Finite Group Databasebeta

← #24918529

Subgroups — #24918529

Order 10624.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128270C211yesmaximal, characteristic
2#19128093C211yesmaximal, characteristic, complemented
3#19128075C211yesmaximal, characteristic, complemented
4#19128067C211yesmaximal, characteristic, complemented
5#19128058C211yesmaximal, characteristic, complemented
6#19127923C212yesmaximal, complemented
7#19127901C212yesmaximal, complemented
8#19127861C212yesmaximal, complemented
9#19127851C212yesmaximal, complemented
10#19127762C211yesmaximal, characteristic, complemented
11#541×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×D332C2211yescharacteristic
14C2×#13050580C2212yescomplemented
15C2×#13050577C2212yes—
16C2×#13050577C2212yes—
17C2×#13050577C2212yescomplemented
18#14581789C2211yescharacteristic
19#14581789C2211yescharacteristic
20#14581780C2211yescharacteristic
21#14581780C2211yescharacteristic
22#14581767—22no—
23#14581762—22no—
24C4⋊(C83⋊C8)C2212yescomplemented
25#14581715C2212yescomplemented
26#14581712C2212yes—
27#14581712C2212yes—
28#14581712C2212yescomplemented
29C22⋊(C83⋊C8)C2212yes—
30#14581703C2211yescharacteristic, complemented
31C4×C83⋊C8C2211yescharacteristic, complemented
32C4⋊Q8×C83C2211yescharacteristic, complemented
33#126×C83—22no—
34#125×C83—22no—
35#125×C83C2211yescharacteristic, complemented
36C22⋊Q8×C83C2212yes—
37C4×Q8×C83C2212yescomplemented
38#117×C83C2211yescharacteristic
39C23×D83—21no—
40#13050590—42no—
41C2×Dic166C2311yescharacteristic
42C2×D332—21no—
43C2×D332—22no—
44C2×D332—22no—
45C2×D332—42no—
46C2×D332C2311yescharacteristic
47C2×D332C2311yescharacteristic
48C2×D332C2311yescharacteristic
49C2×C22⋊D83—21no—
50C2×C4×D83—21no—
51#13050580—42no—
52#13050577—42no—
53#13050577—42no—
54#13050577—42no—
55C2×C83⋊C8C2312yes—
56C2×C83⋊C8C2312yes—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59C2×Q8×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C4⋊C4×C83—22no—
62C4⋊C4×C83—22no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83—22no—
65C4⋊C4×C83D412yes—
66C4⋊C4×C83C2312yes—
67C22⋊C4×C83—22no—
68C22⋊C4×C83D412yes—
69C42×C83—22no—
70C42×C83C2311yescharacteristic
71C42×C83C2311yescharacteristic
72C23×C83C2×D411yescharacteristic
73C22×D83—21no—
74C22×D83—21no—
75C22×D83—21no—
76C22×D83—42no—
77C22×D83—82no—
78Q8×C83—22no—
79Q8×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D412yes—
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2411yescharacteristic
87C22⋊D83—42no—
88C2×Dic83—21no—
89D332—22no—
90D332—22no—
91D332—22no—
92D332—42no—
93D332—82no—
94D332—82no—
95C4×D83—42no—
96Dic166—22no—
97C83⋊C8—22no—
98C83⋊C8—22no—
99C22×C83—21no—
100C22×C83—22no—
101C22×C83C22×D411yescharacteristic
102C2×D83—41no—
103C2×D83—41no—
104C2×D83—41no—
105C2×D83—42no—
106C2×D83—42no—
107C2×D83—42no—
108C2×D83—84no—
109C4×C83—21no—
110C4×C83—22no—
111C4×C83—22no—
112C4×C83—42no—
113C4×C83—42no—
114C4×C83H(5,𝔽2)12yes—
115Dic83—41no—
116C2×C83—41no—
117C2×C83#53511yescharacteristic
118C2×C83#57712yes—
119D83—81no—
120D83—81no—
121D83—81no—
122#3222—831nomaximal, Sylow
123C83#322211yescharacteristic, complemented, Sylow
124#541D8311yescharacteristic, complemented
125#535—831no—
126#497—831no—
127#492—831no—
128#490—831no—
129#488—831no—
130#464—832no—
131#460—832no—
132#452—832no—
133#450—832no—
134#433—831no—
135C2×#41—831no—
136C22×D4—831no—
137C2×Q16—832no—
138C2×SD16—832no—
139C2×SD16—832no—
140C2×SD16—832no—
141C4⋊Q8C2×D8311yescharacteristic, complemented
142#127—831no—
143#127—831no—
144#126—22no—
145#125—22no—
146#125C2×D8311yescharacteristic, complemented
147#124—831no—
148#124—831no—
149C22⋊Q8C2×D8312yes—
150#121—1662no—
151#120—1662no—
152C4×Q8C2×D8312yescomplemented
153#117C2×D8311yescharacteristic
154C4⋊C8—832no—
155#103—832no—
156#102—832no—
157#102—832no—
158#102—832no—
159C22⋊C8—832no—
160#97—831no—
161C4×C8—831no—
162C24—1661no—
163#41—3322no—
164C2×Q8—831no—
165C2×Q8C22×D8312yes—
166C2×D4—3322no—
167C2×D4—831no—
168C2×D4—831no—
169C2×D4—831no—
170C2×D4—1661no—
171C2×D4—1661no—
172C2×D4—1662no—
173C2×D4—1662no—
174C22×C4—1661no—
175C22×C4C22×D8311yescharacteristic
176Q16—3322no—
177SD16—3322no—
178SD16—3322no—
179SD16—3322no—
180C2×C8—832no—
181C2×C8—832no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4C22⋊D8312yes—
187C4⋊C4C22×D8312yes—
188C22⋊C4—22no—
189C22⋊C4—1662no—
190C22⋊C4—1662no—
191C22⋊C4C22⋊D8312yes—
192C42—22no—
193C42C22×D8311yescharacteristic
194C42C22×D8311yescharacteristic
195C23—3322no—
196C23—6642no—
197C23—1661no—
198C23—1661no—
199C23—1661no—
200C23C2×C22⋊D8311yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—1662no—
204D4—3322no—
205D4—3322no—
206D4—6642no—
207D4—6642no—
208D4—1662no—
209D4—1662no—
210D4—1662no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—3322no—
215C2×C4—1661no—
216C2×C4C2×C22⋊D8312yes—
217C2×C4C2×C22⋊D8312yes—
218C2×C4C2×C22⋊D8311yescharacteristic
219C2×C4C23×D8311yescharacteristic
220C8—1662no—
221C8—1662no—
222C22—22no—
223C22—21no—
224C22—3322no—
225C22—3322no—
226C22—3322no—
227C22—3321no—
228C22—3321no—
229C22—3321no—
230C22—6644no—
231C22C22×C22⋊D8311yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—42no—
236C4—42no—
237C4—3321no—
238C4#1458183812yes—
239C2—41no—
240C2—6641no—
241C2—6641no—
242C2—6641no—
243C2#1912826511yescharacteristic
244C2#1912850912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#541×C83C2
socleC22×C83C22×D4
prime radical#19128270C2
hypercenter#541D83
hypocenterC83#3222
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #19128093C2semidirect
3. #19128075C2semidirect
4. #19128067C2semidirect
5. #19128058C2semidirect
6. #19127923C2semidirect
7. #19127901C2semidirect
8. #19127861C2semidirect
9. #19127851C2semidirect
10. #19127762C2semidirect
11. #541×C83C2semidirect
14. C2×#13050580C22semidirect
17. C2×#13050577C22semidirect
24. C4⋊(C83⋊C8)C22semidirect
25. #14581715C22semidirect
28. #14581712C22semidirect
30. #14581703C22semidirect
31. C4×C83⋊C8C22semidirect
32. C4⋊Q8×C83C22semidirect
35. #125×C83C22semidirect
37. C4×Q8×C83C22semidirect
123. C83#3222semidirect
124. #541D83semidirect
141. C4⋊Q8C2×D83semidirect
146. #125C2×D83semidirect
152. C4×Q8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282702
#191280932
#191280752
#191280672
#191280582
#191279232
#191279012
#191278612
#191278512
#191277622
#541×C832
#322283