Finite Group Databasebeta

← #24918525

Subgroups — #24918525

Order 10624.

388 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 109 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#19128307C211yesmaximal, characteristic, complemented
2#19128276C211yesmaximal, characteristic, complemented
3#19128090C211yesmaximal, characteristic, complemented
4#19128075C211yesmaximal, characteristic, complemented
5#19128058C211yesmaximal, characteristic, complemented
6#19127919C212yesmaximal, complemented
7#19127905C212yesmaximal, complemented
8#19127849C212yesmaximal, complemented
9#19127841C212yesmaximal, complemented
10#19127763C211yesmaximal, characteristic, complemented
11#491×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2211yescharacteristic
13C2×Q8×D83C2211yescharacteristic
14C2×#13050579C2212yes—
15C2×#13050577C2212yes—
16C2×#13050577C2212yes—
17C2×D664C2212yes—
18#14581790C2211yescharacteristic
19#14581789C2211yescharacteristic
20#14581781C2211yescharacteristic
21#14581780C2211yescharacteristic
22#14581777—22no—
23#14581777—22no—
24#14581770—22no—
25#14581767—22no—
26#14581767C2212yes—
27#14581754C2212yes—
28#14581749C2211yescharacteristic
29Dic83⋊C8C2212yes—
30#14581715C2212yes—
31#14581712C2212yes—
32(C2×D83)⋊C8C2212yes—
33C4×C83⋊C8C2211yescharacteristic
34C4⋊Q8×C83C2211yescharacteristic
35#125×C83C2211yescharacteristic
36#103×C83C2212yes—
37#103×C83C2212yes—
38#97×C83C2211yescharacteristic
39#13050591—42no—
40C2×Dic166—21no—
41C2×Dic166C2311yescharacteristic
42Q8×D83—42no—
43C2×D332—21no—
44C2×D332—22no—
45C2×D332—22no—
46C2×D332C2312yes—
47C2×D332C2311yescharacteristic
48C2×C4×D83—21no—
49C2×C4×D83—21no—
50C2×C4×D83—22no—
51C2×C4×D83C2311yescharacteristic
52#13050579—42no—
53#13050577—42no—
54#13050577—42no—
55D664—42no—
56C2×C83⋊C8C2312yes—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4D412yes—
64C4×Dic83C2311yescharacteristic
65C2×Q8×C83C2311yescharacteristic
66C2×Q8×C83C2311yescharacteristic
67C2×C8×C83C2312yes—
68C4⋊C4×C83—22no—
69C4⋊C4×C83—22no—
70C4⋊C4×C83C2312yes—
71C42×C83C2311yescharacteristic
72C22×D83—21no—
73C22×D83—22no—
74C22×D83C2×D411yescharacteristic
75Q8×C83—22no—
76Q8×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83C2×D412yes—
81C2×C4×C83C2411yescharacteristic
82C2×Dic83—21no—
83C2×Dic83C2×D412yes—
84C2×Dic83C2×D411yescharacteristic
85D332—22no—
86D332—22no—
87D332—22no—
88D332—42no—
89D332—82no—
90D332—82no—
91C4×D83—22no—
92C4×D83—42no—
93C4×D83—42no—
94C4×D83—82no—
95Dic166—22no—
96Dic166—42no—
97C8×C83—22no—
98C83⋊C8—22no—
99C22×C83C22×D411yescharacteristic
100C2×D83—21no—
101C2×D83—22no—
102C2×D83—41no—
103C2×D83—42no—
104C2×D83—42no—
105C2×D83—42no—
106C2×D83—42no—
107C4×C83—22no—
108C4×C83—41no—
109C4×C83—41no—
110C4×C83—42no—
111C4×C83H(5,𝔽2)12yes—
112Dic83—21no—
113Dic83—22no—
114Dic83—41no—
115C2×C83#53511yescharacteristic
116C2×C83#57712yes—
117D83—41no—
118D83—81no—
119D83—82no—
120#3221—831nomaximal, Sylow
121C83#322111yescharacteristic, complemented, Sylow
122#540—831no—
123#536—831no—
124#496—831no—
125#492—831no—
126#491D8311yescharacteristic, complemented
127#488—831no—
128#463—832no—
129#461—832no—
130#449—832no—
131#448—832no—
132#433—831no—
133C2×#41—831no—
134C22×Q8—831no—
135C2×SD16—832no—
136C2×SD16—832no—
137C2×SD16—832no—
138C2×D8—832no—
139C4⋊Q8—831no—
140C4⋊Q8C2×D8311yescharacteristic
141#127—831no—
142#125C2×D8311yescharacteristic
143#124—831no—
144#124—831no—
145#123—1662no—
146#123—1662no—
147C22⋊Q8—1662no—
148#121—832no—
149#121—1662no—
150C4×D4—832no—
151#117—831no—
152C4⋊C8—832no—
153#103—832no—
154#103C2×D8312yes—
155#103C2×D8312yes—
156#102—832no—
157C22⋊C8—832no—
158#97C2×D8311yescharacteristic
159C4×C8—831no—
160#41—3322no—
161C2×Q8—3322no—
162C2×Q8—831no—
163C2×Q8—1661no—
164C2×Q8C22×D8311yescharacteristic
165C2×Q8C22×D8311yescharacteristic
166C2×D4—831no—
167C2×D4—832no—
168C2×D4—1661no—
169C2×D4—1662no—
170C2×D4—1662no—
171C22×C4—831no—
172C22×C4—1661no—
173C22×C4—1661no—
174C22×C4—1662no—
175SD16—3322no—
176SD16—3322no—
177SD16—3322no—
178D8—3322no—
179C2×C8—832no—
180C2×C8C22×D8312yes—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—832no—
184C4⋊C4—1662no—
185C4⋊C4C22×D8312yes—
186C22⋊C4—832no—
187C22⋊C4—1662no—
188C22⋊C4—1662no—
189C22⋊C4—1662no—
190C22⋊C4—1662no—
191C42—831no—
192C42C22×D8311yescharacteristic
193C23—831no—
194C23—1661no—
195C23—1662no—
196Q8—22no—
197Q8—22no—
198Q8—3322no—
199Q8—1662no—
200D4—3322no—
201D4—6642no—
202D4—6642no—
203D4—1662no—
204D4—1662no—
205D4—1662no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—3322no—
210C2×C4—3322no—
211C2×C4—832no—
212C2×C4—831no—
213C2×C4—6642no—
214C2×C4—1661no—
215C2×C4—1662no—
216C2×C4D4×D8312yes—
217C2×C4C23×D8311yescharacteristic
218C8—22no—
219C8—1662no—
220C22—3322no—
221C22—3322no—
222C22—3322no—
223C22—3321no—
224C22—3322no—
225C22—1662no—
226C22—1661no—
227C22C2×D4×D8311yescharacteristic
228C4—22no—
229C4—42no—
230C4—41no—
231C4—41no—
232C4—3321no—
233C4—1662no—
234C4—1661no—
235C4#1458183812yes—
236C2—3321no—
237C2—6642no—
238C2—6641no—
239C2#1912826911yescharacteristic
240C2#1912851012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#491×C83C2
socleC22×C83C22×D4
hypercenter#491D83
hypocenterC83#3221
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #19128307C2semidirect
2. #19128276C2semidirect
3. #19128090C2semidirect
4. #19128075C2semidirect
5. #19128058C2semidirect
6. #19127919C2semidirect
7. #19127905C2semidirect
8. #19127849C2semidirect
9. #19127841C2semidirect
10. #19127763C2semidirect
11. #491×C83C2semidirect
121. C83#3221semidirect
126. #491D83semidirect

Maximal subgroups

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

subgroupindex
#191283072
#191282762
#191280902
#191280752
#191280582
#191279192
#191279052
#191278492
#191278412
#191277632
#491×C832
#322183