Finite Group Databasebeta

← #24909116

Subgroups — #24909116

Order 10624.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#19128272C211yesmaximal, characteristic
2#19127685C211yesmaximal, characteristic
3#19127490C214yesmaximal, complemented
4#410×C83C211yesmaximal, characteristic
5C22×Dic166C411yescharacteristic, complemented
6C2×#13050574—21no—
7#14581790—21no—
8#14581779—21no—
9#14581773C412yescomplemented
10#14581773C2212yes—
11#14581771—21no—
12#14581771—21no—
13#14581751C411yescharacteristic, complemented
14#14581711—21no—
15#14581709C2212yes—
16C2×#41×C83C2211yescharacteristic
17#117×C83—21no—
18#99×C83—21no—
19#99×C83C2212yes—
20C2×Dic166—21no—
21C2×Dic166—21no—
22C2×Dic166—41no—
23C2×Dic166—41no—
24C22×Dic83C2×C412yes—
25#13050574—22no—
26#13050574—41no—
27C2×C83⋊C8—41no—
28C4⋊Dic83—21no—
29Dic83⋊C4—22no—
30Dic83⋊C4—22no—
31C22⋊Dic83—22no—
32C22⋊Dic83—22no—
33C22⋊Dic83C2×C412yes—
34C4×Dic83—21no—
35#41×C83—41no—
36#41×C83—41no—
37C2×Q8×C83D411yescharacteristic
38C2×D4×C83C2×C411yescharacteristic
39C2×D4×C83D411yescharacteristic
40C2×D4×C83C2311yescharacteristic
41C22×C4×C83C2×C411yescharacteristic
42C22×C4×C83D411yescharacteristic
43C22×C4×C83D411yescharacteristic
44C4⋊C4×C83—41no—
45C22⋊C4×C83—22no—
46C42×C83—41no—
47C23×C83C22⋊C411yescharacteristic
48C23×C83C22×C411yescharacteristic
49C23×C83C2×D411yescharacteristic
50Q8×C83—41no—
51D4×C83—41no—
52D4×C83—41no—
53D4×C83—41no—
54C2×C4×C83—21no—
55C2×C4×C83—21no—
56C2×C4×C83—21no—
57C2×C4×C83—21no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83—42no—
61C2×C4×C83C22⋊C411yescharacteristic
62C2×C4×C83C22⋊C411yescharacteristic
63C2×C4×C83C22⋊C411yescharacteristic
64C2×C4×C83C2×D411yescharacteristic
65C2×Dic83—22no—
66C2×Dic83—22no—
67C2×Dic83—42no—
68Dic166—42no—
69Dic166—41no—
70Dic166—41no—
71C83⋊C8—42no—
72C22×C83—21no—
73C22×C83—21no—
74C22×C83—21no—
75C22×C83—41no—
76C22×C83—41no—
77C22×C83—41no—
78C22×C83C2×C22⋊C411yescharacteristic
79C4×C83—21no—
80C4×C83—21no—
81C4×C83—21no—
82C4×C83—82no—
83C4×C83#9911yescharacteristic
84C4×C83#9911yescharacteristic
85Dic83—42no—
86Dic83—42no—
87C2×C83—21no—
88C2×C83—41no—
89C2×C83—41no—
90C2×C83—41no—
91C2×C83C2×#9911yescharacteristic
92#2123—831nomaximal, Sylow
93C83#212311yescharacteristic, complemented, Sylow
94#536—831no—
95#413—831no—
96#410D8311yescharacteristic
97#352—834no—
98C2×#41C2×D8311yescharacteristic
99C22×Q8—831no—
100C2×M16—1661no—
101C4⋊Q8—1661no—
102#124—1661no—
103#123—832no—
104#123—832no—
105C22⋊Q8—1661no—
106C22⋊Q8—1661no—
107#117—21no—
108#117—831no—
109#101—1661no—
110#100—832no—
111#99—21no—
112#99C2×D8312yes—
113#41—41no—
114#41—41no—
115C2×Q8—3321no—
116C2×Q8—3321no—
117C2×Q8—1661no—
118C2×Q8—1661no—
119C2×Q8D33211yescharacteristic
120C2×D4C4×D8311yescharacteristic
121C2×D4C22⋊D8311yescharacteristic
122C2×D4C22×D8311yescharacteristic
123C22×C4—832no—
124C22×C4C4×D8311yescharacteristic
125C22×C4D33211yescharacteristic
126C22×C4C22⋊D8311yescharacteristic
127M16—3321no—
128M16—1662no—
129C2×C8—3321no—
130C4⋊C4—41no—
131C4⋊C4—1662no—
132C4⋊C4—1661no—
133C4⋊C4—1662no—
134C22⋊C4—22no—
135C22⋊C4—832no—
136C22⋊C4—1662no—
137C22⋊C4—1662no—
138C42—41no—
139C42—1661no—
140C23(C2×D83)⋊C411yescharacteristic
141C23C2×C4×D8311yescharacteristic
142C23C2×C22⋊D8311yescharacteristic
143Q8—41no—
144Q8—3322no—
145Q8—3321no—
146Q8—3321no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—3322no—
158C2×C4—1662no—
159C2×C4—1662no—
160C2×C4(C2×D83)⋊C411yescharacteristic
161C2×C4(C2×D83)⋊C411yescharacteristic
162C2×C4(C2×D83)⋊C411yescharacteristic
163C2×C4C2×D33211yescharacteristic
164C8—3322no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×(C2×D83)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—82no—
176C4—3322no—
177C4—3322no—
178C4#1458170611yescharacteristic
179C4#1458170611yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1458170611yescharacteristic
subgroupG/H
centerC2C2×#14581706
commutatorC23×C83C22×C4
FrattiniC2×D4C22×D83
Fitting#410×C83C2
socleC2×C83C2×#99
prime radicalC2×#41×C83C22
cosocleC2×D4×C83C23
hypercenter#410D83
hypocenterC83#2123
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #19127490C2semidirect
5. C22×Dic166C4semidirect
9. #14581773C4semidirect
13. #14581751C4semidirect
93. C83#2123semidirect

Maximal subgroups

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

subgroupindex
#191282722
#191276852
#191274902
#410×C832
#212383