Finite Group Databasebeta

← #24917938

Subgroups — #24917938

Order 10624.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581824C211yesmaximal, characteristic, complemented
2C2×#14581819C211yesmaximal, characteristic, complemented
3C2×#14581815C211yesmaximal, characteristic, complemented
4C2×SD16×D83C211yesmaximal, characteristic, complemented
5#19128276C211yesmaximal, characteristic, complemented
6#19128269C211yesmaximal, characteristic, complemented
7#19128094C211yesmaximal, characteristic, complemented
8#19128090C211yesmaximal, characteristic, complemented
9#19128087C211yesmaximal, characteristic, complemented
10#19128086C211yesmaximal, characteristic, complemented
11#19128083C211yesmaximal, characteristic, complemented
12#19128075C211yesmaximal, characteristic, complemented
13#19128072C211yesmaximal, characteristic, complemented
14#19127658C211yesmaximal, characteristic, complemented
15#497×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×#13050591C2211yescharacteristic
18C2×Q8×D83C2211yescharacteristic
19C2×D4×D83C2211yescharacteristic
20#14581824—24no—
21#14581819—24no—
22#14581815—24no—
23C2×#13050580C2211yescharacteristic, complemented
24C2×#13050579C2211yescharacteristic
25C2×#13050579C2211yescharacteristic, complemented
26C2×#13050578C2211yescharacteristic
27C2×#13050577C2211yescharacteristic
28C2×#13050577C2211yescharacteristic
29C2×#13050577C2211yescharacteristic, complemented
30C2×#13050577C2211yescharacteristic, complemented
31C2×#13050577C2211yescharacteristic, complemented
32SD16×D83—24no—
33C2×D664C2211yescharacteristic
34C2×D664C2211yescharacteristic, complemented
35C2×#13050575C2211yescharacteristic
36C2×#13050573C2211yescharacteristic
37C2×C8×D83C2211yescharacteristic
38#14581790C2211yescharacteristic, complemented
39#14581789C2211yescharacteristic
40#14581788C2211yescharacteristic
41#14581781C2211yescharacteristic
42#14581781C2211yescharacteristic, complemented
43#14581780C2211yescharacteristic, complemented
44#14581777—22no—
45#14581770—22no—
46#14581765—22no—
47#14581762—22no—
48#14581749C2211yescharacteristic
49#14581702C2211yescharacteristic, complemented
50C4×C83⋊C8C2211yescharacteristic, complemented
51C8×Dic83C2211yescharacteristic, complemented
52C2×Q16×C83C2211yescharacteristic, complemented
53C2×Q16×C83C2211yescharacteristic, complemented
54C2×SD16×C83C2211yescharacteristic
55C2×SD16×C83C2211yescharacteristic
56C4⋊Q8×C83C2211yescharacteristic, complemented
57#124×C83C2211yescharacteristic
58#97×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050591—22no—
61#13050591—22no—
62#13050591—22no—
63#13050591—22no—
64C2×Dic166—21no—
65C2×Dic166C2311yescharacteristic
66Q8×D83—22no—
67Q8×D83—22no—
68C2×D332—21no—
69C2×D332—21no—
70C2×D332—22no—
71C2×D332C2311yescharacteristic
72C2×D332C2311yescharacteristic
73C2×D332C2311yescharacteristic
74C2×C22⋊D83—21no—
75C2×C22⋊D83—22no—
76D4×D83—22no—
77D4×D83—22no—
78C2×C4×D83—21no—
79C2×C4×D83—21no—
80C2×C4×D83—21no—
81C2×C4×D83C2311yescharacteristic
82#13050580—22no—
83#13050579—22no—
84#13050579—22no—
85#13050578—22no—
86#13050577—22no—
87#13050577—22no—
88#13050577—22no—
89#13050577—22no—
90#13050577—22no—
91D664—22no—
92D664—22no—
93#13050575—22no—
94#13050573D414yescomplemented
95C2×C83⋊C8C2311yescharacteristic
96C2×C83⋊C8C2311yescharacteristic
97C8×D83D414yescomplemented
98Dic83⋊C4—22no—
99Dic83⋊C4D412yes—
100(C2×D83)⋊C4—22no—
101(C2×D83)⋊C4—22no—
102(C2×D83)⋊C4—22no—
103(C2×D83)⋊C4D412yes—
104C4×Dic83C2311yescharacteristic
105C2×Q8×C83C2311yescharacteristic
106C2×Q8×C83C2311yescharacteristic
107C2×Q8×C83C2311yescharacteristic
108C2×D4×C83C2311yescharacteristic
109Q16×C83—22no—
110Q16×C83—22no—
111SD16×C83—22no—
112SD16×C83—22no—
113C2×C8×C83C2311yescharacteristic
114C2×C8×C83C2311yescharacteristic
115C4⋊C4×C83—22no—
116C22⋊C4×C83—22no—
117C42×C83C2311yescharacteristic
118C23×C83—21no—
119C22×D83—21no—
120C22×D83—21no—
121C22×D83—21no—
122C22×D83—42no—
123C22×D83—82no—
124C22×D83C2×D411yescharacteristic
125Q8×C83—21no—
126Q8×C83—21no—
127Q8×C83—21no—
128Q8×C83—21no—
129Q8×C83—21no—
130Q8×C83—21no—
131D4×C83—21no—
132D4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83C2×D412yes—
137C2×C4×C83C2411yescharacteristic
138C22⋊D83—42no—
139C22⋊D83—82no—
140C2×Dic83—21no—
141C2×Dic83C2×D412yes—
142C2×Dic83C2×D411yescharacteristic
143D332—21no—
144D332—21no—
145D332—21no—
146D332—21no—
147D332—21no—
148D332—21no—
149D332—42no—
150D332—42no—
151D332—82no—
152C4×D83—42no—
153C4×D83—42no—
154C4×D83—42no—
155C4×D83C2×D412yes—
156C4×D83C2×D412yes—
157Dic166—21no—
158Dic166—21no—
159Dic166—42no—
160C8×C83C2×D412yes—
161C8×C83C2×D412yes—
162C83⋊C8C2×D412yes—
163C83⋊C8C2×D412yes—
164C22×C83—41no—
165C22×C83—42no—
166C22×C83C22×D411yescharacteristic
167C2×D83—22no—
168C2×D83—41no—
169C2×D83—41no—
170C2×D83—41no—
171C2×D83—42no—
172C2×D83—42no—
173C2×D83—42no—
174C2×D83—84no—
175C2×D83#12712yes—
176C4×C83—22no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83C22×D411yescharacteristic
181C4×C83C22×D411yescharacteristic
182Dic83—22no—
183Dic83—41no—
184Dic83#12712yes—
185C2×C83—81no—
186C2×C83C2×#12711yescharacteristic
187C2×C83#57712yes—
188D83—22no—
189D83—81no—
190D83—81no—
191D83—81no—
192#3154—831nomaximal, Sylow
193C83#315411yescharacteristic, complemented, Sylow
194C2×#137—831no—
195C2×AGL(1,ℤ/8)—831no—
196C2×#135—831no—
197C22×SD16—831no—
198#536—831no—
199#535—831no—
200#497—831no—
201#497D8311yescharacteristic, complemented
202#496—831no—
203#496—831no—
204#495—831no—
205#495—831no—
206#492—831no—
207#492—831no—
208#405—831no—
209C2×#41—831no—
210C2×#41—831no—
211C22×Q8—831no—
212C22×D4—831no—
213#137—1664no—
214AGL(1,ℤ/8)—1664no—
215#135—1664no—
216C2×Q16—831no—
217C2×Q16C2×D8311yescharacteristic, complemented
218C2×Q16C2×D8311yescharacteristic, complemented
219C2×SD16—831no—
220C2×SD16—831no—
221C2×SD16—831no—
222C2×SD16—831no—
223C2×SD16—831no—
224C2×SD16—831no—
225C2×SD16—831no—
226C2×SD16—831no—
227C2×SD16—1664no—
228C2×SD16C2×D8311yescharacteristic
229C2×SD16C2×D8311yescharacteristic
230C2×D8—831no—
231C2×D8—831no—
232C2×D8—831no—
233C2×M16—831no—
234C22×C8—831no—
235C4⋊Q8—831no—
236C4⋊Q8C2×D8311yescharacteristic, complemented
237#127—831no—
238#127—831no—
239#124—831no—
240#124—831no—
241#124—831no—
242#124C2×D8311yescharacteristic
243#123—1662no—
244C22⋊Q8—1662no—
245#121—1662no—
246#120—1662no—
247#117—831no—
248#97—831no—
249#97C2×D8311yescharacteristic, complemented
250C4×C8—831no—
251C4×C8—831no—
252C24—1661no—
253#41—1662no—
254#41—1662no—
255#41—1662no—
256#41—1662no—
257C2×Q8—831no—
258C2×Q8—1661no—
259C2×Q8—1662no—
260C2×Q8—1662no—
261C2×Q8C22×D8311yescharacteristic
262C2×Q8C22×D8311yescharacteristic
263C2×Q8C22×D8311yescharacteristic
264C2×D4—831no—
265C2×D4—831no—
266C2×D4—831no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4—1662no—
270C2×D4—1662no—
271C2×D4—1661no—
272C2×D4—1662no—
273C2×D4—1662no—
274C2×D4C22×D8311yescharacteristic
275C22×C4—831no—
276C22×C4—1661no—
277C22×C4—1661no—
278C22×C4—1661no—
279Q16—22no—
280Q16—22no—
281Q16—1662no—
282SD16—22no—
283SD16—22no—
284SD16—1662no—
285SD16—1662no—
286SD16—1662no—
287SD16—1662no—
288SD16—1662no—
289SD16—1662no—
290SD16—1662no—
291SD16—1662no—
292D8—1662no—
293D8—1662no—
294D8—1662no—
295M16—834no—
296C2×C8—831no—
297C2×C8—831no—
298C2×C8—834no—
299C2×C8C22×D8311yescharacteristic
300C2×C8C22×D8311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—832no—
303C4⋊C4—1662no—
304C22⋊C4—22no—
305C22⋊C4—832no—
306C22⋊C4—1662no—
307C22⋊C4—1662no—
308C22⋊C4—1662no—
309C42—831no—
310C42C22×D8311yescharacteristic
311C23—21no—
312C23—3322no—
313C23—831no—
314C23—6642no—
315C23—1661no—
316C23—1661no—
317C23—1661no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—3322no—
325Q8—1661no—
326Q8—1661no—
327D4—21no—
328D4—21no—
329D4—3322no—
330D4—3322no—
331D4—3322no—
332D4—6642no—
333D4—6642no—
334D4—1661no—
335D4—1661no—
336D4—1661no—
337D4—1661no—
338D4—1661no—
339D4—1661no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3322no—
344C2×C4—3322no—
345C2×C4—3322no—
346C2×C4—831no—
347C2×C4—832no—
348C2×C4—832no—
349C2×C4—832no—
350C2×C4—1661no—
351C2×C4D4×D8312yes—
352C2×C4C23×D8311yescharacteristic
353C8—832no—
354C8—832no—
355C8D4×D8312yes—
356C8D4×D8312yes—
357C22—41no—
358C22—42no—
359C22—3321no—
360C22—3321no—
361C22—3321no—
362C22—3322no—
363C22—3322no—
364C22—3322no—
365C22—832no—
366C22—6644no—
367C22—1662no—
368C22C2×D4×D8311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3321no—
374C4—832no—
375C4—1662no—
376C4C2×D4×D8311yescharacteristic
377C4C2×D4×D8311yescharacteristic
378C2—81no—
379C2—6641no—
380C2—6641no—
381C2—6641no—
382C2—1662no—
383C2#127×D8311yescharacteristic
384C2#1912851012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#497×C83C2
socleC22×C83C22×D4
hypercenter#497D83
hypocenterC83#3154
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#14581824C2semidirect
2. C2×#14581819C2semidirect
3. C2×#14581815C2semidirect
4. C2×SD16×D83C2semidirect
5. #19128276C2semidirect
6. #19128269C2semidirect
7. #19128094C2semidirect
8. #19128090C2semidirect
9. #19128087C2semidirect
10. #19128086C2semidirect
11. #19128083C2semidirect
12. #19128075C2semidirect
13. #19128072C2semidirect
14. #19127658C2semidirect
15. #497×C83C2semidirect
23. C2×#13050580C22semidirect
25. C2×#13050579C22semidirect
29. C2×#13050577C22semidirect
30. C2×#13050577C22semidirect
31. C2×#13050577C22semidirect
34. C2×D664C22semidirect
38. #14581790C22semidirect
42. #14581781C22semidirect
43. #14581780C22semidirect
49. #14581702C22semidirect
50. C4×C83⋊C8C22semidirect
51. C8×Dic83C22semidirect
52. C2×Q16×C83C22semidirect
53. C2×Q16×C83C22semidirect
56. C4⋊Q8×C83C22semidirect
58. #97×C83C22semidirect
94. #13050573D4semidirect
97. C8×D83D4semidirect
193. C83#3154semidirect
201. #497D83semidirect
217. C2×Q16C2×D83semidirect
218. C2×Q16C2×D83semidirect
236. C4⋊Q8C2×D83semidirect
249. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818242
C2×#145818192
C2×#145818152
C2×SD16×D832
#191282762
#191282692
#191280942
#191280902
#191280872
#191280862
#191280832
#191280752
#191280722
#191276582
#497×C832
#315483