Finite Group Databasebeta

← #24917294

Subgroups — #24917294

Order 10624.

398 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 149 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#19128329C211yesmaximal, characteristic, complemented
2#19128001C212yesmaximal, complemented
3#19128001C212yesmaximal, complemented
4#19127991C212yesmaximal, complemented
5#19127981C212yesmaximal, complemented
6#19127744C211yesmaximal, characteristic, complemented
7#19127737C211yesmaximal, characteristic, complemented
8#19127728C211yesmaximal, characteristic, complemented
9#19127706C211yesmaximal, characteristic
10C2×#14581713C211yesmaximal, characteristic
11#518×C83C211yesmaximal, characteristic
12C2×#13050574C2211yescharacteristic
13C22×C83⋊C8C2211yescharacteristic
14#14581792C2212yescomplemented
15#14581791—21no—
16#14581784C2212yescomplemented
17#14581783—21no—
18#14581769—22no—
19#14581751C2211yescharacteristic
20C2×C4⋊Dic83C2211yescharacteristic
21#14581725C2211yescharacteristic, complemented
22#14581725C2212yescomplemented
23#14581725C2211yescharacteristic, complemented
24#14581723C2211yescharacteristic, complemented
25#14581723C2212yescomplemented
26#14581723C2211yescharacteristic, complemented
27C4⋊(C83⋊C8)C2212yescomplemented
28C4⋊(C83⋊C8)C2212yescomplemented
29#14581713C2212yes—
30#14581713C2212yes—
31#14581713C2212yes—
32#14581713C2212yes—
33C22×D4×C83C2211yescharacteristic
34C4×D4×C83—21no—
35C4×D4×C83—21no—
36C4×D4×C83C2212yes—
37C4×D4×C83C2212yes—
38#117×C83C2211yescharacteristic
39#117×C83C2211yescharacteristic
40C2×C22⋊C4×C83—22no—
41C2×Dic166—22no—
42C22×Dic83—21no—
43#13050574—22no—
44C2×C83⋊C8—22no—
45C2×C83⋊C8C2311yescharacteristic
46C2×C83⋊C8C2311yescharacteristic
47C2×C83⋊C8C2312yes—
48C4⋊Dic83—22no—
49C4⋊Dic83—21no—
50C4⋊Dic83C2311yescharacteristic
51C4⋊Dic83C2312yes—
52C4⋊Dic83C2311yescharacteristic
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55C22⋊Dic83—21no—
56C4×Dic83—21no—
57C24×C83—21no—
58C2×D4×C83—21no—
59C2×D4×C83—22no—
60C2×D4×C83Q814yescomplemented
61C2×D4×C83C2312yes—
62C22×C4×C83—22no—
63C22×C4×C83—22no—
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83D411yescharacteristic
66C4⋊C4×C83D411yescharacteristic
67C4⋊C4×C83C2311yescharacteristic
68C4⋊C4×C83C2311yescharacteristic
69C22⋊C4×C83—21no—
70C22⋊C4×C83—22no—
71C22⋊C4×C83—22no—
72C22⋊C4×C83D412yes—
73C42×C83—21no—
74C42×C83C2312yes—
75C23×C83—22no—
76C23×C83—24no—
77C23×C83—81no—
78C23×C83—81no—
79C23×C83C2×D411yescharacteristic
80D4×C83—24no—
81D4×C83—22no—
82D4×C83C2×Q814yes—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—82no—
86C2×C4×C83—82no—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D412yes—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×Q812yes—
91C2×C4×C83#4112yes—
92C2×C4×C83C2411yescharacteristic
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—81no—
97Dic166—82no—
98C83⋊C8—21no—
99C83⋊C8—21no—
100C83⋊C8—22no—
101C22×C83—21no—
102C22×C83—21no—
103C22×C83—24no—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—84no—
107C22×C83C22⋊Q812yes—
108C22×C83C22×D411yescharacteristic
109C4×C83—22no—
110C4×C83—22no—
111C4×C83—41no—
112C4×C83—41no—
113C4×C83C22⋊Q812yes—
114C4×C83C22×Q811yescharacteristic
115C4×C83C2×#4111yescharacteristic
116Dic83—41no—
117Dic83—42no—
118Dic83—41no—
119C2×C83—22no—
120C2×C83—44no—
121C2×C83C2×C22⋊Q811yescharacteristic
122C2×C83#57611yescharacteristic
123C2×C83#57711yescharacteristic
124#3081—831nomaximal, Sylow
125C83#308111yescharacteristic, complemented, Sylow
126#544—831no—
127#518D8311yescharacteristic
128#478—832no—
129#478—832no—
130#476—832no—
131#474—832no—
132#428—831no—
133#427—831no—
134#424—831no—
135#418—831no—
136C2×#102—831no—
137C22×D4C2×D8311yescharacteristic
138C2×M16—831no—
139C22×C8—831no—
140C4⋊Q8—832no—
141C4⋊Q8—1661no—
142#125—832no—
143#125—1661no—
144C22⋊Q8—1662no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4C2×D8312yes—
148C4×D4C2×D8312yes—
149#117—831no—
150#117C2×D8311yescharacteristic
151#117C2×D8311yescharacteristic
152C2×C4⋊C4—831no—
153C2×C22⋊C4—22no—
154#107—831no—
155#107—831no—
156#107—832no—
157#106—831no—
158#106—831no—
159#106—832no—
160C4⋊C8—832no—
161C4⋊C8—832no—
162#102—832no—
163#102—832no—
164#102—832no—
165#102—832no—
166C24—21no—
167C2×Q8—1662no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic16614yescomplemented
171C2×D4C22×D8312yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—1661no—
175C22×C4C22×D8311yescharacteristic
176M16—1662no—
177C2×C8—831no—
178C2×C8—832no—
179C2×C8—831no—
180C2×C8—1662no—
181C4⋊C4—832no—
182C4⋊C4—831no—
183C4⋊C4—831no—
184C4⋊C4—1661no—
185C4⋊C4—1662no—
186C4⋊C4—1662no—
187C4⋊C4—1662no—
188C4⋊C4C22⋊D8311yescharacteristic
189C4⋊C4C22⋊D8311yescharacteristic
190C4⋊C4C22×D8311yescharacteristic
191C4⋊C4C22×D8311yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1661no—
196C22⋊C4C22⋊D8312yes—
197C42—21no—
198C42—1661no—
199C42C22×D8312yes—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23C2×C22⋊D8311yescharacteristic
205Q8—6642no—
206D4—24no—
207D4—22no—
208D4C2×Dic16614yes—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—6641no—
214C2×C4—1661no—
215C2×C4—1661no—
216C2×C4—1662no—
217C2×C4C2×C22⋊D8312yes—
218C2×C4C2×C22⋊D8312yes—
219C2×C4C2×C22⋊D8311yescharacteristic
220C2×C4C2×Dic16612yes—
221C2×C4#1305059012yes—
222C2×C4C23×D8311yescharacteristic
223C8—1661no—
224C8—1661no—
225C8—1662no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#1458177112yes—
233C22C22×C22⋊D8311yescharacteristic
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—3322no—
239C4—3321no—
240C4—3321no—
241C4#1458177112yes—
242C4C22×Dic16611yescharacteristic
243C4C2×#1305059011yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#1458177111yescharacteristic
247C2#1912850411yescharacteristic
248C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#518×C83C2
socleC22×C83C22×D4
prime radicalC22×D4×C83C22
hypercenter#518D83
hypocenterC83#3081
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #19128329C2semidirect
2. #19128001C2semidirect
3. #19128001C2semidirect
4. #19127991C2semidirect
5. #19127981C2semidirect
6. #19127744C2semidirect
7. #19127737C2semidirect
8. #19127728C2semidirect
14. #14581792C22semidirect
16. #14581784C22semidirect
21. #14581725C22semidirect
22. #14581725C22semidirect
23. #14581725C22semidirect
24. #14581723C22semidirect
25. #14581723C22semidirect
26. #14581723C22semidirect
27. C4⋊(C83⋊C8)C22semidirect
28. C4⋊(C83⋊C8)C22semidirect
60. C2×D4×C83Q8semidirect
125. C83#3081semidirect
170. C2×D4Dic166semidirect

Maximal subgroups

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

subgroupindex
#191283292
#191280012
#191279912
#191279812
#191277442
#191277372
#191277282
#191277062
C2×#145817132
#518×C832
#308183