Finite Group Databasebeta

← #24911112

Subgroups — #24911112

Order 10624.

1088 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×C22⋊D83C211yesmaximal, characteristic
2C2×#14581778C212yesmaximal, complemented
3C2×#14581769C211yesmaximal, characteristic, complemented
4#19127617C212yesmaximal, complemented
5#19127616C212yesmaximal, complemented
6#19127601C212yesmaximal, complemented
7#19127557C211yesmaximal, characteristic, complemented
8#19127556C211yesmaximal, characteristic, complemented
9#19127556C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic83C211yesmaximal, characteristic, complemented
11#379×C83C211yesmaximal, characteristic, complemented
12C24×D83C2211yescharacteristic
13C22×Dic166C2211yescharacteristic, complemented
14C22×C22⋊D83—28no—
15C22×C22⋊D83—24no—
16C22×C22⋊D83C2212yes—
17C22×C22⋊D83C2212yes—
18C23×Dic83C2211yescharacteristic
19#14581778—28no—
20#14581769—24no—
21C2×C4⋊Dic83C2211yescharacteristic, complemented
22C2×Dic83⋊C4C2211yescharacteristic, complemented
23C2×Dic83⋊C4C2211yescharacteristic, complemented
24C2×C22⋊Dic83C2211yescharacteristic
25C2×C22⋊Dic83C2211yescharacteristic
26C2×C22⋊Dic83C2212yescomplemented
27C2×C22⋊Dic83C2212yescomplemented
28C2×(C2×D83)⋊C4C2212yescomplemented
29C2×(C2×D83)⋊C4C2212yescomplemented
30C2×(C2×D83)⋊C4C2212yescomplemented
31C2×(C2×D83)⋊C4C2212yescomplemented
32C2×C4×Dic83C2212yescomplemented
33C83⋊#95C2212yescomplemented
34C25×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2211yescharacteristic
37C2×C22⋊C4×C83C2212yescomplemented
38C2×C22⋊C4×C83C2212yescomplemented
39C23×D83—22no—
40C23×D83—24no—
41C23×D83—22no—
42C23×D83—22no—
43C23×D83—42no—
44C23×D83C2312yes—
45C2×Dic166—22no—
46C2×Dic166—24no—
47C2×C22⋊D83—232no—
48C2×C22⋊D83—24no—
49C2×C22⋊D83—24no—
50C2×C22⋊D83—24no—
51C2×C22⋊D83—24no—
52C2×C22⋊D83—24no—
53C2×C22⋊D83D418yes—
54C22×Dic83—24no—
55C22×Dic83D414yes—
56C22×Dic83C2311yescharacteristic
57C22×Dic83C2312yes—
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C4⋊Dic83—22no—
62Dic83⋊C4—22no—
63Dic83⋊C4—22no—
64C22⋊Dic83—22no—
65C22⋊Dic83—24no—
66C22⋊Dic83—22no—
67C22⋊Dic83—42no—
68(C2×D83)⋊C4—24no—
69(C2×D83)⋊C4—24no—
70(C2×D83)⋊C4—42no—
71(C2×D83)⋊C4—42no—
72C4×Dic83—24no—
73C24×C83—24no—
74C24×C83—22no—
75C24×C83—22no—
76C24×C83—22no—
77C24×C83—42no—
78C24×C83C2312yes—
79C24×C83C2311yescharacteristic
80C22×C4×C83C2311yescharacteristic
81C22×C4×C83C2312yes—
82C22×C4×C83C2311yescharacteristic
83C22⋊C4×C83—22no—
84C22⋊C4×C83—24no—
85C22⋊C4×C83—22no—
86C22⋊C4×C83—42no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—28no—
90C23×C83—22no—
91C23×C83—22no—
92C23×C83—22no—
93C23×C83—22no—
94C23×C83—22no—
95C23×C83—22no—
96C23×C83—48no—
97C23×C83—48no—
98C23×C83—44no—
99C23×C83—44no—
100C23×C83C2×D414yes—
101C23×C83C2×D412yes—
102C23×C83#4112yes—
103C23×C83#4112yes—
104C23×C83C2411yescharacteristic
105C22×D83—28no—
106C22×D83—22no—
107C22×D83—22no—
108C22×D83—22no—
109C22×D83—22no—
110C22×D83—22no—
111C22×D83—22no—
112C22×D83—48no—
113C22×D83—48no—
114C22×D83—44no—
115C22×D83—44no—
116C22×D83C2×D414yes—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—22no—
122C2×C4×C83—22no—
123C2×C4×C83—22no—
124C22⋊D83—216no—
125C22⋊D83—216no—
126C2×Dic83—22no—
127C2×Dic83—28no—
128C2×Dic83—22no—
129C2×Dic83—22no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—22no—
133C2×Dic83—22no—
134C2×Dic83—22no—
135C2×Dic83C2×D412yes—
136C2×Dic83C2×D414yes—
137C2×Dic83C2×D412yes—
138Dic166—44no—
139C22×C83—22no—
140C22×C83—22no—
141C22×C83—24no—
142C22×C83—24no—
143C22×C83—24no—
144C22×C83—24no—
145C22×C83—24no—
146C22×C83—416no—
147C22×C83—42no—
148C22×C83—42no—
149C22×C83—42no—
150C22×C83—42no—
151C22×C83#12414yescomplemented
152C22×C83C22×D411yescharacteristic
153C22×C83C22×D412yes—
154C22×C83C2×#4111yescharacteristic
155C22×C83C2×#4111yescharacteristic
156C22×C83H(5,𝔽2)12yes—
157C2×D83—24no—
158C2×D83—24no—
159C2×D83—24no—
160C2×D83—416no—
161C2×D83—42no—
162C2×D83—42no—
163C2×D83—42no—
164C2×D83—42no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—42no—
168Dic83—22no—
169Dic83—41no—
170Dic83—41no—
171Dic83—42no—
172Dic83#12014yes—
173C2×C83—24no—
174C2×C83—44no—
175C2×C83C2×#12011yescharacteristic
176C2×C83C2×#12411yescharacteristic
177C2×C83#53511yescharacteristic
178C2×C83#54612yes—
179C2×C83#54612yes—
180D83—44no—
181#2435—831nomaximal, Sylow
182C83#243511yescharacteristic, complemented, Sylow
183C23×D4—831no—
184C2×#124—832no—
185C2×C22⋊Q8—831no—
186#394—832no—
187#394—832no—
188#390—832no—
189#379—831no—
190#379—831no—
191#379—831no—
192#379D8311yescharacteristic, complemented
193C4×C22⋊C4—831no—
194C25—831no—
195C25C2×D8311yescharacteristic
196C22×Q8—831no—
197C22×D4—832no—
198C22×D4—832no—
199C22×D4—1664no—
200C22×D4—1668no—
201C23×C4—831no—
202#124—1668no—
203C22⋊Q8—1664no—
204C2×C4⋊C4—831no—
205C2×C4⋊C4—831no—
206C2×C4⋊C4—831no—
207C2×C22⋊C4—832no—
208C2×C22⋊C4—832no—
209C2×C22⋊C4—832no—
210C2×C22⋊C4—831no—
211C2×C22⋊C4—832no—
212C2×C22⋊C4—831no—
213C2×C22⋊C4—832no—
214C2×C22⋊C4—832no—
215C2×C22⋊C4C2×D8311yescharacteristic
216C2×C22⋊C4C2×D8311yescharacteristic
217C2×C22⋊C4C2×D8312yescomplemented
218C2×C22⋊C4C2×D8312yescomplemented
219C2×C42—832no—
220#95—832no—
221C24—24no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—3322no—
227C24—832no—
228C24—1664no—
229C24—1662no—
230C24—1662no—
231C24—1662no—
232C24C22×D8312yes—
233C24C22×D8311yescharacteristic
234C2×Q8—1662no—
235C2×Q8—1664no—
236C2×D4—838no—
237C2×D4—1664no—
238C2×D4—1664no—
239C2×D4—1664no—
240C2×D4—1664no—
241C2×D4—16632no—
242C2×D4—1664no—
243C22×C4—831no—
244C22×C4—834no—
245C22×C4—831no—
246C22×C4—832no—
247C22×C4—831no—
248C22×C4—831no—
249C22×C4—1664no—
250C22×C4C22×D8312yes—
251C22×C4C22×D8311yescharacteristic
252C22×C4C22×D8311yescharacteristic
253C4⋊C4—1662no—
254C4⋊C4—1662no—
255C4⋊C4—1662no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—3322no—
261C22⋊C4—3322no—
262C22⋊C4—3322no—
263C22⋊C4—1664no—
264C22⋊C4—1664no—
265C22⋊C4—1662no—
266C22⋊C4—1664no—
267C22⋊C4—1662no—
268C42—1664no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—48no—
279C23—48no—
280C23—44no—
281C23—44no—
282C23—3324no—
283C23—3328no—
284C23—3328no—
285C23—3324no—
286C23—834no—
287C23—1668no—
288C23—1662no—
289C23—1662no—
290C23—1662no—
291C23—1662no—
292C23—1662no—
293C23—1662no—
294C23D4×D8314yes—
295C23D4×D8312yes—
296C23#1305058912yes—
297C23#1305059012yes—
298C23C23×D8311yescharacteristic
299Q8—3324no—
300D4—16616no—
301D4—16616no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—834no—
310C2×C4—832no—
311C2×C4—832no—
312C2×C4—1662no—
313C2×C4—1662no—
314C2×C4—1661no—
315C2×C4—1661no—
316C2×C4—1662no—
317C2×C4—1662no—
318C2×C4—1662no—
319C2×C4—1668no—
320C2×C4—1662no—
321C22—24no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—416no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—33216no—
334C22—3322no—
335C22—3322no—
336C22—3322no—
337C22—3322no—
338C22—1664no—
339C22—1664no—
340C22—1664no—
341C22#1458177814yescomplemented
342C22C2×D4×D8312yes—
343C22C2×D4×D8311yescharacteristic
344C22C2×#1305058911yescharacteristic
345C22C2×#1305059011yescharacteristic
346C22#1458183712yes—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—3322no—
351C4—3321no—
352C4—3321no—
353C4—834no—
354C4—1662no—
355C2—24no—
356C2—44no—
357C2—3324no—
358C2#120×D8311yescharacteristic
359C2C2×#1458177811yescharacteristic
360C2#1912826611yescharacteristic
361C2#1912833812yes—
362C2#1912834112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#379×C83C2
prime radicalC23×C22⋊D83C2
hypercenter#379D83
hypocenterC83#2435
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#14581778C2semidirect
3. C2×#14581769C2semidirect
4. #19127617C2semidirect
5. #19127616C2semidirect
6. #19127601C2semidirect
7. #19127557C2semidirect
8. #19127556C2semidirect
9. #19127556C2semidirect
10. C22⋊C4×Dic83C2semidirect
11. #379×C83C2semidirect
13. C22×Dic166C22semidirect
21. C2×C4⋊Dic83C22semidirect
22. C2×Dic83⋊C4C22semidirect
23. C2×Dic83⋊C4C22semidirect
26. C2×C22⋊Dic83C22semidirect
27. C2×C22⋊Dic83C22semidirect
28. C2×(C2×D83)⋊C4C22semidirect
29. C2×(C2×D83)⋊C4C22semidirect
30. C2×(C2×D83)⋊C4C22semidirect
31. C2×(C2×D83)⋊C4C22semidirect
32. C2×C4×Dic83C22semidirect
33. C83⋊#95C22semidirect
37. C2×C22⋊C4×C83C22semidirect
38. C2×C22⋊C4×C83C22semidirect
151. C22×C83#124semidirect
182. C83#2435semidirect
192. #379D83semidirect
217. C2×C22⋊C4C2×D83semidirect
218. C2×C22⋊C4C2×D83semidirect
341. C22#14581778semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D832
C2×#145817782
C2×#145817692
#191276172
#191276162
#191276012
#191275572
#191275562
C22⋊C4×Dic832
#379×C832
#243583