Finite Group Databasebeta

← #24912388

Subgroups — #24912388

Order 10624.

900 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 159 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×#14581774C212yesmaximal, complemented
2C2×#14581763C211yesmaximal, characteristic, complemented
3C2×#14581760C212yesmaximal, complemented
4C2×C22⋊C4×D83C211yesmaximal, characteristic, complemented
5#19127638C211yesmaximal, characteristic, complemented
6#19127637C211yesmaximal, characteristic, complemented
7#19127609C211yesmaximal, characteristic, complemented
8#19127604C212yesmaximal, complemented
9#19127601C212yesmaximal, complemented
10#19127548C211yesmaximal, characteristic, complemented
11#392×C83C211yesmaximal, characteristic, complemented
12C24×D83C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C22⋊D83C2212yes—
15C22×C4×D83C2211yescharacteristic
16C22×C4×D83C2211yescharacteristic
17#14581774—28no—
18#14581763—24no—
19#14581760—28no—
20C2×C4⋊Dic83C2212yescomplemented
21C2×C4⋊Dic83C2211yescharacteristic, complemented
22C2×Dic83⋊C4C2212yescomplemented
23C2×C22⋊Dic83C2211yescharacteristic
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2212yescomplemented
26C2×C22⋊Dic83C2212yescomplemented
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C2×(C2×D83)⋊C4C2212yes—
29C2×(C2×D83)⋊C4C2211yescharacteristic
30C2×(C2×D83)⋊C4C2212yes—
31C22⋊C4×D83—24no—
32C2×C4×Dic83C2212yescomplemented
33C83⋊#95C2211yescharacteristic, complemented
34C22×D4×C83C2212yes—
35C22×D4×C83C2211yescharacteristic, complemented
36C2×C22⋊C4×C83C2211yescharacteristic
37C2×C22⋊C4×C83C2212yescomplemented
38#95×C83C2211yescharacteristic, complemented
39C23×D83—22no—
40C23×D83—22no—
41C23×D83—22no—
42C23×D83—24no—
43C23×D83—42no—
44C23×D83C2311yescharacteristic
45C23×D83C2311yescharacteristic
46C2×C22⋊D83—24no—
47C2×C22⋊D83—24no—
48C2×C22⋊D83—24no—
49C2×C22⋊D83—24no—
50C2×C22⋊D83—24no—
51C2×C22⋊D83—42no—
52C22×Dic83C2311yescharacteristic
53C22×Dic83C2312yes—
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2312yes—
56C2×C4×D83—24no—
57C2×C4×D83—22no—
58C2×C4×D83—42no—
59C2×C4×D83D414yescomplemented
60C4⋊Dic83—22no—
61C4⋊Dic83—24no—
62Dic83⋊C4—24no—
63C22⋊Dic83—24no—
64C22⋊Dic83—24no—
65C22⋊Dic83—24no—
66C22⋊Dic83—22no—
67(C2×D83)⋊C4—24no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4D418yescomplemented
71C4×Dic83—42no—
72C24×C83C2312yes—
73C24×C83C2311yescharacteristic
74C2×D4×C83—24no—
75C2×D4×C83—24no—
76C2×D4×C83—22no—
77C2×D4×C83—42no—
78C2×D4×C83—42no—
79C22×C4×C83C2311yescharacteristic
80C22×C4×C83C2311yescharacteristic
81C22×C4×C83C2312yes—
82C22⋊C4×C83—24no—
83C22⋊C4×C83—22no—
84C23×C83—21no—
85C23×C83—22no—
86C23×C83—21no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—22no—
90C23×C83—22no—
91C23×C83—22no—
92C23×C83—22no—
93C23×C83—22no—
94C23×C83—22no—
95C23×C83#4112yes—
96C23×C83C2411yescharacteristic
97C22×D83—21no—
98C22×D83—21no—
99C22×D83—22no—
100C22×D83—28no—
101C22×D83—21no—
102C22×D83—22no—
103C22×D83—21no—
104C22×D83—22no—
105C22×D83—48no—
106C22×D83—48no—
107C22×D83—44no—
108C22×D83—44no—
109C22×D83C2×D414yes—
110C22×D83C2×D412yes—
111C22×D83#4112yes—
112D4×C83—48no—
113D4×C83—44no—
114C2×C4×C83—22no—
115C2×C4×C83—22no—
116C2×C4×C83—22no—
117C2×C4×C83—21no—
118C2×C4×C83—22no—
119C2×C4×C83C2×D414yes—
120C2×C4×C83C2×D412yes—
121C22⋊D83—48no—
122C22⋊D83—48no—
123C2×Dic83—22no—
124C2×Dic83—22no—
125C2×Dic83—22no—
126C2×Dic83—21no—
127C2×Dic83—22no—
128C2×Dic83—22no—
129C2×Dic83—22no—
130C2×Dic83—22no—
131C2×Dic83C2×D412yes—
132C2×Dic83C2×D414yes—
133C4×D83—44no—
134C4×D83—44no—
135C22×C83—22no—
136C22×C83—24no—
137C22×C83—41no—
138C22×C83—41no—
139C22×C83—42no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83—42no—
143C22×C83—42no—
144C22×C83—42no—
145C22×C83—42no—
146C22×C83—42no—
147C22×C83C22×D411yescharacteristic
148C22×C83C22×D412yes—
149C22×C83C2×#4111yescharacteristic
150C22×C83C2×#4111yescharacteristic
151C22×C83H(5,𝔽2)12yes—
152C2×D83—22no—
153C2×D83—22no—
154C2×D83—24no—
155C2×D83—24no—
156C2×D83—22no—
157C2×D83—24no—
158C2×D83—416no—
159C2×D83—41no—
160C2×D83—41no—
161C2×D83—42no—
162C2×D83#12714yes—
163C4×C83—22no—
164C4×C83—24no—
165C4×C83—41no—
166Dic83—22no—
167Dic83—24no—
168Dic83—41no—
169Dic83—42no—
170C2×C83—42no—
171C2×C83—82no—
172C2×C83#52511yescharacteristic
173C2×C83C2×#12711yescharacteristic
174C2×C83#53411yescharacteristic
175C2×C83#54612yes—
176C2×C83#54612yes—
177D83—24no—
178D83—42no—
179#2539—831nomaximal, Sylow
180C83#253911yescharacteristic, complemented, Sylow
181C2×#123—832no—
182C2×#121—831no—
183C2×#120—832no—
184C22×C22⋊C4—831no—
185#399—831no—
186#399—831no—
187#392—831no—
188#392D8311yescharacteristic, complemented
189#390—832no—
190#390—832no—
191C4×C22⋊C4—831no—
192C25—831no—
193C22×D4—832no—
194C22×D4—832no—
195C22×D4C2×D8312yes—
196C22×D4C2×D8311yescharacteristic, complemented
197C23×C4—831no—
198C23×C4—831no—
199#123—1668no—
200#121—1664no—
201#120—1668no—
202C2×C4⋊C4—831no—
203C2×C4⋊C4—832no—
204C2×C4⋊C4—832no—
205C2×C22⋊C4—832no—
206C2×C22⋊C4—831no—
207C2×C22⋊C4—832no—
208C2×C22⋊C4—832no—
209C2×C22⋊C4—832no—
210C2×C22⋊C4—831no—
211C2×C22⋊C4—831no—
212C2×C22⋊C4—832no—
213C2×C22⋊C4—1664no—
214C2×C22⋊C4C2×D8311yescharacteristic
215C2×C22⋊C4C2×D8312yescomplemented
216C2×C42—832no—
217#95—831no—
218#95C2×D8311yescharacteristic, complemented
219C24—3322no—
220C24—831no—
221C24—831no—
222C24—1664no—
223C24—1662no—
224C24—1662no—
225C24—1662no—
226C24C22×D8312yes—
227C24C22×D8311yescharacteristic
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—3322no—
234C2×D4—1664no—
235C2×D4—1664no—
236C2×D4—1664no—
237C2×D4—1664no—
238C2×D4—1664no—
239C22×C4—3322no—
240C22×C4—834no—
241C22×C4—832no—
242C22×C4—831no—
243C22×C4—831no—
244C22×C4—832no—
245C22×C4—1662no—
246C22×C4—1664no—
247C22×C4C22×D8311yescharacteristic
248C22×C4C22×D8311yescharacteristic
249C22×C4C22×D8312yes—
250C4⋊C4—1664no—
251C4⋊C4—1662no—
252C4⋊C4—1664no—
253C22⋊C4—22no—
254C22⋊C4—24no—
255C22⋊C4—838no—
256C22⋊C4—1662no—
257C22⋊C4—1662no—
258C22⋊C4—1664no—
259C22⋊C4—1664no—
260C22⋊C4—1664no—
261C22⋊C4—1662no—
262C22⋊C4—1664no—
263C42—3322no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—3328no—
276C23—3324no—
277C23—3324no—
278C23—3328no—
279C23—834no—
280C23—832no—
281C23—832no—
282C23—1668no—
283C23—1661no—
284C23—1662no—
285C23—1661no—
286C23—1662no—
287C23—1661no—
288C23—1661no—
289C23—1662no—
290C23#1305058912yes—
291C23C23×D8311yescharacteristic
292D4—48no—
293D4—44no—
294D4—3328no—
295D4—3328no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—3324no—
302C2×C4—3324no—
303C2×C4—834no—
304C2×C4—832no—
305C2×C4—1661no—
306C2×C4—1662no—
307C2×C4—1662no—
308C2×C4—1662no—
309C2×C4—1662no—
310C2×C4—1662no—
311C2×C4—1662no—
312C2×C4—1662no—
313C2×C4D4×D8314yes—
314C2×C4D4×D8312yes—
315C22—24no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—33216no—
328C22—3322no—
329C22—3321no—
330C22—3321no—
331C22—834no—
332C22—1664no—
333C22—1662no—
334C22—1664no—
335C22—1664no—
336C22—1662no—
337C22—1662no—
338C22C2×D4×D8312yes—
339C22C2×D4×D8311yescharacteristic
340C22#41×D8311yescharacteristic
341C22C2×#1305058911yescharacteristic
342C22#1458183712yes—
343C4—24no—
344C4—22no—
345C4—41no—
346C4—3321no—
347C4—3322no—
348C4—1664no—
349C4—1662no—
350C2—42no—
351C2—82no—
352C2—3322no—
353C2—1664no—
354C2#1912821111yescharacteristic
355C2#127×D8311yescharacteristic
356C2#1912826111yescharacteristic
357C2#1912833812yes—
358C2#1912834812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#392×C83C2
hypercenter#392D83
hypocenterC83#2539
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14581774C2semidirect
2. C2×#14581763C2semidirect
3. C2×#14581760C2semidirect
4. C2×C22⋊C4×D83C2semidirect
5. #19127638C2semidirect
6. #19127637C2semidirect
7. #19127609C2semidirect
8. #19127604C2semidirect
9. #19127601C2semidirect
10. #19127548C2semidirect
11. #392×C83C2semidirect
20. C2×C4⋊Dic83C22semidirect
21. C2×C4⋊Dic83C22semidirect
22. C2×Dic83⋊C4C22semidirect
25. C2×C22⋊Dic83C22semidirect
26. C2×C22⋊Dic83C22semidirect
32. C2×C4×Dic83C22semidirect
33. C83⋊#95C22semidirect
35. C22×D4×C83C22semidirect
37. C2×C22⋊C4×C83C22semidirect
38. #95×C83C22semidirect
59. C2×C4×D83D4semidirect
70. (C2×D83)⋊C4D4semidirect
180. C83#2539semidirect
188. #392D83semidirect
196. C22×D4C2×D83semidirect
215. C2×C22⋊C4C2×D83semidirect
218. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817742
C2×#145817632
C2×#145817602
C2×C22⋊C4×D832
#191276382
#191276372
#191276092
#191276042
#191276012
#191275482
#392×C832
#253983