Finite Group Databasebeta

← #24918425

Subgroups — #24918425

Order 10624.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#19128232C211yesmaximal, characteristic, complemented
2C2×#14581779C211yesmaximal, characteristic, complemented
3#19128084C212yesmaximal, complemented
4#19128054C212yesmaximal, complemented
5#19127922C212yesmaximal, complemented
6#19127914C212yesmaximal, complemented
7#19127859C211yesmaximal, characteristic, complemented
8#19127852C211yesmaximal, characteristic, complemented
9#19127839C211yesmaximal, characteristic, complemented
10#19127751C211yesmaximal, characteristic, complemented
11#447×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C2×Dic332C2212yescomplemented
14C2×#13050578C2212yes—
15C2×#13050578C2212yescomplemented
16#14581785—21no—
17#14581783C2211yescharacteristic, complemented
18#14581779—21no—
19#14581779—21no—
20#14581779C2212yes—
21#14581779C2212yes—
22#14581779C2211yescharacteristic, complemented
23#14581773—21no—
24#14581771C2211yescharacteristic
25#14581757C2211yescharacteristic, complemented
26D4×Dic83—21no—
27D4×Dic83C2211yescharacteristic, complemented
28#14581748—21no—
29C2×C22⋊Dic83—21no—
30C2×C22⋊Dic83—21no—
31C2×C4×Dic83C2211yescharacteristic
32Dic83⋊C8C2212yescomplemented
33#14581717C2212yescomplemented
34#14581715C2212yescomplemented
35#14581713C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×D8×C83C2212yes—
40#121×C83C2211yescharacteristic
41#102×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166—42no—
45C2×Dic166C2311yescharacteristic
46C2×Dic166C2312yes—
47C22×Dic83—21no—
48C22×Dic83—21no—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51Dic332—42no—
52#13050578—42no—
53#13050578—42no—
54C2×C83⋊C8C2312yes—
55C4⋊Dic83—21no—
56C4⋊Dic83C2311yescharacteristic
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60C22⋊Dic83—21no—
61C22⋊Dic83—21no—
62C22⋊Dic83—21no—
63C22⋊Dic83—22no—
64C22⋊Dic83—22no—
65C22⋊Dic83—22no—
66C22⋊Dic83—22no—
67C22⋊Dic83—21no—
68C4×Dic83—21no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C4×Dic83C2311yescharacteristic
73C24×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83—21no—
76C2×D4×C83—42no—
77C2×D4×C83C2312yes—
78C2×D4×C83C2311yescharacteristic
79C22×C4×C83C2311yescharacteristic
80D8×C83—42no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83C2311yescharacteristic
83C22⋊C4×C83—21no—
84C23×C83—21no—
85C23×C83—22no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83—42no—
89C23×C83C2×D411yescharacteristic
90D4×C83—22no—
91D4×C83—22no—
92D4×C83—22no—
93D4×C83—42no—
94D4×C83—42no—
95C2×C4×C83—22no—
96C2×C4×C83—21no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C2×Dic83—22no—
100C2×Dic83—21no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—41no—
104C2×Dic83—42no—
105C2×Dic83—42no—
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108C2×Dic83C2×D411yescharacteristic
109C2×Dic83C2×D411yescharacteristic
110Dic166—22no—
111Dic166—22no—
112Dic166—22no—
113Dic166—42no—
114C8×C83—22no—
115C83⋊C8—22no—
116C22×C83—21no—
117C22×C83—21no—
118C22×C83—42no—
119C22×C83—41no—
120C22×C83—41no—
121C22×C83—42no—
122C22×C83—41no—
123C22×C83—42no—
124C22×C83—84no—
125C22×C83#13512yes—
126C22×C83C22×D411yescharacteristic
127C4×C83—21no—
128C4×C83—41no—
129C4×C83H(5,𝔽2)12yes—
130Dic83—21no—
131Dic83—21no—
132Dic83—21no—
133Dic83—21no—
134Dic83—42no—
135Dic83—41no—
136Dic83—41no—
137C2×C83—22no—
138C2×C83—81no—
139C2×C83—82no—
140C2×C83#53511yescharacteristic
141C2×C83C2×#13511yescharacteristic
142C2×C83#57511yescharacteristic
143#3211—831nomaximal, Sylow
144C83#321111yescharacteristic, complemented, Sylow
145#529—831no—
146C2×#124—831no—
147#495—832no—
148#488—832no—
149#464—832no—
150#463—832no—
151#451—831no—
152#450—831no—
153#448—831no—
154#447D8311yescharacteristic
155#431—831no—
156C22×Q8—831no—
157C22×D4C2×D8311yescharacteristic
158C2×Q16—832no—
159C2×SD16—832no—
160C2×SD16—832no—
161C2×D8C2×D8312yes—
162#126—1661no—
163#125—831no—
164#124—832no—
165#124—832no—
166#124—831no—
167#124—1661no—
168#124—1661no—
169#123—1661no—
170C22⋊Q8—831no—
171#121C2×D8311yescharacteristic
172C4×Q8—831no—
173C4×D4—831no—
174C4×D4—1661no—
175#117—1661no—
176C2×C22⋊C4—1661no—
177C2×C22⋊C4—1661no—
178C2×C42—831no—
179C4⋊C8—832no—
180#103—832no—
181#103—832no—
182#102—832no—
183#102C2×D8312yes—
184C22⋊C8—831no—
185C22⋊C8C2×D8311yescharacteristic
186C4×C8—832no—
187C24—21no—
188C2×Q8—3322no—
189C2×Q8—832no—
190C2×Q8—831no—
191C2×Q8—1661no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—42no—
195C2×D4C22×D8312yes—
196C2×D4C22×D8311yescharacteristic
197C22×C4—831no—
198C22×C4—831no—
199C22×C4—1661no—
200C22×C4—1661no—
201C22×C4C22×D8311yescharacteristic
202Q16—3322no—
203SD16—3322no—
204SD16—3322no—
205D8—42no—
206C2×C8—832no—
207C2×C8C22×D8312yes—
208C4⋊C4—831no—
209C4⋊C4—1661no—
210C4⋊C4—1661no—
211C4⋊C4—1661no—
212C4⋊C4—1661no—
213C4⋊C4C22×D8311yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1662no—
216C22⋊C4—1662no—
217C22⋊C4—1662no—
218C22⋊C4—1662no—
219C22⋊C4—1661no—
220C22⋊C4—1661no—
221C22⋊C4—1661no—
222C22⋊C4—1661no—
223C42—831no—
224C42—831no—
225C42—831no—
226C42—831no—
227C42—1661no—
228C23—21no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23D4×D8311yescharacteristic
234Q8—3322no—
235Q8—1662no—
236Q8—1662no—
237Q8—1662no—
238D4—22no—
239D4—22no—
240D4—22no—
241D4—42no—
242D4—42no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—3322no—
246C2×C4—3322no—
247C2×C4—3321no—
248C2×C4—831no—
249C2×C4—831no—
250C2×C4—831no—
251C2×C4—831no—
252C2×C4—1661no—
253C2×C4—1662no—
254C2×C4—1661no—
255C2×C4—1662no—
256C2×C4D4×D8311yescharacteristic
257C2×C4C23×D8311yescharacteristic
258C8—22no—
259C8—1662no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—42no—
264C22—41no—
265C22—42no—
266C22—42no—
267C22—41no—
268C22—84no—
269C22#1458181112yes—
270C22C2×D4×D8311yescharacteristic
271C4—21no—
272C4—41no—
273C4—3321no—
274C4—3322no—
275C4—3321no—
276C4—1661no—
277C4—1661no—
278C4—1661no—
279C4—1661no—
280C4#1458183712yes—
281C2—22no—
282C2—82no—
283C2—81no—
284C2#1912826611yescharacteristic
285C2C2×#1458181111yescharacteristic
286C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#447×C83C2
socleC22×C83C22×D4
hypercenter#447D83
hypocenterC83#3211
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #19128232C2semidirect
2. C2×#14581779C2semidirect
3. #19128084C2semidirect
4. #19128054C2semidirect
5. #19127922C2semidirect
6. #19127914C2semidirect
7. #19127859C2semidirect
8. #19127852C2semidirect
9. #19127839C2semidirect
10. #19127751C2semidirect
13. C2×Dic332C22semidirect
15. C2×#13050578C22semidirect
17. #14581783C22semidirect
22. #14581779C22semidirect
25. #14581757C22semidirect
27. D4×Dic83C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581717C22semidirect
34. #14581715C22semidirect
37. C8×Dic83C22semidirect
144. C83#3211semidirect

Maximal subgroups

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

subgroupindex
#191282322
C2×#145817792
#191280842
#191280542
#191279222
#191279142
#191278592
#191278522
#191278392
#191277512
#447×C832
#321183