Finite Group Databasebeta

← #24917544

Subgroups — #24917544

Order 10624.

390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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#19128237C211yesmaximal, characteristic
2C2×Q8×Dic83C211yesmaximal, characteristic, complemented
3#19128029C211yesmaximal, characteristic, complemented
4#19128015C211yesmaximal, characteristic
5#19128010C212yesmaximal, complemented
6#19127925C212yesmaximal, complemented
7#19127840C211yesmaximal, characteristic
8Q16×Dic83C212yesmaximal, complemented
9#19127786C212yesmaximal, complemented
10#19127751C211yesmaximal, characteristic, complemented
11#451×C83C211yesmaximal, characteristic, complemented
12C2×#13050577C2212yes—
13#14581787—21no—
14#14581783C2211yescharacteristic
15#14581781C2211yescharacteristic
16#14581776—21no—
17#14581766C2211yescharacteristic
18Q8×Dic83—21no—
19Q8×Dic83—21no—
20Q8×Dic83C2212yes—
21Q8×Dic83C2212yes—
22Q8×Dic83C2211yescharacteristic
23#14581754C2211yescharacteristic
24(C22⋊D83)⋊C4—21no—
25#14581750—21no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83C2211yescharacteristic
28C2×C4×Dic83—21no—
29C2×C4×Dic83C2211yescharacteristic
30#14581726C2212yes—
31#14581723C2212yes—
32Dic83⋊C8C2212yes—
33#14581716C2212yes—
34#14581716C2212yes—
35#14581714C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yes—
38C22×Q8×C83C2211yescharacteristic
39C2×Q16×C83C2212yes—
40C22⋊Q8×C83C2211yescharacteristic
41#103×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×D332C2311yescharacteristic
44C2×C22⋊D83—21no—
45C22×Dic83—21no—
46C22×Dic83D411yescharacteristic
47C22×Dic83D411yescharacteristic
48C2×C4×D83—21no—
49#13050577—24no—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83—21no—
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83—22no—
55C4⋊Dic83—22no—
56C4⋊Dic83C2311yescharacteristic
57C4⋊Dic83C2312yes—
58Dic83⋊C4—21no—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63C4×Dic83—22no—
64C4×Dic83—22no—
65C4×Dic83—21no—
66C4×Dic83D411yescharacteristic
67C4×Dic83D411yescharacteristic
68C4×Dic83C2311yescharacteristic
69C4×Dic83C2311yescharacteristic
70C2×Q8×C83—24no—
71C2×Q8×C83—21no—
72C2×Q8×C83C2311yescharacteristic
73C2×Q8×C83C2312yes—
74C22×C4×C83—21no—
75C22×C4×C83C2311yescharacteristic
76Q16×C83—24no—
77C2×C8×C83C2312yes—
78C4⋊C4×C83—21no—
79C4⋊C4×C83C2311yescharacteristic
80C22⋊C4×C83—21no—
81C23×C83C2×D411yescharacteristic
82C22×D83—21no—
83Q8×C83—24no—
84Q8×C83—22no—
85Q8×C83—22no—
86Q8×C83#4114yes—
87C2×C4×C83—24no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2411yescharacteristic
94C22⋊D83—42no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83—42no—
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102C2×Dic83C2×D411yescharacteristic
103C2×Dic83C2×D411yescharacteristic
104D332—22no—
105C4×D83—42no—
106C8×C83—22no—
107C83⋊C8—22no—
108C22×C83—21no—
109C22×C83—21no—
110C22×C83#13512yes—
111C22×C83C22×D411yescharacteristic
112C2×D83—41no—
113C2×D83—41no—
114C2×D83—41no—
115C4×C83—24no—
116C4×C83—21no—
117C4×C83—41no—
118C4×C83—41no—
119C4×C83C2×#4112yes—
120Dic83—21no—
121Dic83—21no—
122Dic83—21no—
123Dic83—42no—
124Dic83—41no—
125Dic83#13712yes—
126C2×C83—22no—
127C2×C83#52511yescharacteristic
128C2×C83C2×#13511yescharacteristic
129C2×C83C2×#13711yescharacteristic
130D83—81no—
131#3108—831nomaximal, Sylow
132C83#310811yescharacteristic, complemented, Sylow
133#529—831no—
134C2×C4×Q8—831no—
135#483—831no—
136#480—831no—
137#479—832no—
138#464—832no—
139#451D8311yescharacteristic, complemented
140#448—831no—
141C4×Q16—832no—
142C4×SD16—832no—
143#431—831no—
144C22×Q8C2×D8311yescharacteristic
145C2×Q16C2×D8312yes—
146C2×SD16—832no—
147#126—1661no—
148#125—831no—
149#124—831no—
150#123—1661no—
151C22⋊Q8C2×D8311yescharacteristic
152#121—831no—
153C4×Q8—832no—
154C4×Q8—832no—
155C4×Q8—831no—
156C4×Q8—1661no—
157C4×Q8—1661no—
158C4×D4—831no—
159C4×D4—1661no—
160#117—1661no—
161C2×C4⋊C4—831no—
162C2×C4⋊C4—1661no—
163C2×C42—831no—
164C2×C42—1661no—
165#107—832no—
166#106—832no—
167C4⋊C8—832no—
168#103—832no—
169#103—832no—
170#103C2×D8312yes—
171#102—832no—
172C22⋊C8—831no—
173C22⋊C8C2×D8311yescharacteristic
174C4×C8—832no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8C22×D8311yescharacteristic
178C2×Q8C22×D8312yes—
179C2×D4—831no—
180C2×D4—1661no—
181C22×C4—21no—
182C22×C4—831no—
183C22×C4—831no—
184C22×C4—1661no—
185C22×C4—1661no—
186C22×C4C22×D8311yescharacteristic
187Q16—24no—
188SD16—1664no—
189C2×C8—832no—
190C2×C8C22×D8312yes—
191C4⋊C4—21no—
192C4⋊C4—832no—
193C4⋊C4—831no—
194C4⋊C4—1662no—
195C4⋊C4—1662no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C4⋊C4—1661no—
200C4⋊C4C22×D8311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—1661no—
203C22⋊C4—1661no—
204C22⋊C4—1661no—
205C22⋊C4—1661no—
206C42—831no—
207C42—831no—
208C42—831no—
209C42—831no—
210C42—1662no—
211C42—1662no—
212C42—1661no—
213C23—1661no—
214C23D4×D8311yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—22no—
218Q8#1305058914yes—
219D4—3322no—
220D4—1662no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—3322no—
227C2×C4—3322no—
228C2×C4—831no—
229C2×C4—831no—
230C2×C4—831no—
231C2×C4—831no—
232C2×C4—1662no—
233C2×C4—1661no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4D4×D8311yescharacteristic
237C2×C4C23×D8311yescharacteristic
238C8—22no—
239C8—1662no—
240C22—21no—
241C22—21no—
242C22—3321no—
243C22—3321no—
244C22—3321no—
245C22#1458181512yes—
246C22C2×D4×D8311yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—3322no—
252C4—3321no—
253C4—832no—
254C4—1661no—
255C4—1661no—
256C4—1661no—
257C4C2×#1305058912yes—
258C2—22no—
259C2—6641no—
260C2#1912820711yescharacteristic
261C2C2×#1458181511yescharacteristic
262C2#137×D8311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#451×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenter#451D83
hypocenterC83#3108
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic83C2semidirect
3. #19128029C2semidirect
5. #19128010C2semidirect
6. #19127925C2semidirect
8. Q16×Dic83C2semidirect
9. #19127786C2semidirect
10. #19127751C2semidirect
11. #451×C83C2semidirect
132. C83#3108semidirect
139. #451D83semidirect

Maximal subgroups

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

subgroupindex
#191282372
C2×Q8×Dic832
#191280292
#191280152
#191280102
#191279252
#191278402
Q16×Dic832
#191277862
#191277512
#451×C832
#310883