Finite Group Databasebeta

← #24918798

Subgroups — #24918798

Order 10624.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128256C211yesmaximal, characteristic, complemented
2#19128238C211yesmaximal, characteristic
3#19128010C212yesmaximal, complemented
4#19127986C211yesmaximal, characteristic, complemented
5#19127982C211yesmaximal, characteristic, complemented
6#19127939C212yesmaximal, complemented
7#19127933C212yesmaximal
8#19127925C212yesmaximal, complemented
9#19127903C211yesmaximal, characteristic, complemented
10C4⋊C8×D83C211yesmaximal, characteristic, complemented
11#462×C83C211yesmaximal, characteristic, complemented
12C2×#13050577C2212yes—
13C2×C8×D83C2212yes—
14#14581792C2211yescharacteristic
15#14581787—21no—
16#14581783—21no—
17#14581783C2211yescharacteristic
18#14581781C2211yescharacteristic
19#14581777—21no—
20#14581772—21no—
21#14581768C2212yes—
22#14581768C2211yescharacteristic
23#14581767C2211yescharacteristic
24Q8×Dic83C2212yes—
25#14581757—21no—
26C4×D332—21no—
27C4×D332C2211yescharacteristic
28#14581750—21no—
29#14581749—21no—
30C42×D83C2211yescharacteristic
31#14581726C2212yes—
32#14581723C2212yes—
33C4⋊(C83⋊C8)C2211yescharacteristic
34Dic83⋊C8C2212yes—
35#14581716C2212yes—
36#14581716C2212yes—
37#14581714C2212yes—
38C2×Q16×C83C2212yes—
39C4⋊Q8×C83C2211yescharacteristic
40C4×Q8×C83C2211yescharacteristic
41C4⋊C8×C83C2211yescharacteristic
42#103×C83C2212yes—
43C2×Dic166—21no—
44C2×D332—21no—
45C2×D332C2311yescharacteristic
46C2×C4×D83—21no—
47C2×C4×D83D411yescharacteristic
48C2×C4×D83D411yescharacteristic
49C2×C4×D83C2311yescharacteristic
50#13050577—42no—
51C2×C83⋊C8C2312yes—
52C8×D83—42no—
53C4⋊Dic83—21no—
54C4⋊Dic83—22no—
55C4⋊Dic83—21no—
56C4⋊Dic83C2311yescharacteristic
57C4⋊Dic83C2312yes—
58Dic83⋊C4—22no—
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67C4×Dic83—22no—
68C4×Dic83D411yescharacteristic
69C4×Dic83D411yescharacteristic
70C4×Dic83C2311yescharacteristic
71C2×Q8×C83C2311yescharacteristic
72C2×Q8×C83C2312yes—
73Q16×C83—42no—
74C2×C8×C83C2312yes—
75C4⋊C4×C83—21no—
76C4⋊C4×C83—21no—
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2311yescharacteristic
79C42×C83—21no—
80C42×C83C2311yescharacteristic
81C22×D83—21no—
82C22×D83C2×D411yescharacteristic
83Q8×C83—21no—
84Q8×C83—22no—
85Q8×C83—21no—
86Q8×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2411yescharacteristic
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83C2×D411yescharacteristic
96C2×Dic83C2×D411yescharacteristic
97C2×Dic83C2×D411yescharacteristic
98D332—21no—
99D332—21no—
100D332—42no—
101C4×D83—22no—
102C4×D83—22no—
103C4×D83—21no—
104C4×D83—21no—
105C4×D83—42no—
106Dic166—42no—
107C8×C83—22no—
108C83⋊C8—22no—
109C22×C83C22×D411yescharacteristic
110C2×D83—21no—
111C2×D83—21no—
112C2×D83—41no—
113C2×D83—41no—
114C2×D83—41no—
115C2×D83#13712yes—
116C4×C83—21no—
117C4×C83—41no—
118C4×C83—42no—
119C4×C83—41no—
120C4×C83#13512yes—
121C4×C83H(5,𝔽2)11yescharacteristic
122C4×C832−1+411yescharacteristic
123Dic83—21no—
124Dic83—21no—
125Dic83—21no—
126Dic83—42no—
127Dic83—41no—
128C2×C83#53711yescharacteristic
129C2×C83C2×#13511yescharacteristic
130C2×C83C2×#13711yescharacteristic
131D83—22no—
132D83—81no—
133#3248—831nomaximal, Sylow
134C83#324811yescharacteristic, complemented, Sylow
135#533—831no—
136#529—831no—
137#479—832no—
138#475—831no—
139#474—831no—
140#467—832no—
141#465—832no—
142#464—832no—
143#462D8311yescharacteristic, complemented
144#461—831no—
145C2×C4⋊C8—831no—
146C2×Q16C2×D8312yes—
147C2×SD16—832no—
148C22×C8—832no—
149C4⋊Q8—831no—
150C4⋊Q8C2×D8311yescharacteristic
151#126—1661no—
152#125—831no—
153#125—1661no—
154#124—831no—
155#123—1661no—
156C22⋊Q8—831no—
157C22⋊Q8—832no—
158C22⋊Q8—1661no—
159#121—831no—
160C4×Q8—832no—
161C4×Q8—1661no—
162C4×Q8C2×D8311yescharacteristic
163C4×D4—831no—
164C4×D4—1661no—
165#117—1661no—
166#117—1661no—
167C2×C42—831no—
168#107—832no—
169#106—832no—
170C4⋊C8—832no—
171C4⋊C8—831no—
172C4⋊C8C2×D8311yescharacteristic
173#103—832no—
174#103—832no—
175#103C2×D8312yes—
176#102—832no—
177C2×Q8—1661no—
178C2×Q8C22×D8311yescharacteristic
179C2×Q8C22×D8312yes—
180C2×D4—831no—
181C2×D4—1661no—
182C22×C4—831no—
183C22×C4—831no—
184C22×C4—831no—
185C22×C4—1661no—
186Q16—42no—
187SD16—3322no—
188C2×C8—3322no—
189C2×C8—832no—
190C2×C8C22×D8312yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—832no—
195C4⋊C4—831no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C4⋊C4—1662no—
200C4⋊C4—1662no—
201C4⋊C4—1661no—
202C4⋊C4C22×D8311yescharacteristic
203C22⋊C4—1661no—
204C22⋊C4—1661no—
205C22⋊C4—1661no—
206C22⋊C4—1662no—
207C22⋊C4—1661no—
208C22⋊C4—1661no—
209C42—21no—
210C42—831no—
211C42—831no—
212C42—831no—
213C42—1662no—
214C42C22×D8311yescharacteristic
215C23—831no—
216C23—1661no—
217Q8—22no—
218Q8—21no—
219Q8—21no—
220Q8—22no—
221Q8—3322no—
222D4—3322no—
223D4—1661no—
224D4—1661no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—3322no—
229C2×C4—831no—
230C2×C4—831no—
231C2×C4—831no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1661no—
235C2×C4—1662no—
236C2×C4—1661no—
237C2×C4—1661no—
238C2×C4D4×D8311yescharacteristic
239C2×C4D4×D8311yescharacteristic
240C2×C4C23×D8311yescharacteristic
241C8—22no—
242C8—1662no—
243C22—3321no—
244C22—3321no—
245C22—3321no—
246C22—832no—
247C22—1661no—
248C22—1661no—
249C22C2×D4×D8311yescharacteristic
250C4—21no—
251C4—41no—
252C4—41no—
253C4—42no—
254C4—3321no—
255C4—3322no—
256C4—1661no—
257C4—1661no—
258C4—1661no—
259C4#1458181512yes—
260C4#1458183711yescharacteristic
261C4#1458183911yescharacteristic
262C2—6641no—
263C2—1662no—
264C2#1912828011yescharacteristic
265C2C2×#1458181511yescharacteristic
266C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#462×C83C2
socleC22×C83C22×D4
prime radical#14581767C22
hypercenter#462D83
hypocenterC83#3248
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128010C2semidirect
4. #19127986C2semidirect
5. #19127982C2semidirect
6. #19127939C2semidirect
8. #19127925C2semidirect
9. #19127903C2semidirect
10. C4⋊C8×D83C2semidirect
11. #462×C83C2semidirect
134. C83#3248semidirect
143. #462D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282382
#191280102
#191279862
#191279822
#191279392
#191279332
#191279252
#191279032
C4⋊C8×D832
#462×C832
#324883