Finite Group Databasebeta

← #24919111

Subgroups — #24919111

Order 10624.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581825C212yesmaximal, complemented
2D4×Dic166C211yesmaximal, characteristic, complemented
3#19128356C211yesmaximal, characteristic
4#19128081C211yesmaximal, characteristic, complemented
5#19127959C212yesmaximal, complemented
6#19127912C211yesmaximal, characteristic, complemented
7#19127907C211yesmaximal, characteristic, complemented
8#19127865C212yesmaximal, complemented
9#19127859C212yesmaximal, complemented
10C4×#13050579C211yesmaximal, characteristic, complemented
11#436×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2212yes—
13C22×Dic166C2212yes—
14#14581825—28no—
15C2×Dic332C2212yes—
16C2×Dic332C2212yes—
17C2×Dic332C2212yes—
18C2×#13050579C2212yes—
19C2×#13050579C2211yescharacteristic
20#14581792C2211yescharacteristic
21#14581792C2211yescharacteristic
22#14581791—21no—
23#14581774—22no—
24#14581771C2212yes—
25#14581771C2212yes—
26#14581769—22no—
27#14581763—21no—
28C4×Dic166C2211yescharacteristic
29C4×D332C2211yescharacteristic
30D4×Dic83—21no—
31C4⋊C4×D83—21no—
32#14581724C2211yescharacteristic
33#14581717C2211yescharacteristic
34#14581717C2212yes—
35#14581717C2212yes—
36#14581714C2211yescharacteristic
37C2×M16×C83C2212yes—
38C4×D4×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40C22⋊C8×C83C2212yes—
41C4×C8×C83C2211yescharacteristic
42#13050590—24no—
43#13050590—42no—
44C2×Dic166—24no—
45C2×Dic166—22no—
46C2×Dic166—42no—
47C2×Dic166C2312yes—
48C2×Dic166C2312yes—
49C2×Dic166C2311yescharacteristic
50C2×D332C2311yescharacteristic
51C2×C22⋊D83—22no—
52C22×Dic83—22no—
53C2×C4×D83—21no—
54C2×C4×D83—22no—
55Dic332—24no—
56Dic332—24no—
57Dic332—42no—
58#13050579—24no—
59#13050579D414yescomplemented
60C4⋊Dic83—21no—
61C4⋊Dic83—21no—
62C4⋊Dic83—21no—
63C4⋊Dic83—21no—
64C4⋊Dic83C2311yescharacteristic
65C4⋊Dic83C2311yescharacteristic
66Dic83⋊C4—21no—
67Dic83⋊C4—22no—
68Dic83⋊C4—22no—
69C22⋊Dic83—22no—
70C22⋊Dic83—22no—
71(C2×D83)⋊C4—21no—
72C4×Dic83—21no—
73C2×D4×C83D411yescharacteristic
74C22×C4×C83C2312yes—
75M16×C83—24no—
76C2×C8×C83C2312yes—
77C2×C8×C83C2311yescharacteristic
78C2×C8×C83C2311yescharacteristic
79C4⋊C4×C83D411yescharacteristic
80C22⋊C4×C83D412yes—
81C42×C83C2311yescharacteristic
82C23×C83C2×D412yes—
83C22×D83—21no—
84D4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D412yes—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2411yescharacteristic
91C22⋊D83—44no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—44no—
98D332—21no—
99D332C2×D412yes—
100C4×D83—44no—
101C4×D83—42no—
102Dic166—21no—
103Dic166—22no—
104Dic166—22no—
105Dic166—22no—
106Dic166—22no—
107Dic166—44no—
108Dic166C2×D412yes—
109C8×C83—22no—
110C8×C83—21no—
111C8×C83C2×D412yes—
112C22×C83—22no—
113C22×C83—22no—
114C22×C83—22no—
115C22×C83C22×D411yescharacteristic
116C2×D83—22no—
117C2×D83—41no—
118C2×D83—41no—
119C4×C83—22no—
120C4×C83—21no—
121C4×C83#13712yescomplemented
122C4×C83C22×D411yescharacteristic
123C4×C83H(5,𝔽2)11yescharacteristic
124Dic83—22no—
125Dic83—42no—
126Dic83—42no—
127Dic83—41no—
128Dic83—41no—
129C2×C83—42no—
130C2×C83D4211yescharacteristic
131C2×C83C2×#13711yescharacteristic
132C2×C83#57811yescharacteristic
133D83—42no—
134#3276—831nomaximal, Sylow
135C83#327611yescharacteristic, complemented, Sylow
136C2×#137—832no—
137D4×Q8—831no—
138#547—831no—
139#494—831no—
140#470—832no—
141#462—831no—
142#461—831no—
143#452—832no—
144#451—832no—
145C4×SD16—831no—
146#436D8311yescharacteristic, complemented
147C2×#41—832no—
148C22×Q8—832no—
149#137—1668no—
150C2×Q16—832no—
151C2×Q16—832no—
152C2×Q16—832no—
153C2×SD16—831no—
154C2×SD16—832no—
155C2×M16C2×D8312yes—
156C4⋊Q8—831no—
157C4⋊Q8—831no—
158C4⋊Q8—1661no—
159#123—1662no—
160C22⋊Q8—832no—
161C22⋊Q8—832no—
162C22⋊Q8—1662no—
163#121—1661no—
164C4×Q8—831no—
165C4×D4—831no—
166C4×D4—1661no—
167C4×D4C2×D8311yescharacteristic
168C2×C4⋊C4—1661no—
169#106—831no—
170C4⋊C8C2×D8311yescharacteristic
171#103—831no—
172#103—832no—
173#103—832no—
174#102—831no—
175C22⋊C8C2×D8312yes—
176C4×C8C2×D8311yescharacteristic
177#41—3322no—
178#41—1664no—
179C2×Q8—3322no—
180C2×Q8—831no—
181C2×Q8—832no—
182C2×Q8—832no—
183C2×Q8—1664no—
184C2×Q8—1662no—
185C2×D4—831no—
186C2×D4—1662no—
187C2×D4D33211yescharacteristic
188C22×C4—1662no—
189C22×C4—1661no—
190C22×C4—1662no—
191C22×C4C22×D8312yes—
192Q16—3322no—
193Q16—1664no—
194Q16—1664no—
195SD16—834no—
196SD16—1664no—
197M16—24no—
198C2×C8C22×D8311yescharacteristic
199C2×C8C22×D8312yes—
200C2×C8C22×D8311yescharacteristic
201C4⋊C4—831no—
202C4⋊C4—831no—
203C4⋊C4—1661no—
204C4⋊C4—1662no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4—1662no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4D33211yescharacteristic
211C22⋊C4—1662no—
212C22⋊C4—1662no—
213C22⋊C4—1661no—
214C22⋊C4D33212yes—
215C42—1661no—
216C42C22×D8311yescharacteristic
217C23—1661no—
218C23C2×D33212yes—
219Q8—3324no—
220Q8—832no—
221Q8—1662no—
222Q8—1662no—
223Q8—1662no—
224Q8—1662no—
225Q8—1661no—
226D4—22no—
227D4—3324no—
228D4—832no—
229D4—1661no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—3324no—
233C2×C4—3324no—
234C2×C4—3322no—
235C2×C4—1661no—
236C2×C4—1661no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1661no—
240C2×C4C2×D33211yescharacteristic
241C2×C4C2×D33212yes—
242C2×C4C2×D33211yescharacteristic
243C2×C4C23×D8311yescharacteristic
244C8—22no—
245C8—21no—
246C8D4×D8312yes—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—3321no—
251C22—3321no—
252C22—1662no—
253C22C22×D33211yescharacteristic
254C4—22no—
255C4—21no—
256C4—3322no—
257C4—3322no—
258C4—3321no—
259C4—3321no—
260C4—1662no—
261C4#1458182512yescomplemented
262C4C2×D4×D8311yescharacteristic
263C4#1458183811yescharacteristic
264C2—42no—
265C2—3322no—
266C2D4×D33211yescharacteristic
267C2C2×#1458182511yescharacteristic
268C2#1912851811yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#436×C83C2
socleC22×C83C22×D4
prime radical#19128356C2
hypercenter#436D83
hypocenterC83#3276
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#14581825C2semidirect
2. D4×Dic166C2semidirect
4. #19128081C2semidirect
5. #19127959C2semidirect
6. #19127912C2semidirect
7. #19127907C2semidirect
8. #19127865C2semidirect
9. #19127859C2semidirect
10. C4×#13050579C2semidirect
11. #436×C83C2semidirect
59. #13050579D4semidirect
121. C4×C83#137semidirect
135. C83#3276semidirect
146. #436D83semidirect
261. C4#14581825semidirect

Maximal subgroups

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

subgroupindex
C2×#145818252
D4×Dic1662
#191283562
#191280812
#191279592
#191279122
#191279072
#191278652
#191278592
C4×#130505792
#436×C832
#327683