Finite Group Databasebeta

← #24919203

Subgroups — #24919203

Order 10624.

482 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 123 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×#14581811C211yesmaximal, characteristic, complemented
2C2×Q16×D83C211yesmaximal, characteristic, complemented
3#19128359C212yesmaximal, complemented
4#19127939C211yesmaximal, characteristic, complemented
5#19127934C211yesmaximal, characteristic, complemented
6#19127920C212yesmaximal, complemented
7#19127860C212yesmaximal, complemented
8#19127838C212yesmaximal, complemented
9C4×Dic332C211yesmaximal, characteristic, complemented
10C8×D332C211yesmaximal, characteristic, complemented
11#495×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2212yes—
13C2×Q8×D83C2212yes—
14#14581811—24no—
15C2×Dic332C2211yescharacteristic
16C2×#13050580C2212yes—
17Q16×D83—24no—
18C2×#13050578C2212yes—
19C2×C8×D83C2211yescharacteristic
20C2×C8×D83C2211yescharacteristic
21#14581778—22no—
22#14581769—22no—
23#14581768C2212yes—
24#14581765—22no—
25#14581763C2212yes—
26C4×Dic166C2212yes—
27C4×D332C2211yescharacteristic
28(C22⋊D83)⋊C4—22no—
29#14581726C2211yescharacteristic
30C4⋊(C83⋊C8)C2211yescharacteristic
31#14581717C2212yes—
32#14581716C2212yes—
33#14581713C2212yes—
34(C2×D83)⋊C8C2211yescharacteristic
35(C2×D83)⋊C8C2211yescharacteristic
36C2×Q16×C83C2211yescharacteristic
37C2×SD16×C83C2212yes—
38C2×D8×C83C2211yescharacteristic
39#124×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41#13050589—24no—
42#13050589—42no—
43C2×Dic166—22no—
44C2×Dic166C2312yes—
45Q8×D83—24no—
46Q8×D83—42no—
47C2×D332D411yescharacteristic
48C2×C22⋊D83—22no—
49C2×C22⋊D83—22no—
50C22×Dic83—22no—
51C2×C4×D83—22no—
52C2×C4×D83C2311yescharacteristic
53C2×C4×D83C2311yescharacteristic
54Dic332D414yescomplemented
55#13050580—24no—
56#13050578—24no—
57C2×C83⋊C8C2311yescharacteristic
58C2×C83⋊C8C2311yescharacteristic
59C8×D83—22no—
60C8×D83—22no—
61C4⋊Dic83D411yescharacteristic
62C4⋊Dic83C2312yes—
63Dic83⋊C4—22no—
64Dic83⋊C4—22no—
65C22⋊Dic83—22no—
66(C2×D83)⋊C4—22no—
67(C2×D83)⋊C4D411yescharacteristic
68(C2×D83)⋊C4D411yescharacteristic
69C4×Dic83—22no—
70C2×Q8×C83C2312yes—
71C2×D4×C83C2312yes—
72Q16×C83—22no—
73SD16×C83—42no—
74D8×C83—22no—
75C2×C8×C83C2311yescharacteristic
76C2×C8×C83C2311yescharacteristic
77C22⋊C4×C83—22no—
78C22⋊C4×C83—22no—
79C42×C83C2311yescharacteristic
80C23×C83—22no—
81C22×D83C2×D411yescharacteristic
82C22×D83C2×D411yescharacteristic
83Q8×C83—22no—
84Q8×C83—22no—
85D4×C83—22no—
86D4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2411yescharacteristic
91C22⋊D83—44no—
92C22⋊D83—44no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—44no—
96C2×Dic83C2×D411yescharacteristic
97C2×Dic83C2×D411yescharacteristic
98D332—22no—
99C4×D83—21no—
100C4×D83—21no—
101C4×D83—21no—
102C4×D83—21no—
103C4×D83—44no—
104Dic166—22no—
105Dic166—44no—
106Dic166C2×D414yes—
107C8×C83—21no—
108C8×C83C2×D412yes—
109C83⋊C8—21no—
110C83⋊C8—21no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83C22×D411yescharacteristic
115C2×D83—21no—
116C2×D83—21no—
117C2×D83—21no—
118C2×D83—21no—
119C2×D83—21no—
120C2×D83#13512yescomplemented
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—42no—
124C4×C83C22×D411yescharacteristic
125C4×C83H(5,𝔽2)11yescharacteristic
126Dic83—24no—
127Dic83—21no—
128Dic83—21no—
129Dic83—42no—
130C2×C83—82no—
131C2×C83D4211yescharacteristic
132C2×C83C2×#13511yescharacteristic
133C2×C83#57811yescharacteristic
134D83—22no—
135D83—41no—
136#3285—831nomaximal, Sylow
137C83#328511yescharacteristic, complemented, Sylow
138C2×#135—831no—
139C22×Q16—831no—
140#548—832no—
141#495D8311yescharacteristic, complemented
142#467—831no—
143#466—831no—
144#464—832no—
145#452—832no—
146#448—832no—
147C4×Q16—831no—
148C8×D4—831no—
149C2×#41—832no—
150C22×Q8—832no—
151#135—1664no—
152C2×Q16—832no—
153C2×Q16—831no—
154C2×Q16—1664no—
155C2×Q16C2×D8311yescharacteristic
156C2×SD16—832no—
157C2×SD16C2×D8312yes—
158C2×D8C2×D8311yescharacteristic
159C22×C8—831no—
160C22×C8—831no—
161#124—1662no—
162#124C2×D8312yes—
163C22⋊Q8—832no—
164C22⋊Q8—1662no—
165#121—832no—
166#121—1662no—
167C4×Q8—832no—
168C4×D4—831no—
169C4×D4—1662no—
170#107—831no—
171C4⋊C8—831no—
172#103—832no—
173#103—832no—
174#102—832no—
175C22⋊C8—831no—
176C22⋊C8—831no—
177C4×C8C2×D8311yescharacteristic
178#41—3322no—
179#41—1664no—
180C2×Q8—3322no—
181C2×Q8—832no—
182C2×Q8—1662no—
183C2×Q8—1664no—
184C2×Q8C22×D8312yes—
185C2×D4—831no—
186C2×D4—1662no—
187C2×D4—1662no—
188C2×D4C22×D8312yes—
189C22×C4—831no—
190C22×C4—831no—
191C22×C4—1662no—
192C22×C4—1662no—
193Q16—22no—
194Q16—834no—
195Q16—1664no—
196SD16—42no—
197SD16—1664no—
198D8—22no—
199C2×C8—831no—
200C2×C8—831no—
201C2×C8—1662no—
202C2×C8—1662no—
203C2×C8C22×D8311yescharacteristic
204C2×C8C22×D8311yescharacteristic
205C4⋊C4—831no—
206C4⋊C4—832no—
207C4⋊C4—1662no—
208C4⋊C4—1662no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—831no—
212C22⋊C4—831no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C42—1662no—
216C42C22×D8311yescharacteristic
217C23—22no—
218C23—831no—
219C23—831no—
220Q8—22no—
221Q8—22no—
222Q8—3324no—
223Q8—834no—
224Q8—1662no—
225D4—22no—
226D4—22no—
227D4—3324no—
228D4—3324no—
229D4—1662no—
230C2×C4—22no—
231C2×C4—3324no—
232C2×C4—3324no—
233C2×C4—831no—
234C2×C4—831no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1661no—
238C2×C4—1661no—
239C2×C4—1661no—
240C2×C4—1661no—
241C2×C4D4×D8311yescharacteristic
242C2×C4D4×D8311yescharacteristic
243C2×C4C23×D8311yescharacteristic
244C8—21no—
245C8—1661no—
246C8—1661no—
247C8D4×D8312yes—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—832no—
252C22—1661no—
253C22—1661no—
254C22—1661no—
255C22—1661no—
256C22—1661no—
257C22C2×D4×D8311yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—3322no—
262C4—1661no—
263C4—1664no—
264C4—1661no—
265C4C2×D4×D8311yescharacteristic
266C4#1458183711yescharacteristic
267C2—82no—
268C2—3321no—
269C2—1662no—
270C2#1912833611yescharacteristic
271C2#135×D8311yescharacteristic
272C2#1912851511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#495×C83C2
socleC22×C83C22×D4
hypercenter#495D83
hypocenterC83#3285
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#14581811C2semidirect
2. C2×Q16×D83C2semidirect
3. #19128359C2semidirect
4. #19127939C2semidirect
5. #19127934C2semidirect
6. #19127920C2semidirect
7. #19127860C2semidirect
8. #19127838C2semidirect
9. C4×Dic332C2semidirect
10. C8×D332C2semidirect
11. #495×C83C2semidirect
54. Dic332D4semidirect
120. C2×D83#135semidirect
137. C83#3285semidirect
141. #495D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818112
C2×Q16×D832
#191283592
#191279392
#191279342
#191279202
#191278602
#191278382
C4×Dic3322
C8×D3322
#495×C832
#328583