Finite Group Databasebeta

← #24917581

Subgroups — #24917581

Order 10624.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C2×C4×Dic166C211yesmaximal, characteristic, complemented
2#19128012C212yesmaximal, complemented
3#19127993C212yesmaximal, complemented
4#19127898C212yesmaximal, complemented
5#19127839C212yesmaximal, complemented
6#19127801C214yesmaximal, complemented
7#19127758C211yesmaximal, characteristic, complemented
8#532×C83C211yesmaximal, characteristic
9C22×Dic166C2211yescharacteristic
10C2×#13050578C2214yes—
11C4×Dic166—21no—
12C4×Dic166—21no—
13C4×Dic166C2212yes—
14C4×Dic166C2212yes—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×Dic83⋊C4—21no—
17C2×C4×Dic83—21no—
18#14581725C2214yes—
19C4⋊(C83⋊C8)C2212yes—
20#14581715C2214yes—
21#14581713C2214yes—
22C22⋊(C83⋊C8)C2212yes—
23#14581703C2212yes—
24C2×#41×C83—21no—
25C4⋊Q8×C83C2211yescharacteristic
26#127×C83C2211yescharacteristic
27#124×C83—21no—
28#121×C83—21no—
29#121×C83C2212yes—
30C4×D4×C83—21no—
31C4×D4×C83C2212yes—
32C2×C42×C83C2211yescharacteristic
33C2×Dic166—21no—
34C2×Dic166—24no—
35C2×Dic166C2312yes—
36C22×Dic83—21no—
37C22×Dic83—21no—
38#13050578—28no—
39C2×C83⋊C8C2314yes—
40C4⋊Dic83—21no—
41C4⋊Dic83C2312yes—
42Dic83⋊C4—22no—
43Dic83⋊C4—22no—
44C4×Dic83—22no—
45C4×Dic83—22no—
46#41×C83—44no—
47C2×Q8×C83—21no—
48C2×D4×C83—22no—
49C2×D4×C83—21no—
50C2×D4×C83—21no—
51C2×D4×C83C2312yes—
52C22×C4×C83—22no—
53C22×C4×C83D411yescharacteristic
54C22×C4×C83D411yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C4⋊C4×C83—21no—
57C4⋊C4×C83C2312yes—
58C22⋊C4×C83—22no—
59C22⋊C4×C83—22no—
60C42×C83D411yescharacteristic
61C42×C83D411yescharacteristic
62C42×C83C2311yescharacteristic
63C42×C83C2311yescharacteristic
64C23×C83—22no—
65C23×C83C2×D411yescharacteristic
66Q8×C83—42no—
67D4×C83—24no—
68D4×C83—42no—
69D4×C83—44no—
70D4×C83—42no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—44no—
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2411yescharacteristic
82C2×Dic83—24no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—42no—
86Dic166—24no—
87Dic166—22no—
88Dic166#4114yescomplemented
89C83⋊C8—24no—
90C22×C83—21no—
91C22×C83—21no—
92C22×C83—42no—
93C22×C83—42no—
94C22×C83—42no—
95C22×C83AGL(1,ℤ/8)12yes—
96C22×C83C22×D411yescharacteristic
97C4×C83—21no—
98C4×C83—21no—
99C4×C83—21no—
100C4×C83—21no—
101C4×C83—42no—
102C4×C83#13712yes—
103C4×C83C2×#4112yes—
104Dic83—24no—
105Dic83—42no—
106C2×C83—22no—
107C2×C83—82no—
108C2×C83#52511yescharacteristic
109C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C83C2×#13711yescharacteristic
111#3112—831nomaximal, Sylow
112C83#311211yescharacteristic, complemented, Sylow
113#532D8311yescharacteristic
114C2×C4×Q8—831no—
115#480—832no—
116#477—832no—
117#460—832no—
118#448—832no—
119#440—834no—
120#432—831no—
121C2×#41—21no—
122C22×Q8—831no—
123C2×SD16—834no—
124C4⋊Q8C2×D8311yescharacteristic
125#127C2×D8311yescharacteristic
126#124—21no—
127#121—21no—
128#121C2×D8312yes—
129C4×Q8—832no—
130C4×Q8—832no—
131C4×Q8—1661no—
132C4×Q8—1661no—
133C4×D4—21no—
134C4×D4C2×D8312yes—
135C2×C4⋊C4—831no—
136C2×C4⋊C4—1661no—
137C2×C42—1661no—
138C2×C42C2×D8311yescharacteristic
139#107—834no—
140C4⋊C8—832no—
141#103—834no—
142#102—834no—
143C22⋊C8—832no—
144#97—832no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—832no—
148C2×Q8—1664no—
149C2×Q8—1661no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D8312yes—
154C22×C4—22no—
155C22×C4—1661no—
156C22×C4—1661no—
157C22×C4C22⋊D8311yescharacteristic
158C22×C4C22⋊D8311yescharacteristic
159C22×C4C22×D8311yescharacteristic
160SD16—1668no—
161C2×C8—834no—
162C4⋊C4—21no—
163C4⋊C4—832no—
164C4⋊C4—1661no—
165C4⋊C4—1662no—
166C4⋊C4—1662no—
167C4⋊C4C22×D8312yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C42—1662no—
171C42—1662no—
172C42C22⋊D8311yescharacteristic
173C42C22⋊D8311yescharacteristic
174C42C22×D8311yescharacteristic
175C42C22×D8311yescharacteristic
176C23—22no—
177C23C2×C22⋊D8311yescharacteristic
178Q8—42no—
179Q8—834no—
180Q8—1664no—
181Q8—1662no—
182D4—24no—
183D4—42no—
184D4—44no—
185D4—42no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—44no—
191C2×C4—3322no—
192C2×C4—1662no—
193C2×C4—1664no—
194C2×C4—1662no—
195C2×C4C2×C22⋊D8311yescharacteristic
196C2×C4C2×C22⋊D8311yescharacteristic
197C2×C4C2×C22⋊D8311yescharacteristic
198C2×C4C2×C22⋊D8311yescharacteristic
199C2×C4C2×C22⋊D8311yescharacteristic
200C2×C4C23×D8311yescharacteristic
201C8—1664no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22#1458181812yes—
208C22C22×C22⋊D8311yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—3322no—
215C4—1664no—
216C4#1458182212yes—
217C4#41×D8312yes—
218C2—22no—
219C2—82no—
220C2#1912820811yescharacteristic
221C2C2×#1458181811yescharacteristic
222C2C2×#1458182211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#532×C83C2
socleC22×C83C22×D4
hypercenter#532D83
hypocenterC83#3112
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic166C2semidirect
2. #19128012C2semidirect
3. #19127993C2semidirect
4. #19127898C2semidirect
5. #19127839C2semidirect
6. #19127801C2semidirect
7. #19127758C2semidirect
88. Dic166#41semidirect
112. C83#3112semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic1662
#191280122
#191279932
#191278982
#191278392
#191278012
#191277582
#532×C832
#311283