Finite Group Databasebeta

← #35216226

Subgroups — #35216226

Order 20864.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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#23782233C211yesmaximal, characteristic
2#23781648C212yesmaximal, complemented
3#23781462C211yesmaximal, characteristic, complemented
4#23781461C212yesmaximal, complemented
5#355×C163C211yesmaximal, characteristic, complemented
6C2×#14579700C2211yescharacteristic
7C2×D4×D163C411yescharacteristic
8M16×D163—22no—
9#19104732C411yescharacteristic
10#19104731C411yescharacteristic
11#19104724C2211yescharacteristic
12#19104708—22no—
13#19104705—22no—
14#19104692C411yescharacteristic
15#19104654C2212yescomplemented
16#19104651—22no—
17#124×C163C2211yescharacteristic
18#101×C163C2212yescomplemented
19C23×D163—21no—
20#14579700—42no—
21C2×D652—22no—
22C2×D652C2×C411yescharacteristic
23C2×D652C2×C411yescharacteristic
24C2×D652D412yes—
25C2×C22⋊D163—22no—
26C2×C22⋊D163—21no—
27D4×D163—42no—
28C2×C4×D163C2×C411yescharacteristic
29C2×C4×D163D412yes—
30#14579683—22no—
31#14579682—42no—
32C8×D163—42no—
33Dic163⋊C4—21no—
34(C2×D163)⋊C4—21no—
35(C2×D163)⋊C4—22no—
36C4×Dic163C2×C411yescharacteristic
37C2×Q8×C163C2311yescharacteristic
38C2×D4×C163C2×C411yescharacteristic
39M16×C163—22no—
40C22⋊C4×C163—22no—
41C42×C163C2×C411yescharacteristic
42C22×D163—21no—
43C22×D163—42no—
44C22×D163—82no—
45C22×D163C22⋊C411yescharacteristic
46C22×D163C22⋊C412yes—
47D652—41no—
48D652—42no—
49D652—41no—
50D652—82no—
51C22⋊D163—42no—
52C22⋊D163—82no—
53C2×Dic163—21no—
54C2×Dic163C22⋊C411yescharacteristic
55C4×D163—22no—
56C4×D163—22no—
57C4×D163—22no—
58C163⋊C8—42no—
59C23×C163—21no—
60Q8×C163—41no—
61D4×C163—41no—
62C2×C4×C163—21no—
63C2×C4×C163C22×C411yescharacteristic
64C2×C4×C163C2×D412yes—
65C8×C163—42no—
66C22×C163—42no—
67C22×C163—41no—
68C22×C163C2×C22⋊C411yescharacteristic
69C2×D163—21no—
70C2×D163—22no—
71C2×D163—42no—
72C2×D163—41no—
73C2×D163—42no—
74C2×D163—41no—
75C2×D163—84no—
76C4×C163—22no—
77C4×C163—21no—
78C4×C163—41no—
79Dic163—41no—
80Dic163#9912yes—
81C2×C163—21no—
82C2×C163—81no—
83C2×C163C2×#9911yescharacteristic
84D163—42no—
85D163—41no—
86D163—81no—
87C163#213411yescharacteristic, complemented, Sylow
88#2134—1631nomaximal, Sylow
89#535—1631no—
90#413—1632no—
91#355—1631no—
92#355—1632no—
93#355D16311yescharacteristic, complemented
94C2×#41—1631no—
95C22×D4—1631no—
96C2×M16—3262no—
97#127—1631no—
98#127—1631no—
99#124—1631no—
100#124C2×D16311yescharacteristic
101#121—3262no—
102#120—3262no—
103#117—1631no—
104#101—1632no—
105#101C2×D16312yescomplemented
106#100—3262no—
107C24—3261no—
108#41—6522no—
109C2×Q8C22×D16311yescharacteristic
110C2×D4—3262no—
111C2×D4—3262no—
112C2×D4—3261no—
113C2×D4—6522no—
114C2×D4—1631no—
115C2×D4—1631no—
116C2×D4—1632no—
117C2×D4C4×D16311yescharacteristic
118C22×C4—1631no—
119C22×C4—1632no—
120M16—22no—
121M16—3262no—
122M16—6522no—
123C2×C8—6522no—
124C4⋊C4—3261no—
125C22⋊C4—22no—
126C22⋊C4—3261no—
127C22⋊C4—3262no—
128C42—1631no—
129C42C4×D16311yescharacteristic
130C23—21no—
131C23—13042no—
132C23—3261no—
133C23—6522no—
134C23—1631no—
135C23—1632no—
136Q8—41no—
137D4—41no—
138D4—13042no—
139D4—13042no—
140D4—6522no—
141D4—6521no—
142D4—6521no—
143D4—6522no—
144C2×C4—21no—
145C2×C4—3262no—
146C2×C4—3262no—
147C2×C4—3262no—
148C2×C4—3261no—
149C2×C4—1631no—
150C2×C4C2×C4×D16311yescharacteristic
151C2×C4D4×D16312yes—
152C8—42no—
153C8—6522no—
154C22—42no—
155C22—41no—
156C22—13044no—
157C22—3261no—
158C22—3262no—
159C22—6522no—
160C22—6522no—
161C22—6521no—
162C22—6521no—
163C22C22⋊C4×D16311yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—6521no—
168C4—1632no—
169C2—21no—
170C2—81no—
171C2—13041no—
172C2—6521no—
173C2—6522no—
174C2#99×D16311yescharacteristic
subgroupG/H
centerC2#99×D163
commutatorC2×C4×C163C22×C4
FrattiniC2×Q8C22×D163
Fitting#355×C163C2
socleC2×C163C2×#99
prime radical#23782233C2
cosocleC2×Q8×C163C23
hypercenter#355D163
hypocenterC163#2134
LCSG ▷ C2×C4×C163 ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #23781648C2semidirect
3. #23781462C2semidirect
4. #23781461C2semidirect
5. #355×C163C2semidirect
15. #19104654C22semidirect
18. #101×C163C22semidirect
87. C163#2134semidirect
93. #355D163semidirect
105. #101C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822332
#237816482
#237814622
#237814612
#355×C1632
#2134163