Finite Group Databasebeta

← #24918783

Subgroups — #24918783

Order 10624.

398 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128001C212yesmaximal, complemented
3#19127993C211yesmaximal, characteristic, complemented
4#19127988C211yesmaximal, characteristic, complemented
5#19127935C212yesmaximal, complemented
6#19127928C212yesmaximal, complemented
7#19127916C212yesmaximal, complemented
8#19127898C211yesmaximal, characteristic, complemented
9#19127893C211yesmaximal, characteristic, complemented
10C2×C4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
11#532×C83C211yesmaximal, characteristic, complemented
12C2×#13050578C2212yes—
13C2×#13050575C2212yes—
14C22×C83⋊C8C2212yes—
15#14581786—21no—
16#14581784C2211yescharacteristic
17#14581780C2211yescharacteristic
18#14581775—21no—
19#14581771C2211yescharacteristic
20#14581766C2211yescharacteristic
21C4×Dic166C2211yescharacteristic
22C4×D332C2211yescharacteristic
23C4×C22⋊D83—21no—
24#14581749—21no—
25#14581725C2212yes—
26#14581723C2212yes—
27C4⋊(C83⋊C8)C2211yescharacteristic
28C4⋊(C83⋊C8)C2211yescharacteristic
29C4⋊(C83⋊C8)C2212yes—
30#14581715C2212yes—
31#14581713C2212yes—
32#14581713C2212yes—
33#14581712C2212yes—
34C2×#41×C83—21no—
35C4⋊Q8×C83C2211yescharacteristic
36#127×C83C2211yescharacteristic
37#124×C83—21no—
38#121×C83—21no—
39#121×C83C2212yes—
40C4×D4×C83—21no—
41C4×D4×C83C2212yes—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83—21no—
47#13050578—42no—
48#13050575—42no—
49C2×C83⋊C8—42no—
50C2×C83⋊C8C2312yes—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—21no—
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58C22⋊Dic83—21no—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63C4×Dic83—21no—
64#41×C83—44no—
65C2×Q8×C83—21no—
66C2×D4×C83—22no—
67C2×D4×C83—21no—
68C2×D4×C83—21no—
69C2×D4×C83C2312yes—
70C22×C4×C83—22no—
71C22×C4×C83D411yescharacteristic
72C22×C4×C83D411yescharacteristic
73C22×C4×C83C2311yescharacteristic
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2312yes—
76C22⋊C4×C83—22no—
77C22⋊C4×C83—22no—
78C42×C83D411yescharacteristic
79C42×C83D411yescharacteristic
80C42×C83C2311yescharacteristic
81C42×C83C2311yescharacteristic
82C23×C83—22no—
83C23×C83C2×D411yescharacteristic
84C22×D83—21no—
85Q8×C83—42no—
86D4×C83—22no—
87D4×C83—22no—
88D4×C83—42no—
89D4×C83—44no—
90D4×C83—42no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—44no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2×D411yescharacteristic
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2×D411yescharacteristic
102C2×C4×C83C2411yescharacteristic
103C22⋊D83—42no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—21no—
107D332—21no—
108D332—21no—
109C4×D83—42no—
110Dic166—21no—
111Dic166—21no—
112C83⋊C8—22no—
113C83⋊C8—22no—
114C22×C83—21no—
115C22×C83—21no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83AGL(1,ℤ/8)12yes—
120C22×C83C22×D411yescharacteristic
121C2×D83—41no—
122C2×D83—41no—
123C2×D83—41no—
124C4×C83—21no—
125C4×C83—21no—
126C4×C83—21no—
127C4×C83—21no—
128C4×C83—42no—
129C4×C83#13512yes—
130C4×C83H(5,𝔽2)11yescharacteristic
131C4×C832−1+411yescharacteristic
132Dic83—41no—
133Dic83—41no—
134Dic83—41no—
135C2×C83—22no—
136C2×C83—82no—
137C2×C83#53711yescharacteristic
138C2×C83C2×#13511yescharacteristic
139C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
140D83—81no—
141#3247—831nomaximal, Sylow
142C83#324711yescharacteristic, complemented, Sylow
143#532D8311yescharacteristic, complemented
144#529—831no—
145#478—832no—
146#477—831no—
147#476—831no—
148#466—832no—
149#465—832no—
150#463—832no—
151#460—831no—
152#459—831no—
153C2×C4⋊C8—831no—
154C2×#41—21no—
155C2×SD16—832no—
156C2×D8—832no—
157C22×C8—832no—
158C4⋊Q8C2×D8311yescharacteristic
159#127C2×D8311yescharacteristic
160#126—1661no—
161#125—831no—
162#124—21no—
163#124—831no—
164#123—1661no—
165C22⋊Q8—831no—
166#121—21no—
167#121—831no—
168#121C2×D8312yes—
169C4×Q8—831no—
170C4×D4—21no—
171C4×D4—831no—
172C4×D4—1661no—
173C4×D4C2×D8312yes—
174#117—1661no—
175C2×C42C2×D8311yescharacteristic
176#107—832no—
177#106—832no—
178C4⋊C8—831no—
179C4⋊C8—831no—
180C4⋊C8—832no—
181#103—832no—
182#102—832no—
183#102—832no—
184#102—832no—
185#41—44no—
186C2×Q8—21no—
187C2×Q8—831no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—831no—
192C2×D4—1661no—
193C2×D4C22×D8312yes—
194C22×C4—22no—
195C22×C4—1661no—
196C22×C4C22⋊D8311yescharacteristic
197C22×C4C22⋊D8311yescharacteristic
198C22×C4C22×D8311yescharacteristic
199SD16—3322no—
200D8—3322no—
201C2×C8—3322no—
202C2×C8—832no—
203C2×C8—832no—
204C4⋊C4—21no—
205C4⋊C4—831no—
206C4⋊C4—831no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4—1661no—
211C4⋊C4C22×D8312yes—
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—1661no—
215C22⋊C4—1661no—
216C22⋊C4—1661no—
217C22⋊C4—1661no—
218C22⋊C4—1661no—
219C42—1661no—
220C42C22⋊D8311yescharacteristic
221C42C22⋊D8311yescharacteristic
222C42C22×D8311yescharacteristic
223C42C22×D8311yescharacteristic
224C23—22no—
225C23—1661no—
226C23C2×C22⋊D8311yescharacteristic
227Q8—42no—
228Q8—1661no—
229Q8—1661no—
230D4—22no—
231D4—22no—
232D4—42no—
233D4—44no—
234D4—42no—
235D4—3322no—
236D4—1661no—
237D4—1661no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—44no—
244C2×C4—3322no—
245C2×C4—1661no—
246C2×C4—1661no—
247C2×C4—1661no—
248C2×C4C2×C22⋊D8311yescharacteristic
249C2×C4C2×C22⋊D8311yescharacteristic
250C2×C4C2×C22⋊D8311yescharacteristic
251C2×C4C2×C22⋊D8311yescharacteristic
252C2×C4C2×C22⋊D8311yescharacteristic
253C2×C4C23×D8311yescharacteristic
254C8—1662no—
255C8—1662no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—3321no—
262C22—3321no—
263C22—3321no—
264C22#1458181812yes—
265C22C22×C22⋊D8311yescharacteristic
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—3321no—
272C4—3321no—
273C4—3321no—
274C4#1458181212yes—
275C4#1458183811yescharacteristic
276C4#1458183911yescharacteristic
277C2—22no—
278C2—82no—
279C2—6641no—
280C2#1912827911yescharacteristic
281C2C2×#1458181211yescharacteristic
282C2C2×#1458181811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#532×C83C2
socleC22×C83C22×D4
hypercenter#532D83
hypocenterC83#3247
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #19128235C2semidirect
2. #19128001C2semidirect
3. #19127993C2semidirect
4. #19127988C2semidirect
5. #19127935C2semidirect
6. #19127928C2semidirect
7. #19127916C2semidirect
8. #19127898C2semidirect
9. #19127893C2semidirect
10. C2×C4⋊(C83⋊C8)C2semidirect
11. #532×C83C2semidirect
142. C83#3247semidirect
143. #532D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191280012
#191279932
#191279882
#191279352
#191279282
#191279162
#191278982
#191278932
C2×C4⋊(C83⋊C8)2
#532×C832
#324783