Finite Group Databasebeta

← #24918631

Subgroups — #24918631

Order 10624.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128235C211yesmaximal, characteristic
2#19128100C212yesmaximal, complemented
3#19128058C212yesmaximal, complemented
4#19128036C211yesmaximal, characteristic, complemented
5#19128031C211yesmaximal, characteristic, complemented
6#19128019C211yesmaximal, characteristic
7#19128014C211yesmaximal, characteristic
8#19128005C212yesmaximal, complemented
9#19127998C212yesmaximal, complemented
10#19127754C211yesmaximal, characteristic, complemented
11C2×#125×C83C211yesmaximal, characteristic, complemented
12#14581786—21no—
13#14581784C2211yescharacteristic
14#14581780C2211yescharacteristic
15#14581775—21no—
16#14581771C2211yescharacteristic
17#14581766C2211yescharacteristic
18C4×Dic166C2211yescharacteristic
19C4×D332C2211yescharacteristic
20C4×C22⋊D83—21no—
21#14581749—21no—
22#14581725C2212yes—
23#14581725C2212yes—
24#14581723C2212yes—
25#14581723C2212yes—
26C4⋊(C83⋊C8)C2212yes—
27#14581715C2212yes—
28#14581715C2212yes—
29#14581712C2212yes—
30#14581712C2212yes—
31C22⋊(C83⋊C8)C2211yescharacteristic
32C22⋊(C83⋊C8)C2211yescharacteristic
33C4×C83⋊C8C2212yes—
34#125×C83—21no—
35#125×C83—21no—
36#125×C83C2212yes—
37#125×C83C2212yes—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83C2211yescharacteristic
41C2×C4⋊C4×C83C2211yescharacteristic
42C2×C42×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83—21no—
47C2×C83⋊C8C2312yes—
48C2×C83⋊C8C2312yes—
49C4⋊Dic83C2311yescharacteristic
50C4⋊Dic83C2311yescharacteristic
51Dic83⋊C4—21no—
52Dic83⋊C4—21no—
53Dic83⋊C4—21no—
54Dic83⋊C4—21no—
55C22⋊Dic83—21no—
56(C2×D83)⋊C4—21no—
57(C2×D83)⋊C4—21no—
58(C2×D83)⋊C4—21no—
59(C2×D83)⋊C4—21no—
60C4×Dic83—21no—
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83D411yescharacteristic
64C22×C4×C83D411yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—22no—
67C4⋊C4×C83—22no—
68C4⋊C4×C83—22no—
69C4⋊C4×C83—22no—
70C4⋊C4×C83—21no—
71C4⋊C4×C83—21no—
72C4⋊C4×C83C2312yes—
73C4⋊C4×C83C2312yes—
74C42×C83D411yescharacteristic
75C42×C83D411yescharacteristic
76C42×C83C2311yescharacteristic
77C42×C83C2311yescharacteristic
78C23×C83C2×D411yescharacteristic
79C22×D83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—41no—
85C2×C4×C83—42no—
86C2×C4×C83—42no—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—42no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—21no—
97D332—22no—
98C4×D83—42no—
99Dic166—22no—
100C83⋊C8—22no—
101C83⋊C8—22no—
102C22×C83—21no—
103C22×C83—21no—
104C22×C83#13512yes—
105C22×C83C22×D411yescharacteristic
106C2×D83—41no—
107C2×D83—41no—
108C2×D83—41no—
109C4×C83—21no—
110C4×C83—21no—
111C4×C83—21no—
112C4×C83—21no—
113C4×C83—21no—
114C4×C83—42no—
115C4×C83—42no—
116C4×C832−1+412yes—
117Dic83—41no—
118Dic83—41no—
119Dic83—41no—
120C2×C83—22no—
121C2×C83#53611yescharacteristic
122C2×C83C2×#13511yescharacteristic
123C2×C83#57511yescharacteristic
124D83—81no—
125#3232—831nomaximal, Sylow
126C83#323211yescharacteristic, complemented, Sylow
127#529—831no—
128C2×#125D8311yescharacteristic, complemented
129#499—832no—
130#488—832no—
131#484—831no—
132#483—831no—
133#481—831no—
134#480—831no—
135#479—832no—
136#478—832no—
137#431—831no—
138#126—1661no—
139#125—21no—
140#125—21no—
141#125—831no—
142#125C2×D8312yes—
143#125C2×D8312yes—
144#124—831no—
145#123—1661no—
146C22⋊Q8—831no—
147#121—831no—
148C4×Q8—831no—
149C4×D4—831no—
150C4×D4—1661no—
151#117—1661no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C2×D8311yescharacteristic
155C2×C4⋊C4C2×D8311yescharacteristic
156C2×C42C2×D8311yescharacteristic
157#107—832no—
158#107—832no—
159#106—832no—
160#106—832no—
161C4⋊C8—832no—
162#103—832no—
163#103—832no—
164#102—832no—
165#102—832no—
166C22⋊C8—831no—
167C22⋊C8—831no—
168C4×C8—832no—
169C2×Q8—831no—
170C2×D4—831no—
171C2×D4—1661no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—1661no—
175C22×C4C22⋊D8311yescharacteristic
176C22×C4C22⋊D8311yescharacteristic
177C22×C4C22×D8311yescharacteristic
178C2×C8—832no—
179C2×C8—832no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—831no—
187C4⋊C4—831no—
188C4⋊C4—1661no—
189C4⋊C4—1661no—
190C4⋊C4—1661no—
191C4⋊C4—1661no—
192C4⋊C4C22×D8312yes—
193C4⋊C4C22×D8312yes—
194C22⋊C4—1661no—
195C22⋊C4—1661no—
196C22⋊C4—1661no—
197C22⋊C4—1661no—
198C22⋊C4—1661no—
199C42—1661no—
200C42C22⋊D8311yescharacteristic
201C42C22⋊D8311yescharacteristic
202C42C22×D8311yescharacteristic
203C42C22×D8311yescharacteristic
204C23—1661no—
205C23C2×C22⋊D8311yescharacteristic
206Q8—1662no—
207D4—3322no—
208D4—1662no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—3322no—
217C2×C4—1661no—
218C2×C4—1661no—
219C2×C4—1661no—
220C2×C4C2×C22⋊D8311yescharacteristic
221C2×C4C2×C22⋊D8311yescharacteristic
222C2×C4C2×C22⋊D8311yescharacteristic
223C2×C4C2×C22⋊D8311yescharacteristic
224C2×C4C2×C22⋊D8311yescharacteristic
225C2×C4C23×D8311yescharacteristic
226C8—1662no—
227C8—1662no—
228C22—21no—
229C22—21no—
230C22—3321no—
231C22—3321no—
232C22—3321no—
233C22#1458181212yes—
234C22C22×C22⋊D8311yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—3321no—
243C4—3321no—
244C4—3321no—
245C4#1458184012yes—
246C2—22no—
247C2—6641no—
248C2#1912827511yescharacteristic
249C2C2×#1458181211yescharacteristic
250C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#125×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenterC2×#125D83
hypocenterC83#3232
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #19128100C2semidirect
3. #19128058C2semidirect
4. #19128036C2semidirect
5. #19128031C2semidirect
8. #19128005C2semidirect
9. #19127998C2semidirect
10. #19127754C2semidirect
11. C2×#125×C83C2semidirect
126. C83#3232semidirect
128. C2×#125D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191281002
#191280582
#191280362
#191280312
#191280192
#191280142
#191280052
#191279982
#191277542
C2×#125×C832
#323283