Finite Group Databasebeta

← #24916977

Subgroups — #24916977

Order 10624.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13050578C211yesmaximal, characteristic, complemented
2#19128272C211yesmaximal, characteristic, complemented
3#19128176C211yesmaximal, characteristic, complemented
4#19127914C214yesmaximal, complemented
5#19127904C214yesmaximal, complemented
6#19127729C211yesmaximal, characteristic, complemented
7#19127711C211yesmaximal, characteristic, complemented
8#19127697C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C2×#13050578—24no—
12C2×#13050578C2214yes—
13C22×C83⋊C8C2211yescharacteristic
14#14581790C2212yescomplemented
15#14581779C2212yescomplemented
16#14581773—22no—
17#14581771—22no—
18D4×Dic83—22no—
19D4×Dic83C2214yes—
20#14581751C2211yescharacteristic
21#14581751C2211yescharacteristic
22C2×C22⋊Dic83—22no—
23Dic83⋊C8C2214yescomplemented
24#14581717C2214yescomplemented
25#14581716C2212yescomplemented
26#14581713C2212yescomplemented
27C2×#41×C83C2211yescharacteristic
28C22×D4×C83C2211yescharacteristic
29AGL(1,ℤ/8)×C83—24no—
30C2×SD16×C83C2212yes—
31C2×D8×C83C2212yes—
32C2×M16×C83C2211yescharacteristic
33C2×Dic166—22no—
34C2×Dic166—22no—
35C2×Dic166—21no—
36C2×Dic166C2312yes—
37C22×Dic83—21no—
38C22×Dic83—24no—
39#13050578—28no—
40C2×C83⋊C8—22no—
41C2×C83⋊C8C2312yes—
42C4⋊Dic83D412yes—
43C4⋊Dic83C2312yes—
44Dic83⋊C4—24no—
45C22⋊Dic83—21no—
46C22⋊Dic83—24no—
47C22⋊Dic83—24no—
48C22⋊Dic83D412yes—
49C4×Dic83—21no—
50C4×Dic83C2312yes—
51C24×C83—21no—
52#41×C83—22no—
53#41×C83—22no—
54C2×Q8×C83C2311yescharacteristic
55C2×D4×C83—21no—
56C2×D4×C83—21no—
57C2×D4×C83—22no—
58C2×D4×C83D414yes—
59C2×D4×C83C2311yescharacteristic
60C2×D4×C83C2312yes—
61C22×C4×C83—21no—
62C22×C4×C83C2311yescharacteristic
63SD16×C83—24no—
64D8×C83—24no—
65M16×C83—22no—
66C2×C8×C83C2312yes—
67C23×C83—22no—
68C23×C83—24no—
69C23×C83—21no—
70C23×C83—82no—
71C23×C83C2×D411yescharacteristic
72Q8×C83—21no—
73Q8×C83—21no—
74D4×C83—24no—
75D4×C83—21no—
76D4×C83—22no—
77D4×C83—21no—
78D4×C83—42no—
79D4×C83C2×D414yes—
80C2×C4×C83—21no—
81C2×C4×C83—42no—
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2×D412yes—
84C2×C4×C83#4112yes—
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—42no—
89C2×Dic83—84no—
90C2×Dic83C2×D414yes—
91Dic166—22no—
92Dic166—22no—
93Dic166—42no—
94C8×C83—22no—
95C83⋊C8—22no—
96C22×C83—22no—
97C22×C83—24no—
98C22×C83—44no—
99C22×C83—41no—
100C22×C83—42no—
101C22×C83—84no—
102C22×C83#12112yes—
103C22×C83C22×D411yescharacteristic
104C4×C83—41no—
105C4×C83#12112yes—
106C4×C83C22×D411yescharacteristic
107C4×C83C2×#4111yescharacteristic
108Dic83—24no—
109Dic83—42no—
110Dic83—42no—
111C2×C83—22no—
112C2×C83—44no—
113C2×C83—81no—
114C2×C83C2×#12111yescharacteristic
115C2×C83#57712yes—
116#3042—831nomaximal, Sylow
117C83#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)D8311yescharacteristic
119C22×SD16—831no—
120#536—831no—
121#518—831no—
122#463—834no—
123#461—834no—
124#424—831no—
125#419—831no—
126#416—831no—
127C2×#41C2×D8311yescharacteristic
128C22×Q8—831no—
129C22×D4C2×D8311yescharacteristic
130AGL(1,ℤ/8)—24no—
131C2×SD16—834no—
132C2×SD16—1664no—
133C2×SD16C2×D8312yes—
134C2×D8C2×D8312yes—
135C2×M16C2×D8311yescharacteristic
136C22×C8—831no—
137C4⋊Q8—832no—
138#124—832no—
139#123—1662no—
140C22⋊Q8—1662no—
141C4×D4—834no—
142C4×D4—1662no—
143#117—831no—
144#117—831no—
145C2×C22⋊C4—1662no—
146C4⋊C8—834no—
147#103—832no—
148#103—834no—
149#102—832no—
150C24—21no—
151#41—22no—
152#41—22no—
153C2×Q8—832no—
154C2×Q8—1662no—
155C2×Q8—1662no—
156C2×Q8—1661no—
157C2×Q8C22×D8311yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4C22⋊D8314yes—
162C2×D4C22×D8311yescharacteristic
163C2×D4C22×D8312yes—
164C22×C4—21no—
165C22×C4—1661no—
166C22×C4—1664no—
167C22×C4C22×D8311yescharacteristic
168SD16—24no—
169SD16—1668no—
170D8—24no—
171M16—22no—
172C2×C8—832no—
173C2×C8—1662no—
174C2×C8C22×D8312yes—
175C4⋊C4—832no—
176C4⋊C4—832no—
177C4⋊C4—1664no—
178C22⋊C4—832no—
179C22⋊C4—1664no—
180C22⋊C4—1664no—
181C22⋊C4—1661no—
182C42—832no—
183C42—1661no—
184C23—22no—
185C23—24no—
186C23—21no—
187C23—82no—
188C23D4×D8311yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—3322no—
192Q8—1662no—
193Q8—1662no—
194D4—24no—
195D4—21no—
196D4—22no—
197D4—21no—
198D4—42no—
199D4C2×C22⋊D8314yes—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—3322no—
203C2×C4—834no—
204C2×C4—6644no—
205C2×C4—1662no—
206C2×C4—1662no—
207C2×C4D4×D8311yescharacteristic
208C2×C4C2×C22⋊D8312yes—
209C2×C4#1305058912yes—
210C2×C4C23×D8311yescharacteristic
211C8—22no—
212C8—1662no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—41no—
217C22—42no—
218C22—84no—
219C22#1458176412yes—
220C22C2×D4×D8311yescharacteristic
221C4—41no—
222C4—3322no—
223C4—3322no—
224C4—1664no—
225C4#1458176412yes—
226C4C22×C22⋊D8311yescharacteristic
227C4C2×#1305058911yescharacteristic
228C2—22no—
229C2—44no—
230C2—81no—
231C2C2×#1458176411yescharacteristic
232C2#1912851112yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radicalC2×AGL(1,ℤ/8)×C83C2
socleC22×C83C22×D4
hypercenterC2×AGL(1,ℤ/8)D83
hypocenterC83#3042
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×#13050578C2semidirect
2. #19128272C2semidirect
3. #19128176C2semidirect
4. #19127914C2semidirect
5. #19127904C2semidirect
6. #19127729C2semidirect
7. #19127711C2semidirect
8. #19127697C2semidirect
14. #14581790C22semidirect
15. #14581779C22semidirect
23. Dic83⋊C8C22semidirect
24. #14581717C22semidirect
25. #14581716C22semidirect
26. #14581713C22semidirect
117. C83#3042semidirect

Maximal subgroups

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

subgroupindex
C22×#130505782
#191282722
#191281762
#191279142
#191279042
#191277292
#191277112
#191276972
C2×AGL(1,ℤ/8)×C832
#304283