Finite Group Databasebeta

← #24918285

Subgroups — #24918285

Order 10624.

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

normal subgroups only

This group has 410 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
1C2×#14581768C211yesmaximal, characteristic
2C22⋊Q8×D83C211yesmaximal, characteristic
3#19128032C211yesmaximal, characteristic, complemented
4#19128030C212yesmaximal, complemented
5#19128029C211yesmaximal, characteristic, complemented
6#19127958C212yesmaximal, complemented
7#19127956C212yesmaximal, complemented
8#19127857C211yesmaximal, characteristic, complemented
9#19127855C212yesmaximal, complemented
10#19127663C211yesmaximal, characteristic
11#451×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C2×#13050580C2212yescomplemented
15C2×#13050573C2212yes—
16#14581772C2211yescharacteristic
17#14581771C2211yescharacteristic
18#14581770—21no—
19#14581769—21no—
20#14581768—21no—
21#14581768—21no—
22#14581768C2212yes—
23#14581768C2211yescharacteristic
24#14581768C2212yes—
25C2×C4⋊Dic83C2211yescharacteristic
26C2×Dic83⋊C4—21no—
27C4⋊C4×D83—21no—
28C4⋊C4×D83C2211yescharacteristic
29C2×(C2×D83)⋊C4—21no—
30C22⋊C4×D83—21no—
31#14581724C2212yescomplemented
32#14581723C2212yescomplemented
33#14581717C2212yescomplemented
34#14581716C2212yescomplemented
35#14581716C2212yescomplemented
36C22⋊(C83⋊C8)C2211yescharacteristic
37(C2×D83)⋊C8C2212yes—
38C22×Q8×C83C2211yescharacteristic
39C2×Q16×C83C2212yescomplemented
40C22⋊Q8×C83C2211yescharacteristic
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83D411yescharacteristic
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46Q8×D83—42no—
47C22×Dic83—21no—
48C22×Dic83D411yescharacteristic
49C2×C4×D83—21no—
50C2×C4×D83—21no—
51C2×C4×D83—42no—
52C2×C4×D83D411yescharacteristic
53C2×C4×D83D411yescharacteristic
54C2×C4×D83C2311yescharacteristic
55C2×C4×D83C2311yescharacteristic
56#13050580—42no—
57#13050573—42no—
58C2×C83⋊C8C2312yes—
59C4⋊Dic83—21no—
60C4⋊Dic83—21no—
61C4⋊Dic83C2312yes—
62C4⋊Dic83C2311yescharacteristic
63Dic83⋊C4—21no—
64Dic83⋊C4—22no—
65Dic83⋊C4—22no—
66Dic83⋊C4—21no—
67Dic83⋊C4—21no—
68C22⋊Dic83—21no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—22no—
71(C2×D83)⋊C4—21no—
72(C2×D83)⋊C4—21no—
73C2×Q8×C83—21no—
74C2×Q8×C83—42no—
75C2×Q8×C83C2312yes—
76C2×Q8×C83C2311yescharacteristic
77C22×C4×C83—21no—
78C22×C4×C83C2311yescharacteristic
79Q16×C83—42no—
80C2×C8×C83C2312yes—
81C4⋊C4×C83—21no—
82C4⋊C4×C83C2311yescharacteristic
83C22⋊C4×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—41no—
88C22×D83C2×D411yescharacteristic
89C22×D83C2×D411yescharacteristic
90Q8×C83—22no—
91Q8×C83—22no—
92Q8×C83—22no—
93Q8×C83—42no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—22no—
98C2×C4×C83—42no—
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2411yescharacteristic
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83—41no—
105C2×Dic83—42no—
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108C4×D83—22no—
109C4×D83—22no—
110C4×D83—41no—
111C4×D83—41no—
112C4×D83—42no—
113C4×D83—42no—
114Dic166—22no—
115Dic166—42no—
116C8×C83—22no—
117C83⋊C8—22no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83#13712yes—
121C22×C83C22×D411yescharacteristic
122C2×D83—21no—
123C2×D83—21no—
124C2×D83—21no—
125C2×D83—21no—
126C2×D83—21no—
127C2×D83—44no—
128C2×D83#13712yes—
129C4×C83—21no—
130C4×C83—41no—
131C4×C83—42no—
132C4×C83—41no—
133C4×C83H(5,𝔽2)12yes—
134Dic83—21no—
135Dic83—21no—
136Dic83—41no—
137Dic83—41no—
138Dic83—42no—
139C2×C83—22no—
140C2×C83#53411yescharacteristic
141C2×C83C2×#13711yescharacteristic
142C2×C83C2×#13711yescharacteristic
143D83—22no—
144D83—41no—
145#3195—831nomaximal, Sylow
146C83#319511yescharacteristic, complemented, Sylow
147C2×C22⋊Q8—831no—
148C2×C22⋊Q8—831no—
149#483—831no—
150#483—832no—
151#483—831no—
152#470—832no—
153#470—832no—
154#451—831no—
155#451—832no—
156#451D8311yescharacteristic, complemented
157#407—831no—
158C22×Q8—831no—
159C22×Q8C2×D8311yescharacteristic
160C23×C4—831no—
161C2×Q16—832no—
162C2×Q16C2×D8312yescomplemented
163C2×M16—832no—
164C22⋊Q8—832no—
165C22⋊Q8—831no—
166C22⋊Q8—832no—
167C22⋊Q8—831no—
168C22⋊Q8—831no—
169C22⋊Q8—1661no—
170C22⋊Q8—1661no—
171C22⋊Q8—1661no—
172C22⋊Q8—1661no—
173C22⋊Q8C2×D8311yescharacteristic
174C2×C4⋊C4—831no—
175C2×C4⋊C4—831no—
176C2×C4⋊C4—1661no—
177C2×C4⋊C4—1661no—
178C2×C22⋊C4—1661no—
179C2×C22⋊C4—1661no—
180#106—832no—
181#106—832no—
182#103—832no—
183#103—832no—
184#103—832no—
185#103C2×D8312yescomplemented
186C22⋊C8—831no—
187C22⋊C8—832no—
188C22⋊C8C2×D8311yescharacteristic
189C24—831no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—3322no—
193C2×Q8—831no—
194C2×Q8—1661no—
195C2×Q8C22×D8311yescharacteristic
196C2×Q8C22×D8312yes—
197C22×C4—21no—
198C22×C4—3322no—
199C22×C4—831no—
200C22×C4—831no—
201C22×C4—831no—
202C22×C4—831no—
203C22×C4—831no—
204C22×C4—1661no—
205C22×C4—1661no—
206C22×C4—1661no—
207C22×C4C22×D8311yescharacteristic
208Q16—42no—
209Q16—3322no—
210M16—3322no—
211C2×C8—832no—
212C2×C8C22×D8312yes—
213C4⋊C4—21no—
214C4⋊C4—831no—
215C4⋊C4—832no—
216C4⋊C4—1661no—
217C4⋊C4—1661no—
218C4⋊C4—1662no—
219C4⋊C4—1662no—
220C4⋊C4—1661no—
221C4⋊C4—1661no—
222C4⋊C4—1661no—
223C4⋊C4C22×D8311yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—1661no—
226C22⋊C4—1661no—
227C22⋊C4—1662no—
228C22⋊C4—1662no—
229C22⋊C4—1661no—
230C23—3321no—
231C23—831no—
232C23—831no—
233C23—1662no—
234C23—1662no—
235C23D4×D8311yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—3322no—
241Q8—1662no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—3321no—
248C2×C4—3321no—
249C2×C4—3322no—
250C2×C4—3322no—
251C2×C4—3322no—
252C2×C4—3321no—
253C2×C4—831no—
254C2×C4—831no—
255C2×C4—1661no—
256C2×C4—1662no—
257C2×C4—1661no—
258C2×C4—1662no—
259C2×C4—1662no—
260C2×C4D4×D8311yescharacteristic
261C2×C4C23×D8311yescharacteristic
262C8—22no—
263C8—1662no—
264C22—21no—
265C22—21no—
266C22—3324no—
267C22—832no—
268C22—1661no—
269C22—1661no—
270C22—1661no—
271C22—1661no—
272C22—1661no—
273C22#1458182412yes—
274C22C2×D4×D8311yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—3321no—
280C4—3322no—
281C4—3321no—
282C4—1661no—
283C4—1661no—
284C4#1458183712yes—
285C2—22no—
286C2—3321no—
287C2—1662no—
288C2#1912826111yescharacteristic
289C2#137×D8311yescharacteristic
290C2C2×#1458182411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#451×C83C2
socleC22×C83C22×D4
prime radicalC23×D83D4
hypercenter#451D83
hypocenterC83#3195
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #19128032C2semidirect
4. #19128030C2semidirect
5. #19128029C2semidirect
6. #19127958C2semidirect
7. #19127956C2semidirect
8. #19127857C2semidirect
9. #19127855C2semidirect
11. #451×C83C2semidirect
14. C2×#13050580C22semidirect
31. #14581724C22semidirect
32. #14581723C22semidirect
33. #14581717C22semidirect
34. #14581716C22semidirect
35. #14581716C22semidirect
39. C2×Q16×C83C22semidirect
41. #103×C83C22semidirect
146. C83#3195semidirect
156. #451D83semidirect
162. C2×Q16C2×D83semidirect
185. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
C22⋊Q8×D832
#191280322
#191280302
#191280292
#191279582
#191279562
#191278572
#191278552
#191276632
#451×C832
#319583