Finite Group Databasebeta

← #24910855

Subgroups — #24910855

Order 10624.

814 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 165 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
2C22×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D83C211yesmaximal, characteristic, complemented
4#19127627C212yesmaximal, complemented
5#19127617C212yesmaximal, complemented
6#19127581C211yesmaximal, characteristic, complemented
7#19127577C212yesmaximal, complemented
8#19127576C211yesmaximal, characteristic, complemented
9#19127564C211yesmaximal, characteristic, complemented
10#19127558C212yesmaximal, complemented
11C2×#123×C83C211yesmaximal, characteristic, complemented
12C24×D83C2211yescharacteristic
13C22×C22⋊D83C2211yescharacteristic
14C22×C22⋊D83C2211yescharacteristic
15C23×Dic83C2211yescharacteristic
16C22×C4×D83C2212yes—
17#14581760—24no—
18C2×C4⋊Dic83C2212yescomplemented
19C2×C22⋊Dic83C2211yescharacteristic
20C2×C22⋊Dic83C2211yescharacteristic
21C2×C22⋊Dic83C2212yescomplemented
22C2×(C2×D83)⋊C4—24no—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2211yescharacteristic
28C22⋊C4×D83—24no—
29C83⋊#95C2212yescomplemented
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—22no—
41C23×D83—24no—
42C23×D83—42no—
43C23×D83D414yes—
44C23×D83C2311yescharacteristic
45C23×D83C2311yescharacteristic
46C2×C22⋊D83—22no—
47C2×C22⋊D83—22no—
48C2×C22⋊D83—24no—
49C2×C22⋊D83—42no—
50C22×Dic83—22no—
51C22×Dic83—42no—
52C22×Dic83C2312yes—
53C22×Dic83C2312yes—
54C22×Dic83C2311yescharacteristic
55C22×Dic83C2311yescharacteristic
56C2×C4×D83—24no—
57C2×C4×D83—24no—
58C2×C4×D83—42no—
59C4⋊Dic83—42no—
60C22⋊Dic83—22no—
61C22⋊Dic83—22no—
62C22⋊Dic83—42no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—24no—
66(C2×D83)⋊C4—24no—
67(C2×D83)⋊C4—24no—
68C24×C83C2311yescharacteristic
69C24×C83C2311yescharacteristic
70C2×D4×C83—22no—
71C2×D4×C83—22no—
72C2×D4×C83—22no—
73C22×C4×C83—22no—
74C22×C4×C83—22no—
75C22×C4×C83—22no—
76C22×C4×C83C2312yes—
77C22×C4×C83C2311yescharacteristic
78C22×C4×C83C2312yes—
79C4⋊C4×C83—24no—
80C22⋊C4×C83—24no—
81C22⋊C4×C83D414yescomplemented
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—21no—
85C23×C83—21no—
86C23×C83—21no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83C2×D412yes—
90C23×C83C2×D412yes—
91C23×C83#4112yes—
92C23×C83C2411yescharacteristic
93C22×D83—21no—
94C22×D83—21no—
95C22×D83—22no—
96C22×D83—28no—
97C22×D83—21no—
98C22×D83—22no—
99C22×D83—21no—
100C22×D83—22no—
101C22×D83—28no—
102C22×D83—44no—
103C22×D83—48no—
104C22×D83—48no—
105C22×D83C2×D412yes—
106C22×D83C2×D412yes—
107C22×D83#4114yes—
108D4×C83—44no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—21no—
113C2×C4×C83—22no—
114C2×C4×C83—21no—
115C2×C4×C83—22no—
116C2×C4×C83—21no—
117C2×C4×C83—44no—
118C2×C4×C83C2×D414yes—
119C22⋊D83—44no—
120C22⋊D83—44no—
121C2×Dic83—22no—
122C2×Dic83—22no—
123C2×Dic83—21no—
124C2×Dic83—21no—
125C2×Dic83—22no—
126C2×Dic83—22no—
127C2×Dic83—22no—
128C2×Dic83—22no—
129C2×Dic83—22no—
130C2×Dic83—22no—
131C2×Dic83—44no—
132C4×D83—48no—
133C22×C83—22no—
134C22×C83—22no—
135C22×C83—24no—
136C22×C83—24no—
137C22×C83—41no—
138C22×C83—41no—
139C22×C83—41no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83#12114yes—
143C22×C83C22×D411yescharacteristic
144C22×C83C22×D411yescharacteristic
145C22×C83C2×#4112yes—
146C22×C83C2×#4111yescharacteristic
147C22×C83H(5,𝔽2)12yes—
148C2×D83—22no—
149C2×D83—22no—
150C2×D83—24no—
151C2×D83—24no—
152C2×D83—22no—
153C2×D83—22no—
154C2×D83—22no—
155C2×D83—416no—
156C2×D83—42no—
157C2×D83—42no—
158C2×D83#12314yescomplemented
159C4×C83—24no—
160C4×C83—42no—
161C4×C83—41no—
162Dic83—41no—
163Dic83—42no—
164Dic83—41no—
165Dic83—42no—
166C2×C83—24no—
167C2×C83—81no—
168C2×C83C2×#12111yescharacteristic
169C2×C83C2×#12311yescharacteristic
170C2×C83#53411yescharacteristic
171C2×C83#54612yes—
172C2×C83#55112yes—
173D83—24no—
174D83—42no—
175#2412—831nomaximal, Sylow
176C83#241211yescharacteristic, complemented, Sylow
177C2×#123D8311yescharacteristic, complemented
178C2×#120—831no—
179C22×C22⋊C4—831no—
180C22×C22⋊C4—831no—
181#397—832no—
182#394—832no—
183#386—831no—
184#386—832no—
185#385—831no—
186#381—831no—
187#380—832no—
188C25—831no—
189C22×D4—831no—
190C22×D4—831no—
191C22×D4C2×D8311yescharacteristic
192C23×C4—831no—
193C23×C4—832no—
194C23×C4C2×D8311yescharacteristic
195#123—24no—
196#120—1664no—
197C2×C4⋊C4—832no—
198C2×C4⋊C4C2×D8312yescomplemented
199C2×C22⋊C4—832no—
200C2×C22⋊C4—831no—
201C2×C22⋊C4—831no—
202C2×C22⋊C4—832no—
203C2×C22⋊C4—832no—
204C2×C22⋊C4—832no—
205C2×C22⋊C4—831no—
206C2×C22⋊C4—831no—
207C2×C22⋊C4—1664no—
208C2×C22⋊C4—1664no—
209C2×C22⋊C4C2×D8311yescharacteristic
210C2×C22⋊C4C2×D8312yescomplemented
211#95—832no—
212#95—832no—
213#95—832no—
214#95—832no—
215C24—3322no—
216C24—834no—
217C24—831no—
218C24—831no—
219C24—1664no—
220C24—1662no—
221C24—1662no—
222C24C22×D8311yescharacteristic
223C24C22×D8311yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—3322no—
228C2×D4—1662no—
229C2×D4—1662no—
230C2×D4—1664no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—3322no—
235C22×C4—3322no—
236C22×C4—831no—
237C22×C4—832no—
238C22×C4—832no—
239C22×C4—831no—
240C22×C4—1664no—
241C22×C4—1662no—
242C22×C4—1664no—
243C22×C4C22×D8312yes—
244C22×C4C22×D8311yescharacteristic
245C22×C4C22×D8312yes—
246C4⋊C4—24no—
247C4⋊C4—3322no—
248C22⋊C4—24no—
249C22⋊C4—3322no—
250C22⋊C4—1664no—
251C22⋊C4—1664no—
252C22⋊C4—1662no—
253C22⋊C4—1662no—
254C22⋊C4—1662no—
255C22⋊C4—1662no—
256C22⋊C4—1664no—
257C22⋊C4C22⋊D8314yescomplemented
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—22no—
265C23—3328no—
266C23—3328no—
267C23—3324no—
268C23—832no—
269C23—834no—
270C23—832no—
271C23—1668no—
272C23—1668no—
273C23—1661no—
274C23—1662no—
275C23—1661no—
276C23—1662no—
277C23—1661no—
278C23—1661no—
279C23—1662no—
280C23D4×D8312yes—
281C23C2×C22⋊D8312yes—
282C23#1305058912yes—
283C23C23×D8311yescharacteristic
284D4—44no—
285D4—3324no—
286D4—3324no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—3328no—
297C2×C4—3324no—
298C2×C4—1662no—
299C2×C4—1661no—
300C2×C4—1661no—
301C2×C4—1662no—
302C2×C4—1662no—
303C2×C4—1662no—
304C2×C4—1662no—
305C2×C4—1662no—
306C2×C4—1662no—
307C2×C4—1662no—
308C2×C4C2×C22⋊D8314yes—
309C22—24no—
310C22—24no—
311C22—22no—
312C22—22no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—33216no—
319C22—3322no—
320C22—3322no—
321C22—834no—
322C22—1662no—
323C22—1662no—
324C22—1662no—
325C22—1664no—
326C22—1664no—
327C22—1662no—
328C22—1662no—
329C22#1458176314yes—
330C22C2×D4×D8311yescharacteristic
331C22C22×C22⋊D8311yescharacteristic
332C22#41×D8312yes—
333C22C2×#1305058911yescharacteristic
334C22#1458183812yes—
335C4—24no—
336C4—42no—
337C4—41no—
338C4—3322no—
339C4—3321no—
340C4—3322no—
341C4—3321no—
342C2—24no—
343C2—81no—
344C2—3322no—
345C2—1664no—
346C2C2×#1458176311yescharacteristic
347C2#123×D8311yescharacteristic
348C2#1912826211yescharacteristic
349C2#1912833912yes—
350C2#1912837412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#123×C83C2
prime radicalC2×#14581760C2
hypercenterC2×#123D83
hypocenterC83#2412
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C22×(C2×D83)⋊C4C2semidirect
3. C2×C22⋊C4×D83C2semidirect
4. #19127627C2semidirect
5. #19127617C2semidirect
6. #19127581C2semidirect
7. #19127577C2semidirect
8. #19127576C2semidirect
9. #19127564C2semidirect
10. #19127558C2semidirect
11. C2×#123×C83C2semidirect
18. C2×C4⋊Dic83C22semidirect
21. C2×C22⋊Dic83C22semidirect
29. C83⋊#95C22semidirect
30. C83⋊#95C22semidirect
31. C83⋊#95C22semidirect
32. C83⋊#95C22semidirect
36. C2×C4⋊C4×C83C22semidirect
38. C2×C22⋊C4×C83C22semidirect
81. C22⋊C4×C83D4semidirect
158. C2×D83#123semidirect
176. C83#2412semidirect
177. C2×#123D83semidirect
198. C2×C4⋊C4C2×D83semidirect
210. C2×C22⋊C4C2×D83semidirect
257. C22⋊C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817602
C22×(C2×D83)⋊C42
C2×C22⋊C4×D832
#191276272
#191276172
#191275812
#191275772
#191275762
#191275642
#191275582
C2×#123×C832
#241283