Finite Group Databasebeta

← #27170357

Subgroups — #27170357

Order 12928.

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

normal subgroups only

This group has 212 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#20265893C211yesmaximal, characteristic
2#20265454C211yesmaximal, characteristic, complemented
3#20265379C211yesmaximal, characteristic, complemented
4#20264926C211yesmaximal, characteristic, complemented
5#20264906C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7#487×C101C211yesmaximal, characteristic, complemented
8C2×#13108061—21no—
9#15692244—22no—
10#15692230—21no—
11#15692230—21no—
12C4×D404C2211yescharacteristic
13C4×D404C2211yescharacteristic
14#15692213—21no—
15#15692213—21no—
16#15692188—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692179—21no—
20#15692175—21no—
21C4⋊Q8×C101C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23#103×C101—22no—
24C4×C8×C101C2211yescharacteristic
25C2×D404—22no—
26C2×D404D411yescharacteristic
27C2×D404D411yescharacteristic
28C2×C4×D101—21no—
29C2×C4×D101—21no—
30#13108061—42no—
31C2×#11717579—21no—
32C2×#11717579—21no—
33C4⋊Dic101D411yescharacteristic
34C4⋊Dic101D411yescharacteristic
35#13108051—22no—
36#13108051—22no—
37#13108051—21no—
38#13108051—21no—
39C4×Dic101—22no—
40C2×Q8×C101—21no—
41C2×Q8×C101D412yes—
42C2×C8×C101—21no—
43C4⋊C4×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C42×C101C2311yescharacteristic
48C22×D101—21no—
49C22×D101—21no—
50Q8×C101—42no—
51Q8×C101—41no—
52Q8×C101—41no—
53C2×C4×C101—22no—
54C2×C4×C101—41no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×Dic101—21no—
59C2×Dic101—21no—
60D404—21no—
61D404—21no—
62D404—21no—
63D404—21no—
64D404—82no—
65C4×D101—41no—
66C4×D101—41no—
67C4×D101—41no—
68C4×D101—41no—
69C8×C101—41no—
70#11717579—41no—
71#11717579—41no—
72C22×C101#12011yescharacteristic
73C2×D101—41no—
74C2×D101—41no—
75C2×D101—41no—
76C2×D101—41no—
77C2×D101—41no—
78C2×D101—41no—
79C4×C101—21no—
80C4×C101—42no—
81C4×C101—81no—
82C4×C101#13511yescharacteristic
83C4×C101#13511yescharacteristic
84C4×C101AGL(1,ℤ/8)11yescharacteristic
85C4×C101#13711yescharacteristic
86Dic101—41no—
87Dic101—41no—
88C2×C101#44911yescharacteristic
89C2×C101#45211yescharacteristic
90C2×C101#45511yescharacteristic
91D101—81no—
92D101—81no—
93#1646—1011nomaximal, Sylow
94C101#164611yescharacteristic, complemented, Sylow
95#555—1011no—
96#487D10111yescharacteristic, complemented
97#474—1011no—
98#461—1011no—
99#333—1011no—
100D4⋊C8—1011no—
101D4⋊C8—1011no—
102C2×SD16—2021no—
103C4⋊Q8C2×D10111yescharacteristic
104C4⋊Q8C2×D10111yescharacteristic
105#124—2022no—
106#121—2021no—
107#121—2021no—
108C4×D4—1011no—
109C4×D4—1011no—
110#117—2021no—
111#117—2021no—
112#107—2021no—
113C4⋊C8—1011no—
114C4⋊C8—1011no—
115#103—22no—
116#103—2021no—
117#102—2021no—
118C4×C8C2×D10111yescharacteristic
119C2×Q8—21no—
120C2×Q8C22⋊D10112yes—
121C2×D4—1011no—
122C2×D4—1011no—
123C2×D4—2022no—
124C22×C4—2021no—
125C22×C4—2021no—
126SD16—4042no—
127C2×C8—21no—
128C2×C8—2021no—
129C2×C8—2021no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1011no—
135C4⋊C4—1011no—
136C22⋊C4—2022no—
137C22⋊C4—2022no—
138C22⋊C4—2021no—
139C22⋊C4—2021no—
140C42—2022no—
141C42C22×D10111yescharacteristic
142C23—2021no—
143C23—2021no—
144Q8—42no—
145Q8—41no—
146Q8—41no—
147D4—8082no—
148D4—2021no—
149D4—2021no—
150D4—2021no—
151D4—2021no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—4041no—
155C2×C4—4041no—
156C2×C4—4041no—
157C2×C4—4041no—
158C2×C4—2021no—
159C2×C4—2021no—
160C2×C4D4×D10111yescharacteristic
161C2×C4D4×D10111yescharacteristic
162C2×C4C2×C22⋊D10111yescharacteristic
163C8—41no—
164C8—4041no—
165C8—4041no—
166C22—4041no—
167C22—4041no—
168C22—4041no—
169C22—4041no—
170C22—4041no—
171C22—4041no—
172C22#1569222311yescharacteristic
173C4—21no—
174C4—42no—
175C4—81no—
176C4—4041no—
177C4—4041no—
178C4#1569227611yescharacteristic
179C4#1569227811yescharacteristic
180C4#1569228211yescharacteristic
181C4#1569228711yescharacteristic
182C2—8081no—
183C2—8081no—
184C2#2026532111yescharacteristic
185C2#2026533811yescharacteristic
186C2#2026535411yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#487×C101C2
socleC22×C101#120
prime radical#20265893C2
hypercenter#487D101
hypocenterC101#1646
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #20265454C2semidirect
3. #20265379C2semidirect
4. #20264926C2semidirect
5. #20264906C2semidirect
6. #20264906C2semidirect
7. #487×C101C2semidirect
94. C101#1646semidirect
96. #487D101semidirect

Maximal subgroups

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

subgroupindex
#202658932
#202654542
#202653792
#202649262
#202649062
#487×C1012
#1646101