Finite Group Databasebeta

← #24919112

Subgroups — #24919112

Order 10624.

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

normal subgroups only

This group has 472 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
1C2×#14581821C212yesmaximal, complemented
2#19128366C211yesmaximal, characteristic, complemented
3#19128355C211yesmaximal, characteristic
4#19128080C211yesmaximal, characteristic, complemented
5#19127958C212yesmaximal, complemented
6#19127912C211yesmaximal, characteristic, complemented
7#19127860C212yesmaximal, complemented
8#19127858C212yesmaximal, complemented
9C4×#13050578C211yesmaximal, characteristic, complemented
10#19127778C211yesmaximal, characteristic, complemented
11#461×C83C211yesmaximal, characteristic, complemented
12C2×#13050589C2212yes—
13C2×Q8×D83C2212yes—
14#14581821—28no—
15C2×Dic332C2212yes—
16C2×#13050580C2212yes—
17C2×#13050580C2212yes—
18C2×#13050578C2211yescharacteristic
19C2×#13050573C2212yes—
20#14581792C2211yescharacteristic
21#14581791—21no—
22#14581776—22no—
23#14581772C2212yes—
24#14581770—22no—
25#14581768C2212yes—
26#14581766—21no—
27C4×Dic166C2211yescharacteristic
28C4×D332C2211yescharacteristic
29#14581754—21no—
30C2×C4⋊Dic83—21no—
31#14581723C2211yescharacteristic
32#14581717C2212yes—
33#14581715C2211yescharacteristic
34#14581713C2211yescharacteristic
35(C2×D83)⋊C8C2212yes—
36C4×C83⋊C8C2211yescharacteristic
37C2×SD16×C83C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40C4⋊C8×C83C2211yescharacteristic
41#103×C83C2212yes—
42#13050589—24no—
43#13050589—42no—
44C2×Dic166—22no—
45C2×Dic166C2312yes—
46C2×Dic166C2311yescharacteristic
47Q8×D83—24no—
48Q8×D83—42no—
49C2×D332D411yescharacteristic
50C2×C22⋊D83—22no—
51C22×Dic83—22no—
52C2×C4×D83—22no—
53C2×C4×D83C2312yes—
54Dic332—24no—
55#13050580—24no—
56#13050580—42no—
57#13050578D414yescomplemented
58#13050573—24no—
59C2×C83⋊C8C2311yescharacteristic
60C2×C83⋊C8C2311yescharacteristic
61C4⋊Dic83—21no—
62C4⋊Dic83—21no—
63C4⋊Dic83—22no—
64C4⋊Dic83D411yescharacteristic
65C4⋊Dic83C2311yescharacteristic
66Dic83⋊C4—21no—
67Dic83⋊C4—22no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4D412yes—
71C4×Dic83—21no—
72C2×Q8×C83C2312yes—
73C2×D4×C83C2311yescharacteristic
74C22×C4×C83—21no—
75SD16×C83—24no—
76C2×C8×C83C2312yes—
77C4⋊C4×C83—21no—
78C4⋊C4×C83—21no—
79C4⋊C4×C83C2311yescharacteristic
80C22⋊C4×C83—21no—
81C42×C83C2311yescharacteristic
82C23×C83—21no—
83C22×D83C2×D412yes—
84Q8×C83—22no—
85Q8×C83—22no—
86D4×C83—21no—
87D4×C83C2×D412yes—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—42no—
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2411yescharacteristic
94C22⋊D83—44no—
95C2×Dic83—21no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83—44no—
99C2×Dic83C2×D412yes—
100D332—22no—
101C4×D83—22no—
102C4×D83—22no—
103C4×D83—44no—
104Dic166—21no—
105Dic166—22no—
106Dic166—22no—
107Dic166—44no—
108Dic166C2×D412yes—
109C8×C83—22no—
110C83⋊C8—21no—
111C83⋊C8C2×D412yes—
112C22×C83—22no—
113C22×C83—41no—
114C22×C83—41no—
115C22×C83C22×D411yescharacteristic
116C2×D83—22no—
117C2×D83—22no—
118C2×D83—22no—
119C4×C83—21no—
120C4×C83—42no—
121C4×C83—41no—
122C4×C83#13712yescomplemented
123C4×C83C22×D411yescharacteristic
124C4×C83H(5,𝔽2)11yescharacteristic
125Dic83—22no—
126Dic83—22no—
127Dic83—42no—
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#461D8311yescharacteristic, complemented
143#452—832no—
144#451—832no—
145C4×SD16—831no—
146#436—831no—
147C2×#41—832no—
148C22×Q8—832no—
149#137—1668no—
150C2×Q16—832no—
151C2×Q16—832no—
152C2×Q16—832no—
153C2×SD16—831no—
154C2×SD16C2×D8312yes—
155C2×M16—832no—
156C4⋊Q8—831no—
157C4⋊Q8—1661no—
158C4⋊Q8C2×D8311yescharacteristic
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—832no—
172#103—831no—
173#103C2×D8312yes—
174#102—831no—
175C22⋊C8—832no—
176C4×C8—831no—
177#41—3322no—
178#41—1664no—
179C2×Q8—3322no—
180C2×Q8—831no—
181C2×Q8—832no—
182C2×Q8—1662no—
183C2×Q8—1664no—
184C2×Q8C22×D8312yes—
185C2×D4—831no—
186C2×D4—1662no—
187C2×D4C22×D8311yescharacteristic
188C22×C4—21no—
189C22×C4—832no—
190C22×C4—1662no—
191C22×C4—1662no—
192Q16—3322no—
193Q16—1664no—
194Q16—1664no—
195SD16—24no—
196SD16—834no—
197M16—1664no—
198C2×C8—831no—
199C2×C8—831no—
200C2×C8C22×D8312yes—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—831no—
204C4⋊C4—831no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4C22×D8311yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—832no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C42—1661no—
216C42C22×D8311yescharacteristic
217C23—21no—
218C23—832no—
219Q8—22no—
220Q8—22no—
221Q8—3324no—
222Q8—832no—
223Q8—1662no—
224Q8—1662no—
225Q8—1661no—
226D4—21no—
227D4—3324no—
228D4—1662no—
229D4D4×D8312yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—3324no—
234C2×C4—3324no—
235C2×C4—832no—
236C2×C4—1661no—
237C2×C4—1662no—
238C2×C4—1661no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4D4×D8311yescharacteristic
242C2×C4D4×D8311yescharacteristic
243C2×C4C23×D8311yescharacteristic
244C8—22no—
245C8—832no—
246C8—1661no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—1662no—
251C22—1662no—
252C22—1662no—
253C22C2×D4×D8311yescharacteristic
254C4—21no—
255C4—42no—
256C4—41no—
257C4—3322no—
258C4—3321no—
259C4—1662no—
260C4—1662no—
261C4#1458182112yescomplemented
262C4C2×D4×D8311yescharacteristic
263C4#1458183811yescharacteristic
264C2—42no—
265C2—3322no—
266C2#1912833511yescharacteristic
267C2C2×#1458182111yescharacteristic
268C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
prime radical#19128355C2
hypercenter#461D83
hypocenterC83#3276
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#14581821C2semidirect
2. #19128366C2semidirect
4. #19128080C2semidirect
5. #19127958C2semidirect
6. #19127912C2semidirect
7. #19127860C2semidirect
8. #19127858C2semidirect
9. C4×#13050578C2semidirect
10. #19127778C2semidirect
11. #461×C83C2semidirect
57. #13050578D4semidirect
122. C4×C83#137semidirect
135. C83#3276semidirect
142. #461D83semidirect
261. C4#14581821semidirect

Maximal subgroups

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

subgroupindex
C2×#145818212
#191283662
#191283552
#191280802
#191279582
#191279122
#191278602
#191278582
C4×#130505782
#191277782
#461×C832
#327683