Finite Group Databasebeta

← #24919691

Subgroups — #24919691

Order 10624.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128425C211yesmaximal, characteristic, complemented
2D4×D332C211yesmaximal, characteristic
3#19128049C211yesmaximal, characteristic, complemented
4#19128028C212yesmaximal, complemented
5#19127948C212yesmaximal, complemented
6#19127902C211yesmaximal, characteristic, complemented
7#19127853C212yesmaximal, complemented
8C4×#13050578C211yesmaximal, characteristic, complemented
9#19127778C211yesmaximal, characteristic, complemented
10#19127704C212yesmaximal, complemented
11#474×C83C211yesmaximal, characteristic, complemented
12C22×D332—21no—
13C2×D4×D83C2212yes—
14C2×#13050579C2212yescomplemented
15C2×#13050578C2211yescharacteristic
16C2×#13050573C2212yescomplemented
17#14581789C2211yescharacteristic
18#14581787—22no—
19#14581781—21no—
20#14581772C2212yescomplemented
21#14581767C2212yes—
22#14581766—21no—
23#14581762—22no—
24Q8×Dic83—21no—
25C4×Dic166C2211yescharacteristic, complemented
26C4×D332C2211yescharacteristic
27#14581750C2212yescomplemented
28#14581723C2211yescharacteristic, complemented
29#14581715C2211yescharacteristic, complemented
30#14581714C2212yescomplemented
31#14581713C2211yescharacteristic
32#14581712C2212yescomplemented
33#14581712C2212yescomplemented
34(C2×D83)⋊C8C2212yescomplemented
35C4×C83⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C83C2211yescharacteristic, complemented
37C4×D4×C83C2211yescharacteristic
38#107×C83C2212yescomplemented
39C4⋊C8×C83C2211yescharacteristic, complemented
40#102×C83C2212yes—
41C23×D83—22no—
42C2×Dic166C2311yescharacteristic
43C2×D332—22no—
44C2×D332—21no—
45C2×D332—21no—
46C2×D332—44no—
47C2×D332D411yescharacteristic
48C2×D332C2312yes—
49C2×C22⋊D83—22no—
50D4×D83—24no—
51D4×D83—42no—
52C2×C4×D83C2312yes—
53#13050579—42no—
54#13050578—22no—
55#13050573—42no—
56C2×C83⋊C8C2311yescharacteristic
57C2×C83⋊C8C2311yescharacteristic
58C4⋊Dic83—22no—
59C4⋊Dic83D411yescharacteristic
60C4⋊Dic83C2311yescharacteristic
61Dic83⋊C4—21no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4D412yes—
66C4×Dic83—21no—
67C4×Dic83—22no—
68C2×Q8×C83—21no—
69C2×D4×C83C2311yescharacteristic
70C22×C4×C83—21no—
71C2×C8×C83C2312yes—
72C4⋊C4×C83—21no—
73C4⋊C4×C83C2311yescharacteristic
74C4⋊C4×C83C2312yes—
75C22⋊C4×C83—21no—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C22×D83—22no—
79C22×D83—44no—
80C22×D83—44no—
81C22×D83—82no—
82C22×D83C2×D412yes—
83Q8×C83—42no—
84D4×C83—21no—
85D4×C83#4112yes—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—42no—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2411yescharacteristic
92C22⋊D83—44no—
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83C2×D412yes—
96D332—22no—
97D332—22no—
98D332—22no—
99D332—44no—
100D332—42no—
101D332—81no—
102C4×D83—22no—
103C4×D83—22no—
104Dic166—21no—
105Dic166—21no—
106C8×C83—22no—
107C83⋊C8—21no—
108C83⋊C8—21no—
109C22×C83—22no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83C22×D411yescharacteristic
113C2×D83—22no—
114C2×D83—22no—
115C2×D83—22no—
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—42no—
119C2×D83—88no—
120C4×C83—21no—
121C4×C83—41no—
122C4×C83—42no—
123C4×C83AGL(1,ℤ/8)12yes—
124C4×C83C2×#4111yescharacteristic
125C4×C83H(5,𝔽2)11yescharacteristic
126Dic83—22no—
127Dic83—41no—
128Dic83—41no—
129C2×C83—42no—
130C2×C83#54611yescharacteristic
131C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C83#57711yescharacteristic
133D83—42no—
134D83—82no—
135#3324—831nomaximal, Sylow
136C83#332411yescharacteristic, complemented, Sylow
137#556—831no—
138D42—831no—
139#486—831no—
140#482—832no—
141#474D8311yescharacteristic, complemented
142#468—832no—
143#460—831no—
144#450—832no—
145C4×SD16—831no—
146#436—831no—
147#418—832no—
148C22×D4—832no—
149C22×D4—1661no—
150C2×SD16—831no—
151C2×SD16—832no—
152C2×M16—832no—
153C4⋊Q8C2×D8311yescharacteristic, complemented
154#127—831no—
155#126—1662no—
156#124—1661no—
157C22⋊Q8—832no—
158#121—832no—
159#121—1661no—
160#120—1662no—
161C4×Q8—831no—
162C4×Q8—1661no—
163C4×D4—831no—
164C4×D4C2×D8311yescharacteristic
165#117—832no—
166#107C2×D8312yescomplemented
167#106—831no—
168C4⋊C8C2×D8311yescharacteristic, complemented
169#103—831no—
170#102—831no—
171#102—832no—
172#102—832no—
173#102—832no—
174#102C2×D8312yes—
175C22⋊C8—832no—
176C4×C8—831no—
177C24—1662no—
178C2×Q8—21no—
179C2×Q8—831no—
180C2×D4—3324no—
181C2×D4—3322no—
182C2×D4—831no—
183C2×D4—832no—
184C2×D4—1662no—
185C2×D4—1662no—
186C2×D4—1664no—
187C2×D4—1661no—
188C2×D4—1661no—
189C2×D4C22×D8311yescharacteristic
190C22×C4—21no—
191C22×C4—832no—
192SD16—3322no—
193SD16—1662no—
194M16—3322no—
195C2×C8—831no—
196C2×C8—831no—
197C2×C8C22×D8312yes—
198C4⋊C4—21no—
199C4⋊C4—831no—
200C4⋊C4—831no—
201C4⋊C4—1662no—
202C4⋊C4—1661no—
203C4⋊C4C22×D8312yes—
204C4⋊C4C22×D8311yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—832no—
207C22⋊C4—1662no—
208C22⋊C4—1662no—
209C22⋊C4—1662no—
210C42—1662no—
211C42—1661no—
212C42C22×D8311yescharacteristic
213C23—21no—
214C23—3324no—
215C23—3324no—
216C23—832no—
217C23—6642no—
218C23—1662no—
219Q8—42no—
220Q8—1661no—
221Q8—1661no—
222D4—21no—
223D4—3324no—
224D4—3324no—
225D4—3322no—
226D4—6641no—
227D4—1662no—
228D4—1662no—
229D4—1662no—
230D4#1305059112yes—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—832no—
235C2×C4—1661no—
236C2×C4—1661no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4D4×D8311yescharacteristic
240C2×C4D4×D8311yescharacteristic
241C2×C4C23×D8311yescharacteristic
242C8—22no—
243C8—1661no—
244C8—1661no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—3322no—
249C22—3322no—
250C22—3322no—
251C22—6648no—
252C22—1662no—
253C22—1662no—
254C22—1662no—
255C22C2×D4×D8311yescharacteristic
256C4—21no—
257C4—41no—
258C4—42no—
259C4—3321no—
260C4—3321no—
261C4—1662no—
262C4#1458181612yes—
263C4C2×#1305059111yescharacteristic
264C4#1458183811yescharacteristic
265C2—42no—
266C2—3322no—
267C2—6642no—
268C2#1912834211yescharacteristic
269C2C2×#1458181611yescharacteristic
270C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#474×C83C2
socleC22×C83C22×D4
prime radicalD4×D332C2
hypercenter#474D83
hypocenterC83#3324
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #19128425C2semidirect
3. #19128049C2semidirect
4. #19128028C2semidirect
5. #19127948C2semidirect
6. #19127902C2semidirect
7. #19127853C2semidirect
8. C4×#13050578C2semidirect
9. #19127778C2semidirect
10. #19127704C2semidirect
11. #474×C83C2semidirect
14. C2×#13050579C22semidirect
16. C2×#13050573C22semidirect
20. #14581772C22semidirect
25. C4×Dic166C22semidirect
27. #14581750C22semidirect
28. #14581723C22semidirect
29. #14581715C22semidirect
30. #14581714C22semidirect
32. #14581712C22semidirect
33. #14581712C22semidirect
34. (C2×D83)⋊C8C22semidirect
35. C4×C83⋊C8C22semidirect
36. C4⋊Q8×C83C22semidirect
38. #107×C83C22semidirect
39. C4⋊C8×C83C22semidirect
136. C83#3324semidirect
141. #474D83semidirect
153. C4⋊Q8C2×D83semidirect
166. #107C2×D83semidirect
168. C4⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284252
D4×D3322
#191280492
#191280282
#191279482
#191279022
#191278532
C4×#130505782
#191277782
#191277042
#474×C832
#332483