Finite Group Databasebeta

← #35216489

Subgroups — #35216489

Order 20864.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#23782465C211yesmaximal, characteristic
2#23782108C212yesmaximal, complemented
3#23781923C211yesmaximal, characteristic, complemented
4#23781476C212yesmaximal, complemented
5#349×C163C211yesmaximal, characteristic, complemented
6C2×#14579699—21no—
7#19104768—22no—
8#19104763—22no—
9#19104757—21no—
10#19104757C2212yes—
11C2×Dic652C2211yescharacteristic, complemented
12#19104714—21no—
13Dic1304—22no—
14#19104680—22no—
15#19104667C2211yescharacteristic, complemented
16#19104660—21no—
17C2×M16×C163C2211yescharacteristic
18M32×C163C2212yescomplemented
19#14579699—22no—
20#14579699—22no—
21#14579699—41no—
22#14579699—41no—
23C2×Dic326—21no—
24C2×Dic326—21no—
25C2×D652—21no—
26C2×C22⋊D163—41no—
27C2×C4×D163—41no—
28Dic652—21no—
29Dic652—22no—
30Dic652D412yes—
31#14579688—22no—
32#14579688—22no—
33D1304—21no—
34D1304D412yes—
35C4⋊Dic163—21no—
36Dic163⋊C4—41no—
37C22⋊Dic163—41no—
38C22×C4×C163D411yescharacteristic
39M16×C163—22no—
40C2×C8×C163D411yescharacteristic
41C2×C8×C163C2311yescharacteristic
42C16×C163—22no—
43C22×D163—41no—
44Dic326—22no—
45Dic326—41no—
46Dic326—41no—
47Dic326—41no—
48D652—22no—
49D652—41no—
50C22⋊D163—42no—
51C22⋊D163—42no—
52C2×Dic163—41no—
53C2×Dic163—41no—
54C2×Dic163—41no—
55C4×D163—42no—
56C4×D163—42no—
57C23×C163D811yescharacteristic
58C2×C4×C163—21no—
59C2×C4×C163—21no—
60C2×C4×C163D811yescharacteristic
61C2×C4×C163C2×D411yescharacteristic
62C8×C163—21no—
63C8×C163C2×D412yes—
64C22×C163—21no—
65C22×C163—41no—
66C22×C163C2×D811yescharacteristic
67C2×D163—42no—
68C2×D163—82no—
69C4×C163—21no—
70C4×C163#12011yescharacteristic
71C4×C163AGL(1,ℤ/8)11yescharacteristic
72Dic163—42no—
73Dic163—81no—
74Dic163—81no—
75C2×C163—21no—
76C2×C163—41no—
77C2×C163#44711yescharacteristic
78D163—82no—
79C163#219111yescharacteristic, complemented, Sylow
80#2191—1631nomaximal, Sylow
81#575—1631no—
82#510—1632no—
83#470—1631no—
84#360—1632no—
85#349D16311yescharacteristic, complemented
86C2×#41—3261no—
87#137—3262no—
88AGL(1,ℤ/8)—3262no—
89#135—3261no—
90#135—1632no—
91C2×Q16—1631no—
92C2×M16C2×D16311yescharacteristic
93C22⋊Q8—3261no—
94Q32—3262no—
95SD32—3262no—
96M32C2×D16312yescomplemented
97#106—1631no—
98#103—3261no—
99#41—3262no—
100#41—3262no—
101#41—6521no—
102#41—6521no—
103C2×Q8—3261no—
104C2×Q8—3261no—
105C2×D4—3261no—
106C2×D4—6521no—
107C22×C4—6521no—
108C22×C4D65211yescharacteristic
109Q16—3261no—
110Q16—3262no—
111Q16—1632no—
112SD16—3262no—
113SD16—3262no—
114D8—3261no—
115D8—1632no—
116M16—22no—
117C2×C8D65211yescharacteristic
118C2×C8C22×D16311yescharacteristic
119C4⋊C4—3261no—
120C4⋊C4—6521no—
121C22⋊C4—6521no—
122C16—22no—
123C23—6521no—
124C23D130411yescharacteristic
125Q8—3262no—
126Q8—6521no—
127Q8—6521no—
128Q8—6521no—
129D4—3262no—
130D4—6521no—
131D4—6522no—
132D4—6522no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—6522no—
136C2×C4—6521no—
137C2×C4—6521no—
138C2×C4—6521no—
139C2×C4—6522no—
140C2×C4D130411yescharacteristic
141C2×C4C2×D65211yescharacteristic
142C8—21no—
143C8D4×D16312yes—
144C22—21no—
145C22—41no—
146C22—13042no—
147C22—6522no—
148C22C2×D130411yescharacteristic
149C4—21no—
150C4—13041no—
151C4—13041no—
152C4—6522no—
153C4#1910470511yescharacteristic
154C4#1910476311yescharacteristic
155C2—21no—
156C2—41no—
157C2—13042no—
158C2#2378180111yescharacteristic
subgroupG/H
centerC2#23781801
commutator, cosocleC2×C8×C163C23
FrattiniC2×C8C22×D163
Fitting#349×C163C2
socleC2×C163#447
prime radical#23782465C2
hypercenter#349D163
hypocenterC163#2191
LCSG ▷ C2×C8×C163 ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #23782108C2semidirect
3. #23781923C2semidirect
4. #23781476C2semidirect
5. #349×C163C2semidirect
11. C2×Dic652C22semidirect
15. #19104667C22semidirect
18. M32×C163C22semidirect
79. C163#2191semidirect
85. #349D163semidirect
96. M32C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824652
#237821082
#237819232
#237814762
#349×C1632
#2191163