Finite Group Databasebeta

← #27170286

Subgroups — #27170286

Order 12928.

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

normal subgroups only

This group has 234 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#20265832C211yesmaximal, characteristic
2#20265528C211yesmaximal, characteristic, complemented
3#20265468C211yesmaximal, characteristic, complemented
4#20265467C211yesmaximal, characteristic, complemented
5#20264924C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×#13108074—21no—
9#15692255C2211yescharacteristic
10#15692255C2211yescharacteristic
11#15692237—22no—
12#15692234—21no—
13#15692226—21no—
14C4×D404C2211yescharacteristic
15C4⋊C4×D101—21no—
16#15692187—21no—
17#15692186—21no—
18C4⋊#11717579C2211yescharacteristic
19#15692180—22no—
20#15692177—21no—
21#15692176—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25#13108074—42no—
26#13108074—42no—
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×D404D411yescharacteristic
30C2×C22⋊D101—22no—
31C2×C4×D101—22no—
32C2×C4×D101—21no—
33C2×#11717579—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101D412yes—
38C4⋊Dic101D411yescharacteristic
39#13108053—22no—
40C22⋊Dic101—22no—
41#13108051—21no—
42C2×D4×C101D411yescharacteristic
43C22×C4×C101—21no—
44C2×C8×C101—21no—
45C2×C8×C101—21no—
46C4⋊C4×C101D411yescharacteristic
47C22⋊C4×C101—21no—
48C42×C101C2311yescharacteristic
49C23×C101—21no—
50C22×D101—21no—
51D4×C101—21no—
52D4×C101—21no—
53C2×C4×C101—21no—
54C2×C4×C101—41no—
55C2×C4×C101—41no—
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C22⋊D101—44no—
60C2×Dic101—21no—
61C2×Dic101—22no—
62C2×Dic101—41no—
63D404—21no—
64D404SD1612yes—
65C4×D101—44no—
66C4×D101—41no—
67C4×D101—41no—
68Dic202—41no—
69Dic202—41no—
70Dic202—42no—
71C8×C101—41no—
72C8×C101—41no—
73#11717579—41no—
74C22×C101—41no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101#12011yescharacteristic
78C2×D101—22no—
79C2×D101—41no—
80C2×D101—41no—
81C4×C101—21no—
82C4×C101—41no—
83C4×C101C2×SD1611yescharacteristic
84C4×C101#13511yescharacteristic
85C4×C101AGL(1,ℤ/8)11yescharacteristic
86C4×C101#13711yescharacteristic
87Dic101—42no—
88Dic101—41no—
89Dic101—81no—
90C2×C101—81no—
91C2×C101#45011yescharacteristic
92C2×C101#45211yescharacteristic
93C2×C101#45411yescharacteristic
94D101—42no—
95#1635—1011nomaximal, Sylow
96C101#163511yescharacteristic, complemented, Sylow
97#547—1011no—
98#487—1011no—
99#476—1011no—
100#476—1011no—
101#332—1011no—
102D4⋊C8—1011no—
103D4⋊C8D10111yescharacteristic, complemented
104C2×#41—2021no—
105C4⋊Q8—1011no—
106C4⋊Q8—1011no—
107#123—2022no—
108C22⋊Q8—2021no—
109#121—2021no—
110C4×D4—1011no—
111C4×D4C2×D10111yescharacteristic
112C2×C4⋊C4—2021no—
113#106—2021no—
114#106—2021no—
115C4⋊C8—1011no—
116C4⋊C8C2×D10111yescharacteristic
117#103—2022no—
118#102—2021no—
119#102—2021no—
120C4×C8C2×D10111yescharacteristic
121#41—4042no—
122#41—4042no—
123C2×Q8—2021no—
124C2×Q8—2021no—
125C2×D4—1011no—
126C2×D4—2022no—
127C2×D4D40411yescharacteristic
128C22×C4—21no—
129C22×C4—2022no—
130C22×C4—2021no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2021no—
134C4⋊C4—1012no—
135C4⋊C4—1011no—
136C4⋊C4—2022no—
137C4⋊C4—2021no—
138C4⋊C4—2021no—
139C4⋊C4—2021no—
140C4⋊C4D40411yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—2022no—
143C22⋊C4—2021no—
144C42C22×D10111yescharacteristic
145C23—21no—
146C23—2021no—
147Q8—4042no—
148Q8—4041no—
149Q8—4041no—
150D4—21no—
151D4—21no—
152D4—1012no—
153D4—4044no—
154D4—2021no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—4044no—
159C2×C4—4041no—
160C2×C4—4041no—
161C2×C4—4041no—
162C2×C4—2021no—
163C2×C4—2022no—
164C2×C4D4×D10111yescharacteristic
165C2×C4D4×D10111yescharacteristic
166C2×C4C2×D40411yescharacteristic
167C8—41no—
168C8—41no—
169C8—4041no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—4041no—
174C22—4041no—
175C22—2022no—
176C22#1569222511yescharacteristic
177C4—21no—
178C4—41no—
179C4—8081no—
180C4—4042no—
181C4—4041no—
182C4SD16×D10111yescharacteristic
183C4#1569227711yescharacteristic
184C4#1569227911yescharacteristic
185C4#1569228811yescharacteristic
186C2—81no—
187C2—4042no—
188C2#2026532911yescharacteristic
189C2#2026534111yescharacteristic
190C2#2026535111yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
prime radical#20265832C2
hypercenterD4⋊C8D101
hypocenterC101#1635
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20265528C2semidirect
3. #20265468C2semidirect
4. #20265467C2semidirect
5. #20264924C2semidirect
6. #20264906C2semidirect
7. D4⋊C8×C101C2semidirect
96. C101#1635semidirect
103. D4⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202658322
#202655282
#202654682
#202654672
#202649242
#202649062
D4⋊C8×C1012
#1635101