Finite Group Databasebeta

← #24918428

Subgroups — #24918428

Order 10624.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128235C211yesmaximal, characteristic, complemented
2#19128083C212yesmaximal, complemented
3#19128055C212yesmaximal, complemented
4#19127920C212yesmaximal, complemented
5#19127913C212yesmaximal, complemented
6#19127857C211yesmaximal, characteristic, complemented
7#19127852C211yesmaximal, characteristic, complemented
8#19127840C211yesmaximal, characteristic, complemented
9#19127835C211yesmaximal, characteristic
10#19127754C211yesmaximal, characteristic, complemented
11C2×#124×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yescomplemented
13C2×#13050578C2212yes—
14C2×#13050577C2212yescomplemented
15C2×#13050575C2212yes—
16#14581786—21no—
17#14581784C2211yescharacteristic, complemented
18#14581780C2211yescharacteristic, complemented
19#14581775—21no—
20#14581771C2211yescharacteristic
21#14581766C2211yescharacteristic
22C4×Dic166C2211yescharacteristic, complemented
23C4×D332C2211yescharacteristic, complemented
24C4×C22⋊D83—21no—
25#14581749—21no—
26C4⋊(C83⋊C8)C2212yescomplemented
27#14581716C2212yescomplemented
28#14581716C2212yescomplemented
29#14581713C2212yes—
30#14581713C2212yes—
31C22⋊(C83⋊C8)C2211yescharacteristic
32C22⋊(C83⋊C8)C2211yescharacteristic
33C4×C83⋊C8C2212yescomplemented
34C22×Q8×C83C2211yescharacteristic
35C22×D4×C83C2211yescharacteristic
36#124×C83—21no—
37#124×C83—21no—
38#124×C83C2212yes—
39#124×C83C2212yes—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—21no—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83—21no—
47#13050580—42no—
48#13050578—42no—
49#13050577—42no—
50#13050575—42no—
51C2×C83⋊C8C2312yes—
52C2×C83⋊C8C2312yes—
53C4⋊Dic83C2311yescharacteristic
54C4⋊Dic83C2311yescharacteristic
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59C22⋊Dic83—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64C4×Dic83—21no—
65C24×C83—21no—
66C2×Q8×C83—21no—
67C2×Q8×C83—42no—
68C2×Q8×C83C2312yes—
69C2×D4×C83—21no—
70C2×D4×C83—42no—
71C2×D4×C83C2312yes—
72C22×C4×C83—21no—
73C22×C4×C83D411yescharacteristic
74C22×C4×C83D411yescharacteristic
75C22×C4×C83C2311yescharacteristic
76C22⋊C4×C83—22no—
77C22⋊C4×C83—22no—
78C22⋊C4×C83—22no—
79C22⋊C4×C83—22no—
80C42×C83D411yescharacteristic
81C42×C83D411yescharacteristic
82C42×C83C2311yescharacteristic
83C42×C83C2311yescharacteristic
84C23×C83—22no—
85C23×C83—42no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83C2×D411yescharacteristic
89C22×D83—21no—
90Q8×C83—22no—
91Q8×C83—22no—
92Q8×C83—42no—
93D4×C83—22no—
94D4×C83—22no—
95D4×C83—42no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—41no—
100C2×C4×C83—42no—
101C2×C4×C83C2×D411yescharacteristic
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2×D411yescharacteristic
104C2×C4×C83C2×D411yescharacteristic
105C2×C4×C83C2×D411yescharacteristic
106C2×C4×C83C2411yescharacteristic
107C22⋊D83—42no—
108C2×Dic83—21no—
109C2×Dic83—21no—
110C2×Dic83—21no—
111D332—22no—
112C4×D83—42no—
113Dic166—22no—
114C83⋊C8—22no—
115C83⋊C8—22no—
116C22×C83—21no—
117C22×C83—21no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—84no—
122C22×C83#13512yes—
123C22×C83C22×D411yescharacteristic
124C2×D83—41no—
125C2×D83—41no—
126C2×D83—41no—
127C4×C83—21no—
128C4×C83—21no—
129C4×C83—21no—
130C4×C83—21no—
131C4×C83—21no—
132C4×C83—42no—
133C4×C83H(5,𝔽2)12yes—
134Dic83—41no—
135Dic83—41no—
136Dic83—41no—
137C2×C83—22no—
138C2×C83—82no—
139C2×C83#53511yescharacteristic
140C2×C83C2×#13511yescharacteristic
141C2×C83#57511yescharacteristic
142D83—81no—
143#3211—831nomaximal, Sylow
144C83#321111yescharacteristic, complemented, Sylow
145#529—831no—
146C2×#124D8311yescharacteristic, complemented
147#495—832no—
148#488—832no—
149#464—832no—
150#463—832no—
151#451—831no—
152#450—831no—
153#448—831no—
154#447—831no—
155#431—831no—
156C22×Q8C2×D8311yescharacteristic
157C22×D4C2×D8311yescharacteristic
158C2×Q16—832no—
159C2×SD16—832no—
160C2×SD16—832no—
161C2×D8—832no—
162#126—1661no—
163#125—831no—
164#124—21no—
165#124—21no—
166#124—831no—
167#124C2×D8312yes—
168#124C2×D8312yes—
169#123—1661no—
170C22⋊Q8—831no—
171#121—831no—
172C4×Q8—831no—
173C4×D4—831no—
174C4×D4—1661no—
175#117—1661no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42C2×D8311yescharacteristic
179C4⋊C8—832no—
180#103—832no—
181#103—832no—
182#102—832no—
183#102—832no—
184C22⋊C8—831no—
185C22⋊C8—831no—
186C4×C8—832no—
187C24—21no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8—831no—
191C2×Q8C22×D8312yes—
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—831no—
195C2×D4—1661no—
196C2×D4C22×D8312yes—
197C22×C4—21no—
198C22×C4—1661no—
199C22×C4C22⋊D8311yescharacteristic
200C22×C4C22⋊D8311yescharacteristic
201C22×C4C22×D8311yescharacteristic
202Q16—3322no—
203SD16—3322no—
204SD16—3322no—
205D8—3322no—
206C2×C8—832no—
207C2×C8—832no—
208C4⋊C4—831no—
209C4⋊C4—831no—
210C4⋊C4—1661no—
211C4⋊C4—1661no—
212C4⋊C4—1661no—
213C4⋊C4—1661no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—1661no—
219C22⋊C4—1661no—
220C22⋊C4—1661no—
221C22⋊C4—1661no—
222C22⋊C4—1661no—
223C42—1661no—
224C42C22⋊D8311yescharacteristic
225C42C22⋊D8311yescharacteristic
226C42C22×D8311yescharacteristic
227C42C22×D8311yescharacteristic
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—1661no—
233C23C2×C22⋊D8311yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—42no—
237Q8—1662no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—3322no—
242D4—1662no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—3322no—
249C2×C4—1661no—
250C2×C4—1661no—
251C2×C4—1661no—
252C2×C4C2×C22⋊D8311yescharacteristic
253C2×C4C2×C22⋊D8311yescharacteristic
254C2×C4C2×C22⋊D8311yescharacteristic
255C2×C4C2×C22⋊D8311yescharacteristic
256C2×C4C2×C22⋊D8311yescharacteristic
257C2×C4C23×D8311yescharacteristic
258C8—1662no—
259C8—1662no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—3321no—
267C22—3321no—
268C22—3321no—
269C22#1458181212yes—
270C22C22×C22⋊D8311yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—42no—
277C4—3321no—
278C4—3321no—
279C4—3321no—
280C4#1458183712yes—
281C2—22no—
282C2—82no—
283C2—6641no—
284C2#1912826811yescharacteristic
285C2C2×#1458181211yescharacteristic
286C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#124×C83C2
socleC22×C83C22×D4
prime radical#19127835C2
hypercenterC2×#124D83
hypocenterC83#3211
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #19128235C2semidirect
2. #19128083C2semidirect
3. #19128055C2semidirect
4. #19127920C2semidirect
5. #19127913C2semidirect
6. #19127857C2semidirect
7. #19127852C2semidirect
8. #19127840C2semidirect
10. #19127754C2semidirect
11. C2×#124×C83C2semidirect
12. C2×#13050580C22semidirect
14. C2×#13050577C22semidirect
17. #14581784C22semidirect
18. #14581780C22semidirect
22. C4×Dic166C22semidirect
23. C4×D332C22semidirect
26. C4⋊(C83⋊C8)C22semidirect
27. #14581716C22semidirect
28. #14581716C22semidirect
33. C4×C83⋊C8C22semidirect
144. C83#3211semidirect
146. C2×#124D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191280832
#191280552
#191279202
#191279132
#191278572
#191278522
#191278402
#191278352
#191277542
C2×#124×C832
#321183