Finite Group Databasebeta

← #27170322

Subgroups — #27170322

Order 12928.

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

normal subgroups only

This group has 250 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#20265847C211yesmaximal, characteristic
2#20265454C211yesmaximal, characteristic, complemented
3#20265384C211yesmaximal, characteristic, complemented
4#20264921C211yesmaximal, characteristic, complemented
5#20264909C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7#486×C101C211yesmaximal, characteristic, complemented
8C2×#13108073—21no—
9C2×#13108064—21no—
10#15692251—21no—
11#15692227—22no—
12#15692226—21no—
13C4×Dic202C2211yescharacteristic
14C4×D404C2211yescharacteristic
15D4×Dic101—21no—
16#15692188—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692179—21no—
20#15692175—21no—
21C4⋊Q8×C101C2211yescharacteristic
22#127×C101C2211yescharacteristic
23#102×C101—22no—
24C4×C8×C101C2211yescharacteristic
25#13108073—42no—
26#13108073—42no—
27C2×Dic202D411yescharacteristic
28C2×D404D411yescharacteristic
29C2×C22⋊D101—22no—
30C2×C22⋊D101—22no—
31C22×Dic101—22no—
32C2×C4×D101—21no—
33#13108064—42no—
34C2×#11717579—21no—
35C2×#11717579—21no—
36C4⋊Dic101D411yescharacteristic
37C4⋊Dic101D411yescharacteristic
38#13108053—21no—
39C22⋊Dic101—22no—
40#13108051—21no—
41C4×Dic101—21no—
42C2×Q8×C101—21no—
43C2×D4×C101—21no—
44C2×D4×C101—21no—
45C2×D4×C101D412yes—
46C2×C8×C101—21no—
47C4⋊C4×C101—21no—
48C4⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—22no—
51C22×D101—21no—
52Q8×C101—41no—
53Q8×C101—41no—
54D4×C101—42no—
55D4×C101—41no—
56D4×C101—41no—
57D4×C101—41no—
58D4×C101—41no—
59C2×C4×C101—41no—
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C22⋊D101—44no—
64C22⋊D101—82no—
65C2×Dic101—21no—
66C2×Dic101—21no—
67C2×Dic101—21no—
68C2×Dic101—44no—
69D404—21no—
70D404—21no—
71C4×D101—41no—
72C4×D101—41no—
73Dic202—21no—
74Dic202D812yes—
75C8×C101—41no—
76#11717579—41no—
77#11717579—41no—
78C22×C101—42no—
79C22×C101—42no—
80C22×C101—42no—
81C22×C101#12011yescharacteristic
82C2×D101—41no—
83C2×D101—41no—
84C2×D101—41no—
85C4×C101—21no—
86C4×C101—81no—
87C4×C101C2×D811yescharacteristic
88C4×C101#13511yescharacteristic
89C4×C101AGL(1,ℤ/8)11yescharacteristic
90C4×C101#13711yescharacteristic
91Dic101—22no—
92Dic101—41no—
93Dic101—41no—
94C2×C101—82no—
95C2×C101#44711yescharacteristic
96C2×C101#45211yescharacteristic
97C2×C101#45511yescharacteristic
98D101—81no—
99#1641—1011nomaximal, Sylow
100C101#164111yescharacteristic, complemented, Sylow
101#550—1011no—
102#486D10111yescharacteristic, complemented
103#474—1011no—
104#462—1011no—
105#331—1011no—
106Q8⋊C8—1011no—
107D4⋊C8—1011no—
108C2×#41—2021no—
109C2×Q16—2021no—
110C4⋊Q8C2×D10111yescharacteristic
111#127—2021no—
112#127C2×D10111yescharacteristic
113#121—2021no—
114#121—2022no—
115C4×Q8—1011no—
116C4×D4—1011no—
117C4×D4—2021no—
118#107—2021no—
119C4⋊C8—1011no—
120C4⋊C8—1011no—
121#103—2021no—
122#102—22no—
123#102—2021no—
124C4×C8C2×D10111yescharacteristic
125#41—4042no—
126#41—4042no—
127C2×Q8—21no—
128C2×Q8—1011no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—1011no—
132C2×D4—2022no—
133C2×D4—2022no—
134C2×D4C22⋊D10112yes—
135C22×C4—2022no—
136C22×C4—2021no—
137Q16—4042no—
138C2×C8—21no—
139C2×C8—2021no—
140C2×C8—2021no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1011no—
144C4⋊C4—1011no—
145C4⋊C4—2021no—
146C22⋊C4—2022no—
147C22⋊C4—2021no—
148C42—2021no—
149C42C22×D10111yescharacteristic
150C23—22no—
151C23—2021no—
152Q8—41no—
153Q8—41no—
154Q8—1012no—
155Q8—2021no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—8082no—
162D4—4044no—
163D4—2021no—
164D4—2021no—
165C2×C4—41no—
166C2×C4—4044no—
167C2×C4—4041no—
168C2×C4—4041no—
169C2×C4—2021no—
170C2×C4—2021no—
171C2×C4—2021no—
172C2×C4D4×D10111yescharacteristic
173C2×C4D4×D10111yescharacteristic
174C2×C4C2×C22⋊D10111yescharacteristic
175C8—41no—
176C8—4041no—
177C8—4041no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—4041no—
182C22—4041no—
183C22—4041no—
184C22#1569222311yescharacteristic
185C4—21no—
186C4—81no—
187C4—4041no—
188C4—4041no—
189C4—2022no—
190C4D8×D10111yescharacteristic
191C4#1569227611yescharacteristic
192C4#1569227911yescharacteristic
193C4#1569228411yescharacteristic
194C2—82no—
195C2—8081no—
196C2#2026530911yescharacteristic
197C2#2026533611yescharacteristic
198C2#2026535411yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#486×C101C2
socleC22×C101#120
prime radical#20265847C2
hypercenter#486D101
hypocenterC101#1641
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #20265454C2semidirect
3. #20265384C2semidirect
4. #20264921C2semidirect
5. #20264909C2semidirect
6. #20264906C2semidirect
7. #486×C101C2semidirect
100. C101#1641semidirect
102. #486D101semidirect

Maximal subgroups

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

subgroupindex
#202658472
#202654542
#202653842
#202649212
#202649092
#202649062
#486×C1012
#1641101