Finite Group Databasebeta

← #24918630

Subgroups — #24918630

Order 10624.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128232C211yesmaximal, characteristic
2C2×#14581783C211yesmaximal, characteristic, complemented
3#19128101C212yesmaximal, complemented
4#19128054C212yesmaximal, complemented
5#19128036C211yesmaximal, characteristic, complemented
6#19128032C211yesmaximal, characteristic, complemented
7#19128012C211yesmaximal, characteristic
8#19128011C212yesmaximal, complemented
9#19128003C212yesmaximal, complemented
10#19127751C211yesmaximal, characteristic, complemented
11#481×C83C211yesmaximal, characteristic
12#14581785—21no—
13#14581783—21no—
14#14581783—21no—
15#14581783C2211yescharacteristic
16#14581783C2212yes—
17#14581783C2212yes—
18#14581779C2211yescharacteristic
19#14581773—21no—
20#14581771C2211yescharacteristic
21#14581757C2211yescharacteristic
22D4×Dic83—21no—
23D4×Dic83C2211yescharacteristic
24#14581748—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83C2211yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×C4×Dic83C2211yescharacteristic
29#14581725C2212yes—
30#14581725C2212yes—
31#14581724C2212yes—
32Dic83⋊C8C2212yes—
33#14581717C2212yes—
34#14581715C2212yes—
35#14581713C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yes—
38#121×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#106×C83C2212yes—
41#102×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C22×Dic83—21no—
45C22×Dic83—21no—
46C22×Dic83D411yescharacteristic
47C22×Dic83D411yescharacteristic
48C2×C83⋊C8C2312yes—
49C4⋊Dic83—21no—
50C4⋊Dic83—22no—
51C4⋊Dic83—22no—
52C4⋊Dic83—21no—
53C4⋊Dic83C2311yescharacteristic
54C4⋊Dic83C2312yes—
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58Dic83⋊C4—22no—
59Dic83⋊C4—22no—
60C22⋊Dic83—21no—
61C22⋊Dic83—21no—
62C22⋊Dic83—21no—
63C22⋊Dic83—21no—
64C4×Dic83—21no—
65C4×Dic83D411yescharacteristic
66C4×Dic83D411yescharacteristic
67C4×Dic83C2311yescharacteristic
68C4×Dic83C2311yescharacteristic
69C2×D4×C83—21no—
70C2×D4×C83C2311yescharacteristic
71C22×C4×C83—21no—
72C22×C4×C83C2311yescharacteristic
73C2×C8×C83C2312yes—
74C4⋊C4×C83—21no—
75C4⋊C4×C83C2312yes—
76C4⋊C4×C83C2311yescharacteristic
77C22⋊C4×C83—21no—
78C23×C83—21no—
79C23×C83C2×D411yescharacteristic
80D4×C83—22no—
81D4×C83—42no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—42no—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2411yescharacteristic
88C2×Dic83—22no—
89C2×Dic83—21no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—41no—
93C2×Dic83—42no—
94C2×Dic83—42no—
95C2×Dic83C2×D411yescharacteristic
96C2×Dic83C2×D411yescharacteristic
97C2×Dic83C2×D411yescharacteristic
98C2×Dic83C2×D411yescharacteristic
99Dic166—22no—
100C8×C83—22no—
101C83⋊C8—22no—
102C22×C83—21no—
103C22×C83—21no—
104C22×C83—41no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83#13512yes—
108C22×C83C22×D411yescharacteristic
109C4×C83—21no—
110C4×C83—41no—
111C4×C83—42no—
112C4×C832−1+412yes—
113Dic83—21no—
114Dic83—21no—
115Dic83—21no—
116Dic83—21no—
117Dic83—41no—
118Dic83—42no—
119Dic83—41no—
120C2×C83—22no—
121C2×C83—81no—
122C2×C83#53611yescharacteristic
123C2×C83C2×#13511yescharacteristic
124C2×C83#57511yescharacteristic
125#3232—831nomaximal, Sylow
126C83#323211yescharacteristic, complemented, Sylow
127#529—831no—
128C2×#125—831no—
129#499—832no—
130#488—832no—
131#484—831no—
132#483—831no—
133#481D8311yescharacteristic
134#480—831no—
135#479—832no—
136#478—832no—
137#431—831no—
138#126—1661no—
139#125—832no—
140#125—832no—
141#125—831no—
142#125—1661no—
143#125—1661no—
144#124—831no—
145#123—1661no—
146C22⋊Q8—831no—
147#121C2×D8311yescharacteristic
148C4×Q8—831no—
149C4×D4—831no—
150C4×D4—1661no—
151#117—1661no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—1661no—
154C2×C4⋊C4—1661no—
155C2×C4⋊C4C2×D8311yescharacteristic
156C2×C42—831no—
157#107—832no—
158#107—832no—
159#106—832no—
160#106C2×D8312yes—
161C4⋊C8—832no—
162#103—832no—
163#103—832no—
164#102—832no—
165#102C2×D8312yes—
166C22⋊C8—831no—
167C22⋊C8C2×D8311yescharacteristic
168C4×C8—832no—
169C2×Q8—831no—
170C2×D4—21no—
171C2×D4C22×D8311yescharacteristic
172C22×C4—21no—
173C22×C4—831no—
174C22×C4—831no—
175C22×C4—1661no—
176C22×C4—1661no—
177C22×C4C22×D8311yescharacteristic
178C2×C8—832no—
179C2×C8C22×D8312yes—
180C4⋊C4—21no—
181C4⋊C4—832no—
182C4⋊C4—831no—
183C4⋊C4—1662no—
184C4⋊C4—1662no—
185C4⋊C4—1662no—
186C4⋊C4—1662no—
187C4⋊C4—1661no—
188C4⋊C4—1661no—
189C4⋊C4—1661no—
190C4⋊C4—1661no—
191C4⋊C4—1661no—
192C4⋊C4C22×D8312yes—
193C4⋊C4C22×D8311yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—1661no—
196C22⋊C4—1661no—
197C22⋊C4—1661no—
198C22⋊C4—1661no—
199C42—831no—
200C42—831no—
201C42—831no—
202C42—831no—
203C42—1661no—
204C23—21no—
205C23D4×D8311yescharacteristic
206Q8—1662no—
207D4—22no—
208D4—42no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—3322no—
214C2×C4—3322no—
215C2×C4—3321no—
216C2×C4—831no—
217C2×C4—831no—
218C2×C4—831no—
219C2×C4—831no—
220C2×C4—1661no—
221C2×C4—1661no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4D4×D8311yescharacteristic
225C2×C4C23×D8311yescharacteristic
226C8—22no—
227C8—1662no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22#1458181312yes—
234C22C2×D4×D8311yescharacteristic
235C4—21no—
236C4—41no—
237C4—42no—
238C4—3322no—
239C4—3321no—
240C4—3321no—
241C4—1661no—
242C4—1661no—
243C4—1661no—
244C4—1661no—
245C4#1458183912yes—
246C2—22no—
247C2—81no—
248C2#1912827311yescharacteristic
249C2C2×#1458181311yescharacteristic
250C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#481×C83C2
socleC22×C83C22×D4
prime radical#121×C83C22
hypercenter#481D83
hypocenterC83#3232
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#14581783C2semidirect
3. #19128101C2semidirect
4. #19128054C2semidirect
5. #19128036C2semidirect
6. #19128032C2semidirect
8. #19128011C2semidirect
9. #19128003C2semidirect
10. #19127751C2semidirect
126. C83#3232semidirect

Maximal subgroups

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

subgroupindex
#191282322
C2×#145817832
#191281012
#191280542
#191280362
#191280322
#191280122
#191280112
#191280032
#191277512
#481×C832
#323283