Finite Group Databasebeta

← #24918464

Subgroups — #24918464

Order 10624.

400 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128255C211yesmaximal, characteristic, complemented
2#124×D83C211yesmaximal, characteristic
3#19128094C211yesmaximal, characteristic, complemented
4#19128092C211yesmaximal, characteristic, complemented
5#19128062C211yesmaximal, characteristic, complemented
6#19128059C211yesmaximal, characteristic, complemented
7#19127924C211yesmaximal, characteristic, complemented
8#19127923C211yesmaximal, characteristic, complemented
9#19127920C211yesmaximal, characteristic, complemented
10#19127858C211yesmaximal, characteristic, complemented
11#19127855C211yesmaximal, characteristic, complemented
12#19127853C211yesmaximal, characteristic
13#19127850C211yesmaximal, characteristic
14#19127757C211yesmaximal, characteristic, complemented
15#464×C83C211yesmaximal, characteristic, complemented
16C2×Q8×D83C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×Dic332C2211yescharacteristic, complemented
19C2×#13050580C2211yescharacteristic, complemented
20C2×#13050580C2211yescharacteristic, complemented
21C2×#13050579C2211yescharacteristic
22C2×#13050578C2211yescharacteristic
23C2×#13050577C2211yescharacteristic
24#14581791C2211yescharacteristic, complemented
25#14581790C2211yescharacteristic, complemented
26#14581782—21no—
27#14581781C2211yescharacteristic
28#14581780C2211yescharacteristic
29#14581779C2211yescharacteristic
30#14581778—21no—
31#14581778—21no—
32#14581772C2211yescharacteristic
33#14581770—21no—
34#14581768C2211yescharacteristic
35C4×Dic166—21no—
36C4×Dic166C2211yescharacteristic, complemented
37#14581749—21no—
38C22⋊C4×D83—21no—
39C22⋊C4×D83—21no—
40C42×D83C2211yescharacteristic
41C4⋊(C83⋊C8)C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581712C2211yescharacteristic
48(C2×D83)⋊C8C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50#14581702C2211yescharacteristic, complemented
51#14581702C2211yescharacteristic, complemented
52C2×Q16×C83C2211yescharacteristic, complemented
53C2×SD16×C83C2211yescharacteristic
54#124×C83C2211yescharacteristic
55C4×Q8×C83C2211yescharacteristic, complemented
56C4⋊C8×C83C2211yescharacteristic, complemented
57#103×C83C2211yescharacteristic, complemented
58#102×C83C2211yescharacteristic
59C23×D83—21no—
60C2×Dic166—21no—
61C2×Dic166—21no—
62C2×Dic166C2311yescharacteristic
63C2×Dic166C2311yescharacteristic
64Q8×D83—41no—
65Q8×D83—41no—
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68D4×D83—41no—
69D4×D83—41no—
70C2×C4×D83—21no—
71C2×C4×D83D411yescharacteristic
72C2×C4×D83D411yescharacteristic
73C2×C4×D83C2311yescharacteristic
74Dic332—41no—
75#13050580—41no—
76#13050580—41no—
77#13050579—41no—
78#13050578—41no—
79#13050577—41no—
80C2×C83⋊C8C2311yescharacteristic
81C2×C83⋊C8C2311yescharacteristic
82C4⋊Dic83—21no—
83C4⋊Dic83C2311yescharacteristic
84Dic83⋊C4—21no—
85Dic83⋊C4—21no—
86Dic83⋊C4—21no—
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97C4×Dic83—21no—
98C4×Dic83D411yescharacteristic
99C4×Dic83D411yescharacteristic
100C4×Dic83C2311yescharacteristic
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C2×D4×C83C2311yescharacteristic
104Q16×C83—41no—
105SD16×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C42×C83—21no—
113C42×C83C2311yescharacteristic
114C23×C83—21no—
115C22×D83—21no—
116C22×D83—42no—
117C22×D83—42no—
118C22×D83—42no—
119C22×D83C2×D411yescharacteristic
120Q8×C83—21no—
121Q8×C83—21no—
122Q8×C83—21no—
123Q8×C83—21no—
124D4×C83—21no—
125D4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2411yescharacteristic
132C22⋊D83—42no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83C2×D411yescharacteristic
137C2×Dic83C2×D411yescharacteristic
138C2×Dic83C2×D411yescharacteristic
139D332—21no—
140D332—21no—
141C4×D83—22no—
142C4×D83—21no—
143C4×D83—21no—
144C4×D83—41no—
145C4×D83—42no—
146Dic166—21no—
147Dic166—21no—
148Dic166—21no—
149Dic166—21no—
150Dic166—42no—
151Dic166—42no—
152C8×C83—21no—
153C8×C83—21no—
154C83⋊C8—21no—
155C83⋊C8—21no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83C22×D411yescharacteristic
160C2×D83—21no—
161C2×D83—21no—
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—84no—
166C2×D83#13712yes—
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83H(5,𝔽2)11yescharacteristic
173C4×C83H(5,𝔽2)11yescharacteristic
174Dic83—21no—
175Dic83—21no—
176Dic83—21no—
177Dic83—41no—
178Dic83—41no—
179Dic83—41no—
180C2×C83—81no—
181C2×C83#53511yescharacteristic
182C2×C83C2×#13711yescharacteristic
183C2×C83#57511yescharacteristic
184D83—22no—
185D83—81no—
186#3215—831nomaximal, Sylow
187C83#321511yescharacteristic, complemented, Sylow
188#533—831no—
189C2×#124—831no—
190#497—831no—
191#497—831no—
192#489—831no—
193#489—831no—
194#464—831no—
195#464—831no—
196#464—831no—
197#464D8311yescharacteristic, complemented
198#451—831no—
199#451—831no—
200#450—831no—
201#450—831no—
202#432—831no—
203C22×Q8—831no—
204C22×D4—831no—
205C2×Q16—831no—
206C2×Q16—831no—
207C2×Q16—831no—
208C2×Q16C2×D8311yescharacteristic, complemented
209C2×SD16—831no—
210C2×SD16—831no—
211C2×SD16—831no—
212C2×SD16C2×D8311yescharacteristic
213C4⋊Q8—831no—
214C4⋊Q8—831no—
215#125—1661no—
216#124—831no—
217#124—831no—
218#124—831no—
219#124—1661no—
220#124—1661no—
221#124C2×D8311yescharacteristic
222C22⋊Q8—831no—
223C22⋊Q8—831no—
224C22⋊Q8—1661no—
225C4×Q8—831no—
226C4×Q8—1661no—
227C4×Q8C2×D8311yescharacteristic, complemented
228#117—1661no—
229C2×C22⋊C4—1661no—
230C2×C22⋊C4—1661no—
231C2×C42—831no—
232C4⋊C8—831no—
233C4⋊C8C2×D8311yescharacteristic, complemented
234#103—831no—
235#103—831no—
236#103—831no—
237#103C2×D8311yescharacteristic, complemented
238#102—831no—
239#102—831no—
240#102—831no—
241#102C2×D8311yescharacteristic
242C22⋊C8—831no—
243C22⋊C8—831no—
244#97—831no—
245#97—831no—
246C24—1661no—
247C2×Q8—3321no—
248C2×Q8—3321no—
249C2×Q8—831no—
250C2×Q8—831no—
251C2×Q8—1661no—
252C2×Q8—1661no—
253C2×Q8C22×D8311yescharacteristic
254C2×Q8C22×D8311yescharacteristic
255C2×D4—3321no—
256C2×D4—3321no—
257C2×D4—831no—
258C2×D4—1661no—
259C2×D4C22×D8311yescharacteristic
260C22×C4—831no—
261C22×C4—831no—
262C22×C4—831no—
263C22×C4—1661no—
264Q16—41no—
265Q16—3321no—
266Q16—3321no—
267Q16—3321no—
268SD16—41no—
269SD16—3321no—
270SD16—3321no—
271SD16—3321no—
272C2×C8—831no—
273C2×C8—831no—
274C2×C8C22×D8311yescharacteristic
275C2×C8C22×D8311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—831no—
278C4⋊C4—1661no—
279C4⋊C4—1661no—
280C4⋊C4—1661no—
281C4⋊C4—1661no—
282C4⋊C4—1661no—
283C4⋊C4—1661no—
284C4⋊C4C22×D8311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C42—21no—
296C42—831no—
297C42—831no—
298C42—831no—
299C42—1661no—
300C42C22×D8311yescharacteristic
301C23—21no—
302C23—3322no—
303C23—3322no—
304C23—3322no—
305C23—831no—
306C23—1661no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—3322no—
312Q8—3322no—
313Q8—1661no—
314Q8—1661no—
315Q8—1661no—
316Q8—1661no—
317D4—21no—
318D4—21no—
319D4—3322no—
320D4—1661no—
321D4—1661no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—3321no—
326C2×C4—3322no—
327C2×C4—831no—
328C2×C4—831no—
329C2×C4—831no—
330C2×C4—1662no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4—1661no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4D4×D8311yescharacteristic
337C2×C4D4×D8311yescharacteristic
338C2×C4C23×D8311yescharacteristic
339C8—21no—
340C8—21no—
341C8—1661no—
342C8—1661no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3321no—
347C22—3321no—
348C22—3321no—
349C22—832no—
350C22—6644no—
351C22—1661no—
352C22—1661no—
353C22C2×D4×D8311yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—3321no—
360C4—3321no—
361C4—3321no—
362C4—1661no—
363C4—1661no—
364C4—1661no—
365C4#1458183711yescharacteristic
366C4#1458183811yescharacteristic
367C2—81no—
368C2—6641no—
369C2—1662no—
370C2#1912826711yescharacteristic
371C2#137×D8311yescharacteristic
372C2#1912849811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#464×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×D83C22
hypercenter#464D83
hypocenterC83#3215
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #19128255C2semidirect
3. #19128094C2semidirect
4. #19128092C2semidirect
5. #19128062C2semidirect
6. #19128059C2semidirect
7. #19127924C2semidirect
8. #19127923C2semidirect
9. #19127920C2semidirect
10. #19127858C2semidirect
11. #19127855C2semidirect
14. #19127757C2semidirect
15. #464×C83C2semidirect
18. C2×Dic332C22semidirect
19. C2×#13050580C22semidirect
20. C2×#13050580C22semidirect
24. #14581791C22semidirect
25. #14581790C22semidirect
36. C4×Dic166C22semidirect
41. C4⋊(C83⋊C8)C22semidirect
42. #14581717C22semidirect
43. #14581716C22semidirect
44. #14581715C22semidirect
50. #14581702C22semidirect
51. #14581702C22semidirect
52. C2×Q16×C83C22semidirect
55. C4×Q8×C83C22semidirect
56. C4⋊C8×C83C22semidirect
57. #103×C83C22semidirect
187. C83#3215semidirect
197. #464D83semidirect
208. C2×Q16C2×D83semidirect
227. C4×Q8C2×D83semidirect
233. C4⋊C8C2×D83semidirect
237. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282552
#124×D832
#191280942
#191280922
#191280622
#191280592
#191279242
#191279232
#191279202
#191278582
#191278552
#191278532
#191278502
#191277572
#464×C832
#321583