Finite Group Databasebeta

← #24920060

Subgroups — #24920060

Order 10624.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128416C211yesmaximal, characteristic, complemented
2#19128356C211yesmaximal, characteristic, complemented
3#19128099C211yesmaximal, characteristic, complemented
4#19128024C212yesmaximal, complemented
5#19128018C212yesmaximal, complemented
6#19127991C211yesmaximal, characteristic, complemented
7#19127981C211yesmaximal, characteristic, complemented
8#19127951C212yesmaximal, complemented
9#19127780C211yesmaximal, characteristic, complemented
10#19127742C212yesmaximal, complemented
11C4×D8×C83C211yesmaximal, characteristic, complemented
12C2×#13050590—21no—
13C2×#13050573C2212yes—
14#14581792C2211yescharacteristic
15#14581792C2211yescharacteristic
16#14581778—22no—
17#14581774—22no—
18#14581771—21no—
19#14581766—21no—
20#14581763C2212yes—
21#14581763C2212yes—
22C4×D332C2211yescharacteristic
23#14581751—21no—
24#14581750C2212yes—
25C4⋊C4×D83C2212yes—
26#14581725C2212yes—
27#14581724C2212yes—
28#14581724C2212yes—
29#14581723C2212yes—
30C4⋊(C83⋊C8)C2211yescharacteristic
31#14581713C2212yes—
32#14581713C2212yes—
33(C2×D83)⋊C8C2212yes—
34C2×D8×C83C2211yescharacteristic
35C4×D4×C83C2211yescharacteristic
36C4×D4×C83C2211yescharacteristic
37#107×C83C2211yescharacteristic
38#102×C83C2211yescharacteristic
39#102×C83C2211yescharacteristic
40C4×C8×C83C2211yescharacteristic
41#13050590—44no—
42C2×Dic166—21no—
43C2×Dic166—21no—
44C2×D332D411yescharacteristic
45C2×C22⋊D83—22no—
46C2×C22⋊D83—22no—
47C2×C4×D83—22no—
48C2×C4×D83C2312yes—
49#13050573—24no—
50C2×C83⋊C8C2312yes—
51C4⋊Dic83—21no—
52C4⋊Dic83—21no—
53C4⋊Dic83D411yescharacteristic
54C4⋊Dic83C2312yes—
55C4⋊Dic83C2312yes—
56Dic83⋊C4—22no—
57C22⋊Dic83—22no—
58C22⋊Dic83—22no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4D412yes—
61C4×Dic83—22no—
62C2×D4×C83C2311yescharacteristic
63C2×D4×C83C2311yescharacteristic
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66D8×C83—22no—
67C2×C8×C83C2311yescharacteristic
68C2×C8×C83C2311yescharacteristic
69C4⋊C4×C83C2311yescharacteristic
70C4⋊C4×C83C2311yescharacteristic
71C22⋊C4×C83—21no—
72C22⋊C4×C83—21no—
73C42×C83C2311yescharacteristic
74C23×C83—21no—
75C23×C83—21no—
76C22×D83C2×D412yes—
77D4×C83—21no—
78D4×C83—21no—
79D4×C83—21no—
80D4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—42no—
84C2×C4×C83—42no—
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2411yescharacteristic
88C22⋊D83—44no—
89C22⋊D83—82no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83C2×D412yes—
93D332—22no—
94C4×D83—22no—
95C4×D83—22no—
96C4×D83—44no—
97Dic166—42no—
98Dic166—81no—
99C8×C83—21no—
100C8×C83#4112yescomplemented
101C83⋊C8—22no—
102C22×C83—41no—
103C22×C83—41no—
104C22×C83—41no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83C22×D411yescharacteristic
109C2×D83—22no—
110C2×D83—22no—
111C2×D83—22no—
112C4×C83—21no—
113C4×C83—41no—
114C4×C83—41no—
115C4×C83AGL(1,ℤ/8)12yes—
116C4×C83C2×#4111yescharacteristic
117C4×C832−1+411yescharacteristic
118Dic83—22no—
119Dic83—42no—
120Dic83—42no—
121C2×C83—81no—
122C2×C83—81no—
123C2×C83#54711yescharacteristic
124C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C83#57711yescharacteristic
126D83—42no—
127#3354—831nomaximal, Sylow
128C83#335411yescharacteristic, complemented, Sylow
129#555—831no—
130#547—831no—
131#498—831no—
132#482—832no—
133#481—832no—
134#476—831no—
135#474—831no—
136#469—832no—
137C4×D8D8311yescharacteristic, complemented
138#436—831no—
139#428—832no—
140C2×#41—1661no—
141C2×D8C2×D8311yescharacteristic
142C2×M16—832no—
143C4⋊Q8—831no—
144C4⋊Q8—831no—
145#124—1662no—
146#123—1662no—
147C22⋊Q8—1661no—
148#121—832no—
149#121—832no—
150#121—1661no—
151C4×D4—831no—
152C4×D4C2×D8311yescharacteristic
153C4×D4C2×D8311yescharacteristic
154#117—832no—
155#117—1661no—
156C2×C4⋊C4—832no—
157#107—832no—
158#107C2×D8311yescharacteristic
159#106—832no—
160#106—832no—
161#106—832no—
162C4⋊C8—831no—
163#102—832no—
164#102—832no—
165#102C2×D8311yescharacteristic
166#102C2×D8311yescharacteristic
167C22⋊C8—832no—
168C4×C8C2×D8311yescharacteristic
169#41—3324no—
170C2×Q8—1661no—
171C2×Q8—1661no—
172C2×D4—831no—
173C2×D4—1662no—
174C2×D4—1662no—
175C2×D4C22×D8311yescharacteristic
176C2×D4C22×D8311yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—832no—
180C22×C4—1662no—
181D8—22no—
182M16—1664no—
183C2×C8—832no—
184C2×C8C22×D8311yescharacteristic
185C2×C8C22×D8311yescharacteristic
186C4⋊C4—831no—
187C4⋊C4—832no—
188C4⋊C4—832no—
189C4⋊C4—1662no—
190C4⋊C4—1661no—
191C4⋊C4—1661no—
192C4⋊C4C22×D8311yescharacteristic
193C4⋊C4C22×D8311yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—832no—
197C22⋊C4—1662no—
198C22⋊C4—1662no—
199C22⋊C4—1662no—
200C42—1662no—
201C42C22×D8311yescharacteristic
202C23—21no—
203C23—21no—
204C23—832no—
205Q8—3322no—
206Q8—6641no—
207D4—21no—
208D4—21no—
209D4—21no—
210D4—21no—
211D4—3324no—
212D4—6642no—
213D4—1662no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—3324no—
219C2×C4—832no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4D4×D8311yescharacteristic
225C2×C4D4×D8311yescharacteristic
226C2×C4C23×D8311yescharacteristic
227C8—21no—
228C8—1662no—
229C8#1305059012yescomplemented
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—1662no—
237C22—1662no—
238C22—1662no—
239C22C2×D4×D8311yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—3322no—
244C4—3322no—
245C4—1662no—
246C4#1458181612yes—
247C4C2×#1305059011yescharacteristic
248C4#1458183911yescharacteristic
249C2—81no—
250C2—81no—
251C2—3322no—
252C2#1912835611yescharacteristic
253C2C2×#1458181611yescharacteristic
254C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×D8×C83C2
socleC22×C83C22×D4
hypercenterC4×D8D83
hypocenterC83#3354
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #19128416C2semidirect
2. #19128356C2semidirect
3. #19128099C2semidirect
4. #19128024C2semidirect
5. #19128018C2semidirect
6. #19127991C2semidirect
7. #19127981C2semidirect
8. #19127951C2semidirect
9. #19127780C2semidirect
10. #19127742C2semidirect
11. C4×D8×C83C2semidirect
100. C8×C83#41semidirect
128. C83#3354semidirect
137. C4×D8D83semidirect
229. C8#13050590semidirect

Maximal subgroups

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

subgroupindex
#191284162
#191283562
#191280992
#191280242
#191280182
#191279912
#191279812
#191279512
#191277802
#191277422
C4×D8×C832
#335483