Finite Group Databasebeta

← #24918005

Subgroups — #24918005

Order 10624.

380 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 157 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#19128258C211yesmaximal, characteristic
2#19128107C214yesmaximal, complemented
3#19128104C212yesmaximal, complemented
4#19128097C212yesmaximal, complemented
5#19127741C214yesmaximal
6C4×#13050574C211yesmaximal, characteristic
7C2×C4⋊Q8×C83C211yesmaximal, characteristic
8C2×#13050574C2212yes—
9#14581792C2211yescharacteristic
10#14581792C2211yescharacteristic
11#14581784C2212yes—
12#14581771—21no—
13#14581771—21no—
14C4×Dic166—21no—
15C4×Dic166—21no—
16#14581751C2212yes—
17#14581725C2218yes—
18#14581723C2218yes—
19#14581703C2212yes—
20C4×C83⋊C8C2212yes—
21C22×Q8×C83—21no—
22C4⋊Q8×C83—21no—
23C4⋊Q8×C83—21no—
24C4⋊Q8×C83C2212yes—
25C4⋊Q8×C83C2212yes—
26C2×C4⋊C4×C83—21no—
27C2×C4⋊C4×C83C2212yes—
28C2×C42×C83C2211yescharacteristic
29C2×Dic166—21no—
30C2×Dic166—21no—
31#13050574Q818yescomplemented
32C2×C83⋊C8C2314yes—
33C4⋊Dic83—21no—
34C4⋊Dic83—21no—
35C4⋊Dic83C2312yes—
36C4⋊Dic83C2312yes—
37Dic83⋊C4—22no—
38Dic83⋊C4—22no—
39C22⋊Dic83—22no—
40C4×Dic83—22no—
41C2×Q8×C83—22no—
42C2×Q8×C83—22no—
43C2×Q8×C83—44no—
44C22×C4×C83—22no—
45C22×C4×C83D411yescharacteristic
46C22×C4×C83D411yescharacteristic
47C22×C4×C83C2311yescharacteristic
48C4⋊C4×C83—22no—
49C4⋊C4×C83—22no—
50C4⋊C4×C83—22no—
51C4⋊C4×C83C2314yes—
52C42×C83D411yescharacteristic
53C42×C83D411yescharacteristic
54C42×C83C2311yescharacteristic
55C42×C83C2311yescharacteristic
56C23×C83C2×D411yescharacteristic
57Q8×C83—44no—
58Q8×C83—44no—
59C2×C4×C83—22no—
60C2×C4×C83—22no—
61C2×C4×C83—24no—
62C2×C4×C83—44no—
63C2×C4×C83C2×D411yescharacteristic
64C2×C4×C83C2×D411yescharacteristic
65C2×C4×C83C2×D411yescharacteristic
66C2×C4×C83C2×D411yescharacteristic
67C2×C4×C83C2×D411yescharacteristic
68C2×C4×C83C2×Q814yes—
69C2×C4×C83C2411yescharacteristic
70C2×Dic83—22no—
71C2×Dic83—22no—
72Dic166—42no—
73Dic166—42no—
74C83⋊C8C2×Q818yescomplemented
75C22×C83—21no—
76C22×C83—21no—
77C22×C83C4⋊Q812yes—
78C22×C83C22×D411yescharacteristic
79C4×C83—21no—
80C4×C83—21no—
81C4×C83—44no—
82C4×C83C4⋊Q812yes—
83C4×C83AGL(1,ℤ/8)12yes—
84C4×C83#13712yes—
85C4×C83C22×Q812yes—
86Dic83—42no—
87Dic83—42no—
88C2×C83—22no—
89C2×C83C2×C4⋊Q811yescharacteristic
90C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
91C2×C83C2×#13711yescharacteristic
92#3165—831nomaximal, Sylow
93C83#316511yescharacteristic, complemented, Sylow
94#533—831no—
95C2×C4⋊Q8D8311yescharacteristic
96#501—834no—
97#500—832no—
98#498—832no—
99#428—834no—
100C4×M16—831no—
101C22×Q8—21no—
102C2×M16—832no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—831no—
106C4⋊Q8—831no—
107C4⋊Q8C2×D8312yes—
108C4⋊Q8C2×D8312yes—
109#125—832no—
110C22⋊Q8—1661no—
111C22⋊Q8—1661no—
112C4×Q8—1661no—
113C4×Q8—1661no—
114#117—832no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×D8312yes—
117C2×C42C2×D8311yescharacteristic
118#107—838no—
119#106—838no—
120#97—832no—
121C4×C8—832no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—44no—
125C2×Q8—1661no—
126C2×Q8—1661no—
127C22×C4—22no—
128C22×C4C22⋊D8311yescharacteristic
129C22×C4C22⋊D8311yescharacteristic
130C22×C4C22×D8311yescharacteristic
131M16—838no—
132C2×C8—834no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—832no—
137C4⋊C4—832no—
138C4⋊C4—1662no—
139C4⋊C4—1662no—
140C4⋊C4—1661no—
141C4⋊C4—1661no—
142C4⋊C4C22×D8314yes—
143C22⋊C4—1662no—
144C42—1662no—
145C42C22⋊D8311yescharacteristic
146C42C22⋊D8311yescharacteristic
147C42C22×D8311yescharacteristic
148C42C22×D8311yescharacteristic
149C23C2×C22⋊D8311yescharacteristic
150Q8—44no—
151Q8—44no—
152Q8—3322no—
153Q8—3322no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—44no—
158C2×C4—1662no—
159C2×C4—1662no—
160C2×C4C2×C22⋊D8311yescharacteristic
161C2×C4C2×C22⋊D8311yescharacteristic
162C2×C4C2×C22⋊D8311yescharacteristic
163C2×C4C2×C22⋊D8311yescharacteristic
164C2×C4C2×C22⋊D8311yescharacteristic
165C2×C4Q8×D8314yes—
166C2×C4C23×D8311yescharacteristic
167C8—838no—
168C22—21no—
169C22—21no—
170C22#1458179012yes—
171C22C22×C22⋊D8311yescharacteristic
172C4—21no—
173C4—21no—
174C4—44no—
175C4—3322no—
176C4—3322no—
177C4#1458179012yes—
178C4#1458181812yes—
179C4#1458182312yes—
180C4C2×Q8×D8312yes—
181C2—22no—
182C2C2×#1458179011yescharacteristic
183C2C2×#1458181811yescharacteristic
184C2C2×#1458182311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C4⋊Q8×C83C2
socleC22×C83C22×D4
prime radicalC23×C83C2×D4
hypercenterC2×C4⋊Q8D83
hypocenterC83#3165
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #19128107C2semidirect
3. #19128104C2semidirect
4. #19128097C2semidirect
31. #13050574Q8semidirect
74. C83⋊C8C2×Q8semidirect
93. C83#3165semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191281072
#191281042
#191280972
#191277412
C4×#130505742
C2×C4⋊Q8×C832
#316583