Finite Group Databasebeta

← #27170245

Subgroups — #27170245

Order 12928.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265900C211yesmaximal, characteristic, complemented
2#20265524C211yesmaximal, characteristic, complemented
3#20265377C211yesmaximal, characteristic, complemented
4#20265373C211yesmaximal, characteristic
5#20264924C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7Q8⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×#13108061—21no—
9C2×D808—21no—
10#15692255C2211yescharacteristic
11#15692252C2211yescharacteristic
12#15692250—22no—
13#15692243—21no—
14#15692231—21no—
15C4×Dic202—21no—
16C4×D404C2211yescharacteristic
17#15692213—21no—
18C4⋊#11717579C2211yescharacteristic
19#15692177—22no—
20#15692177—21no—
21#15692175—21no—
22C4×Q8×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202—21no—
26C2×D404—21no—
27C2×D404—21no—
28C2×D404—21no—
29C2×D404D411yescharacteristic
30C2×C4×D101—21no—
31#13108061—42no—
32D808—42no—
33C2×#11717579—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101D411yescharacteristic
36C4⋊Dic101D412yes—
37#13108053—22no—
38#13108051—22no—
39#13108051—22no—
40#13108051—21no—
41C4×Dic101—22no—
42C2×Q8×C101D411yescharacteristic
43C2×C8×C101—21no—
44C2×C8×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101D411yescharacteristic
47C42×C101—21no—
48C42×C101C2311yescharacteristic
49C22×D101—21no—
50C22×D101—41no—
51Q8×C101—21no—
52Q8×C101—21no—
53C2×C4×C101—21no—
54C2×C4×C101—21no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×Dic101—21no—
59C2×Dic101—22no—
60D404—21no—
61D404—21no—
62D404—42no—
63D404—41no—
64D404—41no—
65D404—41no—
66D404—41no—
67C4×D101—41no—
68C4×D101—41no—
69Dic202—41no—
70Dic202—41no—
71C8×C101—41no—
72C8×C101—41no—
73#11717579—41no—
74C22×C101#12011yescharacteristic
75C2×D101—41no—
76C2×D101—41no—
77C2×D101—41no—
78C2×D101—81no—
79C2×D101—81no—
80C2×D101—81no—
81C4×C101—21no—
82C4×C101—41no—
83C4×C101—41no—
84C4×C101#13511yescharacteristic
85C4×C101#13511yescharacteristic
86C4×C101AGL(1,ℤ/8)11yescharacteristic
87C4×C101AGL(1,ℤ/8)11yescharacteristic
88Dic101—42no—
89Dic101—41no—
90C2×C101#44911yescharacteristic
91C2×C101#44911yescharacteristic
92C2×C101#45411yescharacteristic
93D101—81no—
94D101—161no—
95#1628—1011nomaximal, Sylow
96C101#162811yescharacteristic, complemented, Sylow
97#556—1011no—
98#486—1011no—
99#460—1011no—
100#459—1011no—
101#332—1011no—
102Q8⋊C8D10111yescharacteristic, complemented
103D4⋊C8—1011no—
104C2×SD16—2021no—
105C2×D8—2021no—
106C4⋊Q8—1011no—
107#127—1011no—
108#126—2022no—
109#124—2021no—
110C22⋊Q8—2021no—
111C4×Q8—2021no—
112C4×Q8C2×D10111yescharacteristic
113C4×D4—1011no—
114#117—2021no—
115C4⋊C8—1011no—
116C4⋊C8C2×D10111yescharacteristic
117#102—2021no—
118#102—2022no—
119#102—2021no—
120C4×C8C2×D10111yescharacteristic
121C2×Q8—2021no—
122C2×Q8D40411yescharacteristic
123C2×D4—1011no—
124C2×D4—2021no—
125C2×D4—2021no—
126C2×D4—2021no—
127C22×C4—2021no—
128SD16—4042no—
129D8—4042no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—2021no—
133C4⋊C4—21no—
134C4⋊C4—1011no—
135C4⋊C4—1012no—
136C4⋊C4—2022no—
137C4⋊C4—2021no—
138C4⋊C4D40411yescharacteristic
139C22⋊C4—2022no—
140C22⋊C4—2022no—
141C22⋊C4—2021no—
142C42—21no—
143C42—2022no—
144C42C22×D10111yescharacteristic
145C23—4041no—
146C23—2021no—
147Q8—21no—
148Q8—21no—
149Q8—4041no—
150Q8—4041no—
151D4—4042no—
152D4—4041no—
153D4—4041no—
154D4—4041no—
155D4—4041no—
156D4—2021no—
157D4—2021no—
158C2×C4—21no—
159C2×C4—21no—
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—8081no—
171C22—8081no—
172C22—8081no—
173C22—4041no—
174C22—4041no—
175C22—4041no—
176C22#1569222511yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—4041no—
181C4—4042no—
182C4#1569227711yescharacteristic
183C4#1569227811yescharacteristic
184C4#1569228211yescharacteristic
185C4#1569228311yescharacteristic
186C2—8081no—
187C2—16161no—
188C2#2026532411yescharacteristic
189C2#2026532511yescharacteristic
190C2#2026535111yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingQ8⋊C8×C101C2
socleC22×C101#120
prime radical#20265373C2
hypercenterQ8⋊C8D101
hypocenterC101#1628
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #20265900C2semidirect
2. #20265524C2semidirect
3. #20265377C2semidirect
5. #20264924C2semidirect
6. #20264906C2semidirect
7. Q8⋊C8×C101C2semidirect
96. C101#1628semidirect
102. Q8⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202659002
#202655242
#202653772
#202653732
#202649242
#202649062
Q8⋊C8×C1012
#1628101