Finite Group Databasebeta

← #24919877

Subgroups — #24919877

Order 10624.

352 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128421C211yesmaximal, characteristic, complemented
2#19128416C211yesmaximal, characteristic
3#19128099C211yesmaximal, characteristic, complemented
4#19128043C212yesmaximal, complemented
5#19128026C212yesmaximal, complemented
6#19127997C211yesmaximal, characteristic, complemented
7#19127982C211yesmaximal, characteristic, complemented
8#19127930C212yesmaximal, complemented
9C4×#13050579C211yesmaximal, characteristic, complemented
10#19127739C212yesmaximal, complemented
11C8×D4×C83C211yesmaximal, characteristic, complemented
12C2×#13050579C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581785—22no—
16#14581779—21no—
17#14581778—22no—
18#14581771C2212yes—
19#14581766C2212yes—
20#14581763—21no—
21C4×Dic166C2211yescharacteristic
22#14581757—21no—
23C4×D332C2211yescharacteristic
24#14581751C2212yes—
25#14581751C2212yes—
26#14581750—21no—
27#14581726C2212yes—
28#14581726C2212yes—
29#14581724C2212yes—
30#14581724C2212yes—
31#14581724C2211yescharacteristic
32#14581717C2212yes—
33#14581717C2211yescharacteristic
34#14581714C2212yes—
35#14581714C2211yescharacteristic
36C22×C8×C83C2212yes—
37C4×D4×C83C2211yescharacteristic
38C4⋊C8×C83C2211yescharacteristic
39C22⋊C8×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41C2×Dic166—21no—
42C2×Dic166—21no—
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—22no—
46C2×C4×D83—21no—
47#13050579—22no—
48C4⋊Dic83—21no—
49C4⋊Dic83—21no—
50C4⋊Dic83C2312yes—
51C4⋊Dic83C2312yes—
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—22no—
55Dic83⋊C4—21no—
56C22⋊Dic83—22no—
57C22⋊Dic83—22no—
58C22⋊Dic83—22no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—21no—
61C4×Dic83—22no—
62C4×Dic83—21no—
63C4×Dic83—22no—
64C2×D4×C83D411yescharacteristic
65C22×C4×C83C2312yes—
66C2×C8×C83—24no—
67C2×C8×C83C2311yescharacteristic
68C2×C8×C83C2312yes—
69C2×C8×C83C2311yescharacteristic
70C4⋊C4×C83D411yescharacteristic
71C22⋊C4×C83D412yes—
72C42×C83C2311yescharacteristic
73C23×C83C2×D412yes—
74C22×D83—21no—
75D4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D412yes—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2411yescharacteristic
82C22⋊D83—82no—
83C2×Dic83—21no—
84C2×Dic83—21no—
85C2×Dic83—22no—
86C2×Dic83—21no—
87C2×Dic83—22no—
88D332—21no—
89D332—21no—
90C4×D83—42no—
91Dic166—21no—
92Dic166—21no—
93Dic166—42no—
94Dic166—81no—
95C8×C83—22no—
96C8×C83—21no—
97C8×C83#4112yes—
98C22×C83—22no—
99C22×C83—22no—
100C22×C83—22no—
101C22×C83C22×D411yescharacteristic
102C2×D83—41no—
103C2×D83—41no—
104C2×D83—41no—
105C4×C83—22no—
106C4×C83—21no—
107C4×C83#13512yes—
108C4×C83C2×#4111yescharacteristic
109C4×C832−1+411yescharacteristic
110Dic83—41no—
111Dic83—42no—
112Dic83—41no—
113Dic83—42no—
114Dic83—41no—
115C2×C83—42no—
116C2×C83#54711yescharacteristic
117C2×C83C2×#13511yescharacteristic
118C2×C83#57711yescharacteristic
119D83—81no—
120#3340—831nomaximal, Sylow
121C83#334011yescharacteristic, complemented, Sylow
122#556—831no—
123#555—831no—
124#498—831no—
125#485—832no—
126#482—832no—
127#477—831no—
128#474—831no—
129#465—832no—
130C4×SD16—831no—
131C8×D4D8311yescharacteristic, complemented
132#427—832no—
133C2×SD16—831no—
134C22×C8C2×D8312yes—
135C4⋊Q8—831no—
136C4⋊Q8—831no—
137#126—1662no—
138#124—1661no—
139#124—1662no—
140C22⋊Q8—832no—
141#121—832no—
142#121—1661no—
143C4×Q8—831no—
144C4×Q8—1661no—
145C4×D4—831no—
146C4×D4C2×D8311yescharacteristic
147#117—832no—
148#117—832no—
149#117—1661no—
150#107—832no—
151#107—832no—
152#106—831no—
153#106—832no—
154#106—832no—
155C4⋊C8C2×D8311yescharacteristic
156#103—831no—
157#103—832no—
158#102—831no—
159#102—832no—
160C22⋊C8C2×D8312yes—
161C4×C8C2×D8311yescharacteristic
162C2×Q8—831no—
163C2×Q8—1661no—
164C2×Q8—1661no—
165C2×D4—831no—
166C2×D4—1662no—
167C2×D4D33211yescharacteristic
168C22×C4—1661no—
169C22×C4C22×D8312yes—
170SD16—1662no—
171C2×C8—24no—
172C2×C8C22×D8311yescharacteristic
173C2×C8C22×D8312yes—
174C2×C8C22×D8311yescharacteristic
175C4⋊C4—832no—
176C4⋊C4—831no—
177C4⋊C4—832no—
178C4⋊C4—831no—
179C4⋊C4—1661no—
180C4⋊C4—1661no—
181C4⋊C4—1662no—
182C4⋊C4—1661no—
183C4⋊C4D33211yescharacteristic
184C22⋊C4—1662no—
185C22⋊C4—1662no—
186C22⋊C4—1662no—
187C22⋊C4—1662no—
188C22⋊C4—1661no—
189C22⋊C4D33212yes—
190C42—1662no—
191C42—1662no—
192C42—1661no—
193C42C22×D8311yescharacteristic
194C23—1661no—
195C23C2×D33212yes—
196Q8—3322no—
197Q8—6641no—
198Q8—1661no—
199Q8—1661no—
200D4—22no—
201D4—6642no—
202D4—1661no—
203D4—1661no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—3322no—
207C2×C4—1661no—
208C2×C4—1661no—
209C2×C4—1662no—
210C2×C4—1662no—
211C2×C4—1661no—
212C2×C4C2×D33212yes—
213C2×C4C2×D33211yescharacteristic
214C2×C4C2×D33211yescharacteristic
215C2×C4C23×D8311yescharacteristic
216C8—22no—
217C8—21no—
218C8#1305058912yes—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—3321no—
223C22—3321no—
224C22—3321no—
225C22C22×D33211yescharacteristic
226C4—22no—
227C4—21no—
228C4—3321no—
229C4—3321no—
230C4—3322no—
231C4—3322no—
232C4—3321no—
233C4#1458181412yes—
234C4C2×#1305058911yescharacteristic
235C4#1458183911yescharacteristic
236C2—42no—
237C2—6641no—
238C2#1912835511yescharacteristic
239C2C2×#1458181411yescharacteristic
240C2#1912851311yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC8×D4×C83C2
socleC22×C83C22×D4
prime radical#19128416C2
hypercenterC8×D4D83
hypocenterC83#3340
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #19128421C2semidirect
3. #19128099C2semidirect
4. #19128043C2semidirect
5. #19128026C2semidirect
6. #19127997C2semidirect
7. #19127982C2semidirect
8. #19127930C2semidirect
9. C4×#13050579C2semidirect
10. #19127739C2semidirect
11. C8×D4×C83C2semidirect
121. C83#3340semidirect
131. C8×D4D83semidirect

Maximal subgroups

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

subgroupindex
#191284212
#191284162
#191280992
#191280432
#191280262
#191279972
#191279822
#191279302
C4×#130505792
#191277392
C8×D4×C832
#334083