋊Dic83C2211yescharacteristic, complemented
23C2×(C2×D83)⋊C4—24no—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×(C2×D83)⋊C4C2212yes—
29C2×C4×Dic83C2211yescharacteristic, complemented
30C83⋊#95C2212yescomplemented
31C83⋊#95C2212yescomplemented
32C83⋊#95C2212yescomplemented
33C22×D4×C83C2211yescharacteristic
34C23×C4×C83C2211yescharacteristic
35#123×C83—24no—
36C2×C4⋊C4×C83C2212yescomplemented
37C2×C22⋊C4×C83C2211yescharacteristic
38C2×C22⋊C4×C83C2212yescomplemented
39C23×D83—22no—
40C23×D83—24no—
41C23×D83—22no—
42C23×D83—22no—
43C23×D83—42no—
44C23×D83C2312yes—
45C2×C22⋊D83—24no—
46C2×C22⋊D83—24no—
47C2×C22⋊D83—42no—
48C22×Dic83—22no—
49C22×Dic83—42no—
50C22×Dic83—42no—
51C22×Dic83D414yescomplemented
52C22×Dic83C2311yescharacteristic
53C22×Dic83C2312yes—
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2311yescharacteristic
56C22×Dic83C2311yescharacteristic
57C4⋊Dic83—22no—
58Dic83⋊C4—42no—
59C22⋊Dic83—22no—
60C22⋊Dic83—22no—
61C22⋊Dic83—22no—
62(C2×D83)⋊C4—24no—
63(C2×D83)⋊C4—24no—
64(C2×D83)⋊C4—24no—
65(C2×D83)⋊C4—24no—
66(C2×D83)⋊C4—42no—
67C4×Dic83—22no—
68C24×C83C2311yescharacteristic
69C24×C83C2311yescharacteristic
70C2×D4×C83—22no—
71C2×D4×C83—22no—
72C2×D4×C83D414yes—
73C22×C4×C83—22no—
74C22×C4×C83—22no—
75C22×C4×C83—22no—
76C22×C4×C83C2312yes—
77C22×C4×C83C2312yes—
78C22×C4×C83C2311yescharacteristic
79C4⋊C4×C83—24no—
80C22⋊C4×C83—22no—
81C22⋊C4×C83—24no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—21no—
85C23×C83—21no—
86C23×C83—22no—
87C23×C83—22no—
88C23×C83C2×D412yes—
89C23×C83C2×D412yes—
90C23×C83C2×D412yes—
91C23×C83#4112yes—
92C23×C83C2411yescharacteristic
93C22×D83—28no—
94C22×D83—22no—
95C22×D83—22no—
96C22×D83—22no—
97C22×D83—22no—
98C22×D83—22no—
99C22×D83—22no—
100C22×D83—48no—
101C22×D83—48no—
102C22×D83—44no—
103C22×D83—44no—
104C22×D83#4114yes—
105D4×C83—28no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—44no—
115C2×C4×C83C2×D412yes—
116C22⋊D83—48no—
117C2×Dic83—22no—
118C2×Dic83—22no—
119C2×Dic83—21no—
120C2×Dic83—22no—
121C2×Dic83—21no—
122C2×Dic83—22no—
123C2×Dic83—22no—
124C2×Dic83—22no—
125C2×Dic83—22no—
126C2×Dic83—44no—
127C2×Dic83—44no—
128C2×Dic83C2×D412yes—
129C2×Dic83C2×D412yes—
130C22×C83—22no—
131C22×C83—22no—
132C22×C83—24no—
133C22×C83—24no—
134C22×C83—22no—
135C22×C83—24no—
136C22×C83—41no—
137C22×C83—41no—
138C22×C83—42no—
139C22×C83#12114yes—
140C22×C83C22×D411yescharacteristic
141C22×C83C22×D411yescharacteristic
142C22×C83C2×#4112yes—
143C22×C83C2×#4111yescharacteristic
144C22×C83H(5,𝔽2)12yes—
145C2×D83—24no—
146C2×D83—24no—
147C2×D83—24no—
148C2×D83—416no—
149C2×D83—42no—
150C2×D83—42no—
151C2×D83—42no—
152C2×D83—42no—
153C4×C83—22no—
154C4×C83—42no—
155C4×C83—42no—
156Dic83—22no—
157Dic83—22no—
158Dic83—41no—
159Dic83—41no—
160Dic83—42no—
161C2×C83—24no—
162C2×C83—42no—
163C2×C83C2×#12111yescharacteristic
164C2×C83#52511yescharacteristic
165C2×C83#53411yescharacteristic
166C2×C83#54612yes—
167C2×C83#55112yes—
168D83—44no—
169#2534—831nomaximal, Sylow
170C83#253411yescharacteristic, complemented, Sylow
171C2×#123D8311yescharacteristic, complemented
172C2×#120—831no—
173C2×C4×D4—831no—
174C22×C22⋊C4—831no—
175#396—832no—
176#394—832no—
177#388—832no—
178#386—832no—
179#381—831no—
180#380—831no—
181#379—831no—
182C25—831no—
183C22×D4—832no—
184C22×D4C2×D8311yescharacteristic
185C23×C4—831no—
186C23×C4—831no—
187C23×C4C2×D8311yescharacteristic
188#123—24no—
189#120—1664no—
190C4×D4—1664no—
191C2×C4⋊C4—832no—
192C2×C4⋊C4—831no—
193C2×C4⋊C4C2×D8312yescomplemented
194C2×C22⋊C4—832no—
195C2×C22⋊C4—831no—
196C2×C22⋊C4—831no—
197C2×C22⋊C4—831no—
198C2×C22⋊C4—832no—
199C2×C22⋊C4—832no—
200C2×C22⋊C4—832no—
201C2×C22⋊C4—832no—
202C2×C22⋊C4—1664no—
203C2×C22⋊C4C2×D8311yescharacteristic
204C2×C22⋊C4C2×D8312yescomplemented
205C2×C42—831no—
206#95—832no—
207#95—832no—
208#95—832no—
209C24—3322no—
210C24—832no—
211C24—1664no—
212C24—1662no—
213C24—1662no—
214C24—1662no—
215C24C22×D8311yescharacteristic
216C24C22×D8311yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—3322no—
220C2×D4—1664no—
221C2×D4—1664no—
222C2×D4C22⋊D8314yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—3322no—
227C22×C4—3322no—
228C22×C4—834no—
229C22×C4—831no—
230C22×C4—831no—
231C22×C4—832no—
232C22×C4—831no—
233C22×C4—831no—
234C22×C4—1662no—
235C22×C4C22×D8312yes—
236C22×C4C22×D8312yes—
237C22×C4C22×D8311yescharacteristic
238C4⋊C4—24no—
239C4⋊C4—3322no—
240C4⋊C4—1662no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—3322no—
244C22⋊C4—1664no—
245C22⋊C4—1664no—
246C22⋊C4—1664no—
247C22⋊C4—1662no—
248C22⋊C4—1662no—
249C22⋊C4—1662no—
250C22⋊C4—1664no—
251C42—1662no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—3324no—
259C23—3328no—
260C23—3328no—
261C23—3324no—
262C23—834no—
263C23—1668no—
264C23—1662no—
265C23—1662no—
266C23—1662no—
267C23—1662no—
268C23—1662no—
269C23—1662no—
270C23D4×D8312yes—
271C23C2×C22⋊D8312yes—
272C23C2×C22⋊D8312yes—
273C23#1305058912yes—
274C23C23×D8311yescharacteristic
275D4—28no—
276D4—3328no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—3324no—
287C2×C4—3324no—
288C2×C4—832no—
289C2×C4—832no—
290C2×C4—1661no—
291C2×C4—1661no—
292C2×C4—1662no—
293C2×C4—1662no—
294C2×C4—1662no—
295C2×C4—1662no—
296C2×C4—1662no—
297C2×C4—1662no—
298C2×C4—1662no—
299C2×C4C2×C22⋊D8312yes—
300C22—24no—
301C22—24no—
302C22—22no—
303C22—22no—
304C22—24no—
305C22—22no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—33216no—
310C22—3322no—
311C22—3322no—
312C22—3322no—
313C22—3322no—
314C22—1664no—
315C22—1664no—
316C22—1664no—
317C22#1458176414yes—
318C22C2×D4×D8311yescharacteristic
319C22C22×C22⋊D8311yescharacteristic
320C22#41×D8312yes—
321C22C2×#1305058911yescharacteristic
322C22#1458183812yes—
323C4—22no—
324C4—42no—
325C4—42no—
326C4—3322no—
327C4—3321no—
328C4—3321no—
329C4—1662no—
330C4—1662no—
331C2—24no—
332C2—42no—
333C2—3324no—
334C2C2×#1458176411yescharacteristic
335C2#1912820911yescharacteristic
336C2#1912826211yescharacteristic
337C2#1912833912yes—
338C2#1912837412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#123×C83C2
prime radicalC2×#14581760C2
hypercenterC2×#123D83
hypocenterC83#2534
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic83C2semidirect
3. C22×(C2×D83)⋊C4C2semidirect
4. #19127623C2semidirect
5. #19127616C2semidirect
6. #19127591C2semidirect
7. #19127578C2semidirect
8. #19127564C2semidirect
9. #19127559C2semidirect
10. #19127556C2semidirect
11. C2×#123×C83C2semidirect
15. C23×Dic83C22semidirect
18. C2×C4⋊Dic83C22semidirect
19. C2×Dic83⋊C4C22semidirect
22. C2×C22⋊Dic83C22semidirect
29. C2×C4×Dic83C22semidirect
30. C83⋊#95C22semidirect
31. C83⋊#95C22semidirect
32. C83⋊#95C22semidirect
36. C2×C4⋊C4×C83C22semidirect
38. C2×C22⋊C4×C83C22semidirect
51. C22×Dic83D4semidirect
170. C83#2534semidirect
171. C2×#123D83semidirect
193. C2×C4⋊C4C2×D83semidirect
204. C2×C22⋊C4C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817602
C2×D4×Dic832
C22×(C2×D83)⋊C42
#191276232
#191276162
#191275912
#191275782
#191275642
#191275592
#191275562
C2×#123×C832
#253483