Finite Group Databasebeta

← #24911675

Subgroups — #24911675

Order 10624.

476 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581748C211yesmaximal, characteristic
2#19127634C211yesmaximal, characteristic, complemented
3#19127632C211yesmaximal, characteristic, complemented
4#19127629C211yesmaximal, characteristic
5#19127629C211yesmaximal, characteristic
6#19127597C211yesmaximal, characteristic, complemented
7#19127596C211yesmaximal, characteristic, complemented
8#19127594C211yesmaximal, characteristic
9#19127585C211yesmaximal, characteristic, complemented
10#19127582C211yesmaximal, characteristic, complemented
11#19127582C211yesmaximal, characteristic, complemented
12#19127573C211yesmaximal, characteristic
13#19127573C211yesmaximal, characteristic
14C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
15C2×#126×C83C211yesmaximal, characteristic
16C23×Dic83C2211yescharacteristic
17#14581748—24no—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×C4×Dic83C2211yescharacteristic
31C2×C4×Dic83C2211yescharacteristic
32C2×C4×Dic83C2211yescharacteristic
33C83⋊#95C2211yescharacteristic
34C83⋊#95C2211yescharacteristic
35C83⋊#95C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45#126×C83—24no—
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C4⋊C4×C83C2211yescharacteristic
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52C2×C42×C83C2211yescharacteristic
53C22×Dic83—22no—
54C22×Dic83—22no—
55C22×Dic83—22no—
56C22×Dic83C2311yescharacteristic
57C22×Dic83C2311yescharacteristic
58C22×Dic83C2311yescharacteristic
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C4⋊Dic83—22no—
65Dic83⋊C4—22no—
66Dic83⋊C4—22no—
67Dic83⋊C4—22no—
68Dic83⋊C4—22no—
69Dic83⋊C4—22no—
70Dic83⋊C4—22no—
71Dic83⋊C4—22no—
72Dic83⋊C4—22no—
73C22⋊Dic83—22no—
74C22⋊Dic83—22no—
75C22⋊Dic83—22no—
76C4×Dic83—22no—
77C4×Dic83—22no—
78C4×Dic83—22no—
79C24×C83C2311yescharacteristic
80C22×C4×C83C2311yescharacteristic
81C22×C4×C83C2311yescharacteristic
82C22×C4×C83C2311yescharacteristic
83C22×C4×C83C2311yescharacteristic
84C22×C4×C83C2311yescharacteristic
85C22×C4×C83C2311yescharacteristic
86C4⋊C4×C83—22no—
87C4⋊C4×C83—22no—
88C4⋊C4×C83—22no—
89C22⋊C4×C83—22no—
90C22⋊C4×C83—22no—
91C22⋊C4×C83D414yes—
92C42×C83—22no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83C2×D412yes—
100C23×C83C2411yescharacteristic
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83C2×D412yes—
118C2×C4×C83C2×D412yes—
119C2×Dic83—28no—
120C2×Dic83—21no—
121C2×Dic83—21no—
122C2×Dic83—21no—
123C2×Dic83—21no—
124C2×Dic83—21no—
125C2×Dic83—21no—
126C2×Dic83—21no—
127C2×Dic83—21no—
128C2×Dic83—21no—
129C2×Dic83—21no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83#4112yes—
141C2×Dic83#4112yes—
142C2×Dic83#4112yes—
143C2×Dic83#4112yes—
144C22×C83—22no—
145C22×C83—22no—
146C22×C83—22no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83C22×D411yescharacteristic
152C22×C83C2×#4111yescharacteristic
153C22×C83C2×#4111yescharacteristic
154C22×C83C2×#4111yescharacteristic
155C22×C83C2×#4111yescharacteristic
156C22×C832−1+411yescharacteristic
157C22×C832−1+411yescharacteristic
158C4×C83—22no—
159C4×C83—22no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—41no—
164Dic83—22no—
165Dic83—22no—
166Dic83—22no—
167Dic83—22no—
168Dic83—41no—
169Dic83—41no—
170Dic83—41no—
171Dic83—41no—
172C2×C83—42no—
173C2×C83#52511yescharacteristic
174C2×C83#52911yescharacteristic
175C2×C83#54111yescharacteristic
176C2×C83#54711yescharacteristic
177C2×C83#54711yescharacteristic
178C2×C83#55211yescharacteristic
179C2×C83#55611yescharacteristic
180#2478—831nomaximal, Sylow
181C83#247811yescharacteristic, complemented, Sylow
182C2×#126D8311yescharacteristic
183C2×#117—831no—
184#398—831no—
185#398—831no—
186#397—831no—
187#397—831no—
188#389—831no—
189#389—831no—
190#388—831no—
191#387—831no—
192#387—831no—
193#387—831no—
194#385—831no—
195#385—831no—
196C4×C4⋊C4—831no—
197C23×C4—831no—
198#126—24no—
199#117—1664no—
200C2×C4⋊C4—831no—
201C2×C4⋊C4—831no—
202C2×C4⋊C4—831no—
203C2×C4⋊C4—831no—
204C2×C4⋊C4—831no—
205C2×C4⋊C4—831no—
206C2×C4⋊C4—831no—
207C2×C4⋊C4—831no—
208C2×C4⋊C4—831no—
209C2×C4⋊C4C2×D8311yescharacteristic
210C2×C4⋊C4C2×D8311yescharacteristic
211C2×C4⋊C4C2×D8311yescharacteristic
212C2×C22⋊C4—831no—
213C2×C22⋊C4—831no—
214C2×C22⋊C4—831no—
215C2×C22⋊C4C2×D8311yescharacteristic
216C2×C22⋊C4C2×D8311yescharacteristic
217C2×C22⋊C4C2×D8311yescharacteristic
218C2×C42—831no—
219C2×C42—831no—
220C2×C42—831no—
221C2×C42C2×D8311yescharacteristic
222#95—831no—
223#95—831no—
224#95—831no—
225#95—831no—
226#95—831no—
227#95—831no—
228#95—831no—
229#95—831no—
230#95—831no—
231#95—831no—
232#95—831no—
233#95—831no—
234C24C22×D8311yescharacteristic
235C22×C4—831no—
236C22×C4—831no—
237C22×C4—831no—
238C22×C4—831no—
239C22×C4—831no—
240C22×C4—831no—
241C22×C4—831no—
242C22×C4—831no—
243C22×C4—1662no—
244C22×C4—1662no—
245C22×C4—1662no—
246C22×C4C22×D8311yescharacteristic
247C22×C4C22×D8311yescharacteristic
248C22×C4C22×D8311yescharacteristic
249C22×C4C22×D8311yescharacteristic
250C22×C4C22×D8311yescharacteristic
251C22×C4C22×D8311yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—1662no—
256C4⋊C4—1662no—
257C4⋊C4—1662no—
258C4⋊C4—1662no—
259C4⋊C4—1662no—
260C4⋊C4—1662no—
261C4⋊C4—1662no—
262C4⋊C4—1662no—
263C4⋊C4—1662no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—1662no—
267C22⋊C4—1662no—
268C22⋊C4—1662no—
269C22⋊C4C22⋊D8314yes—
270C42—22no—
271C42—1662no—
272C42—1662no—
273C42—1662no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23C2×C22⋊D8312yes—
281C23C23×D8311yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—832no—
299C2×C4—832no—
300C2×C4—832no—
301C2×C4—832no—
302C2×C4—1661no—
303C2×C4—1661no—
304C2×C4—1661no—
305C2×C4—1661no—
306C2×C4—1661no—
307C2×C4—1661no—
308C2×C4—1661no—
309C2×C4—1661no—
310C2×C4—1661no—
311C2×C4—1661no—
312C2×C4—1661no—
313C2×C4—1661no—
314C2×C4—1668no—
315C2×C4—1661no—
316C2×C4—1661no—
317C2×C4—1661no—
318C2×C4—1661no—
319C2×C4—1661no—
320C2×C4—1661no—
321C2×C4—1661no—
322C2×C4—1661no—
323C2×C4C2×C22⋊D8312yes—
324C2×C4C2×C22⋊D8312yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C22⋊D8311yescharacteristic
333C22#41×D8311yescharacteristic
334C22#41×D8311yescharacteristic
335C22#41×D8311yescharacteristic
336C22#41×D8311yescharacteristic
337C22#1458183911yescharacteristic
338C22#1458183911yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—3321no—
346C4—3321no—
347C4—3321no—
348C4—3321no—
349C4—1662no—
350C4—1662no—
351C4—1662no—
352C4—1662no—
353C2—42no—
354C2#1912820811yescharacteristic
355C2#1912823411yescharacteristic
356C2#1912830811yescharacteristic
357C2#1912835011yescharacteristic
358C2#1912835011yescharacteristic
359C2#1912838711yescharacteristic
360C2#1912841811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#126×C83C2
prime radicalC24×C83C23
hypercenterC2×#126D83
hypocenterC83#2478
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. #19127634C2semidirect
3. #19127632C2semidirect
6. #19127597C2semidirect
7. #19127596C2semidirect
9. #19127585C2semidirect
10. #19127582C2semidirect
11. #19127582C2semidirect
14. C4×Dic83⋊C4C2semidirect
181. C83#2478semidirect

Maximal subgroups

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

subgroupindex
C2×#145817482
#191276342
#191276322
#191276292
#191275972
#191275962
#191275942
#191275852
#191275822
#191275732
C4×Dic83⋊C42
C2×#126×C832
#247883