Finite Group Databasebeta

← #24919793

Subgroups — #24919793

Order 10624.

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

normal subgroups only

This group has 430 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
1#19128371C211yesmaximal, characteristic, complemented
2#19128356C211yesmaximal, characteristic, complemented
3#19128047C211yesmaximal, characteristic, complemented
4#19128018C212yesmaximal, complemented
5#19127987C211yesmaximal, characteristic, complemented
6#19127951C212yesmaximal, complemented
7#19127843C212yesmaximal, complemented
8C4×#13050578C211yesmaximal, characteristic, complemented
9#19127778C211yesmaximal, characteristic, complemented
10#19127699C212yesmaximal, complemented
11#459×C83C211yesmaximal, characteristic, complemented
12C2×#13050590—21no—
13C2×#13050589C2212yes—
14C2×#13050578C2211yescharacteristic
15C2×#13050573C2212yes—
16#14581792C2211yescharacteristic
17#14581788—21no—
18#14581774—22no—
19#14581771—21no—
20#14581764—22no—
21#14581763C2212yes—
22#14581763C2212yes—
23C4×Dic166C2211yescharacteristic
24C4×D332C2211yescharacteristic
25D4×Dic83—21no—
26C4⋊C4×D83C2212yes—
27#14581724C2212yes—
28#14581723C2211yescharacteristic
29#14581717C2212yes—
30#14581715C2211yescharacteristic
31#14581713C2212yes—
32#14581713C2211yescharacteristic
33#14581713C2212yes—
34(C2×D83)⋊C8C2212yes—
35C4×C83⋊C8C2211yescharacteristic
36C2×D8×C83C2212yes—
37#127×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40#102×C83C2212yes—
41#13050590—44no—
42#13050589—24no—
43#13050589—42no—
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46C2×D332D411yescharacteristic
47C2×C22⋊D83—22no—
48C2×C22⋊D83—22no—
49C2×C22⋊D83—22no—
50C22×Dic83—22no—
51C2×C4×D83—22no—
52C2×C4×D83C2312yes—
53#13050578—22no—
54#13050573—42no—
55C2×C83⋊C8C2311yescharacteristic
56C2×C83⋊C8C2311yescharacteristic
57C4⋊Dic83—21no—
58C4⋊Dic83D411yescharacteristic
59C4⋊Dic83C2311yescharacteristic
60C4⋊Dic83C2312yes—
61Dic83⋊C4—21no—
62Dic83⋊C4—22no—
63C22⋊Dic83—22no—
64C22⋊Dic83—22no—
65(C2×D83)⋊C4D412yes—
66C4×Dic83—21no—
67C2×D4×C83—21no—
68C2×D4×C83—21no—
69C2×D4×C83C2311yescharacteristic
70C2×D4×C83C2312yes—
71C22×C4×C83—21no—
72D8×C83—42no—
73C2×C8×C83C2312yes—
74C4⋊C4×C83C2311yescharacteristic
75C22⋊C4×C83—21no—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C23×C83—22no—
79C22×D83C2×D412yes—
80D4×C83—21no—
81D4×C83—22no—
82D4×C83—21no—
83D4×C83—22no—
84D4×C83—42no—
85D4×C83—81no—
86C2×C4×C83—21no—
87C2×C4×C83—42no—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2411yescharacteristic
91C22⋊D83—44no—
92C22⋊D83—44no—
93C22⋊D83—82no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—44no—
98C2×Dic83C2×D412yes—
99D332—22no—
100C4×D83—22no—
101C4×D83—22no—
102C4×D83—44no—
103Dic166—21no—
104Dic166—42no—
105Dic166#4112yescomplemented
106C8×C83—22no—
107C83⋊C8—21no—
108C83⋊C8—21no—
109C22×C83—42no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83—41no—
115C22×C83C22×D411yescharacteristic
116C2×D83—22no—
117C2×D83—22no—
118C2×D83—22no—
119C4×C83—21no—
120C4×C83—41no—
121C4×C83AGL(1,ℤ/8)12yes—
122C4×C83C2×#4111yescharacteristic
123C4×C83H(5,𝔽2)11yescharacteristic
124Dic83—22no—
125Dic83—22no—
126Dic83—42no—
127Dic83—41no—
128C2×C83—82no—
129C2×C83—81no—
130C2×C83#54611yescharacteristic
131C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C83#57711yescharacteristic
133D83—42no—
134#3332—831nomaximal, Sylow
135C83#333211yescharacteristic, complemented, Sylow
136#550—831no—
137#547—831no—
138#486—831no—
139#481—832no—
140#475—831no—
141#469—832no—
142#459D8311yescharacteristic, complemented
143#449—832no—
144C4×SD16—831no—
145#436—831no—
146#417—832no—
147C2×#41—832no—
148C2×#41—1661no—
149C2×SD16—831no—
150C2×D8C2×D8312yes—
151C2×M16—832no—
152C4⋊Q8—831no—
153#127—1661no—
154#127C2×D8311yescharacteristic
155#123—1662no—
156C22⋊Q8—1661no—
157#121—832no—
158#121—832no—
159#121—1662no—
160C4×Q8—831no—
161C4×D4—831no—
162C4×D4—1661no—
163C4×D4C2×D8311yescharacteristic
164C2×C4⋊C4—832no—
165#106—832no—
166#106—831no—
167C4⋊C8C2×D8311yescharacteristic
168#103—832no—
169#103—831no—
170#102—832no—
171#102—831no—
172#102—832no—
173#102C2×D8312yes—
174C22⋊C8—832no—
175C4×C8—831no—
176#41—3324no—
177#41—3322no—
178#41—1664no—
179C2×Q8—831no—
180C2×Q8—1661no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—831no—
184C2×D4—1662no—
185C2×D4—1662no—
186C2×D4—1662no—
187C2×D4C22×D8312yes—
188C2×D4C22×D8311yescharacteristic
189C22×C4—21no—
190C22×C4—832no—
191C22×C4—1662no—
192C22×C4—1662no—
193SD16—1662no—
194D8—42no—
195M16—3322no—
196C2×C8—831no—
197C2×C8—831no—
198C2×C8C22×D8312yes—
199C4⋊C4—831no—
200C4⋊C4—832no—
201C4⋊C4—831no—
202C4⋊C4—1662no—
203C4⋊C4—1661no—
204C4⋊C4—1661no—
205C4⋊C4C22×D8311yescharacteristic
206C22⋊C4—21no—
207C22⋊C4—832no—
208C22⋊C4—1662no—
209C22⋊C4—1662no—
210C42—1661no—
211C42C22×D8311yescharacteristic
212C23—21no—
213C23—22no—
214C23—832no—
215Q8—3322no—
216Q8—832no—
217Q8—1661no—
218D4—21no—
219D4—22no—
220D4—21no—
221D4—22no—
222D4—42no—
223D4—81no—
224D4—3324no—
225D4—3324no—
226D4—6642no—
227D4—1662no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—3324no—
231C2×C4—3324no—
232C2×C4—832no—
233C2×C4—1661no—
234C2×C4—1662no—
235C2×C4—1661no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4D4×D8311yescharacteristic
239C2×C4D4×D8311yescharacteristic
240C2×C4C23×D8311yescharacteristic
241C8—22no—
242C8—1661no—
243C8—1661no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—1662no—
251C22—1662no—
252C22—1662no—
253C22C2×D4×D8311yescharacteristic
254C4—21no—
255C4—41no—
256C4—3322no—
257C4—3321no—
258C4—1662no—
259C4—1662no—
260C4#1458181612yes—
261C4#41×D8311yescharacteristic
262C4#1458183711yescharacteristic
263C2—82no—
264C2—81no—
265C2—3322no—
266C2#1912834811yescharacteristic
267C2C2×#1458181611yescharacteristic
268C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#459×C83C2
socleC22×C83C22×D4
hypercenter#459D83
hypocenterC83#3332
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #19128371C2semidirect
2. #19128356C2semidirect
3. #19128047C2semidirect
4. #19128018C2semidirect
5. #19127987C2semidirect
6. #19127951C2semidirect
7. #19127843C2semidirect
8. C4×#13050578C2semidirect
9. #19127778C2semidirect
10. #19127699C2semidirect
11. #459×C83C2semidirect
105. Dic166#41semidirect
135. C83#3332semidirect
142. #459D83semidirect

Maximal subgroups

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

subgroupindex
#191283712
#191283562
#191280472
#191280182
#191279872
#191279512
#191278432
C4×#130505782
#191277782
#191276992
#459×C832
#333283