Finite Group Databasebeta

← #27170257

Subgroups — #27170257

Order 12928.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265849C211yesmaximal, characteristic
2#20265551C211yesmaximal, characteristic, complemented
3#20265464C211yesmaximal, characteristic, complemented
4#20264920C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7#477×C101C211yesmaximal, characteristic, complemented
8C2×#13108075—21no—
9C2×#13108061—22no—
10#15692252—21no—
11#15692252C2211yescharacteristic
12#15692230—22no—
13#15692230—21no—
14C4×D404—21no—
15C4×D404C2211yescharacteristic
16#15692186—21no—
17C4⋊#11717579C2211yescharacteristic
18#15692175—21no—
19C4×#11717579C2211yescharacteristic
20C4⋊Q8×C101C2211yescharacteristic
21C4×Q8×C101C2211yescharacteristic
22#107×C101—21no—
23C4⋊C8×C101C2211yescharacteristic
24#103×C101—21no—
25#13108075—42no—
26#13108075—42no—
27C2×D404—22no—
28C2×D404—21no—
29C2×D404—21no—
30C2×D404—22no—
31C2×D404D411yescharacteristic
32C2×D404D412yes—
33C2×C4×D101—22no—
34C2×C4×D101—21no—
35#13108061—44no—
36C2×#11717579—21no—
37C2×#11717579—21no—
38C4⋊Dic101D411yescharacteristic
39#13108051—22no—
40#13108051—21no—
41C2×Q8×C101—21no—
42C2×Q8×C101D411yescharacteristic
43C2×C8×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101—21no—
47C4⋊C4×C101D411yescharacteristic
48C42×C101—21no—
49C42×C101C2311yescharacteristic
50C22×D101—21no—
51C22×D101—22no—
52Q8×C101—21no—
53Q8×C101—42no—
54Q8×C101SD1612yescomplemented
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101C2×D411yescharacteristic
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×D411yescharacteristic
61C2×Dic101—21no—
62D404—21no—
63D404—21no—
64D404—42no—
65D404—44no—
66D404—41no—
67D404—41no—
68D404—41no—
69D404—41no—
70D404—82no—
71C4×D101—44no—
72C4×D101—41no—
73C4×D101—41no—
74C8×C101—41no—
75#11717579—22no—
76#11717579—41no—
77C22×C101#12011yescharacteristic
78C2×D101—42no—
79C2×D101—41no—
80C2×D101—41no—
81C2×D101—42no—
82C2×D101—42no—
83C2×D101—41no—
84C4×C101—22no—
85C4×C101—21no—
86C4×C101—41no—
87C4×C101—81no—
88C4×C101C2×SD1611yescharacteristic
89C4×C101#13511yescharacteristic
90C4×C101AGL(1,ℤ/8)11yescharacteristic
91C4×C101#13711yescharacteristic
92Dic101—41no—
93C2×C101#45011yescharacteristic
94C2×C101#45211yescharacteristic
95C2×C101#45311yescharacteristic
96D101—81no—
97D101—82no—
98#1631—1011nomaximal, Sylow
99C101#163111yescharacteristic, complemented, Sylow
100#550—1011no—
101#492—1011no—
102#477D10111yescharacteristic, complemented
103#476—1011no—
104#331—1011no—
105Q8⋊C8—1011no—
106D4⋊C8—1011no—
107C2×#41—2021no—
108C2×SD16—2022no—
109C4⋊Q8C2×D10111yescharacteristic
110#127—1011no—
111#127—2021no—
112#121—2022no—
113#121—2021no—
114C4×Q8C2×D10111yescharacteristic
115C4×D4—1011no—
116C4×D4—2021no—
117#107—21no—
118#106—2021no—
119C4⋊C8—1011no—
120C4⋊C8C2×D10111yescharacteristic
121#103—21no—
122#102—2021no—
123C4×C8—1011no—
124#41—4042no—
125#41—4042no—
126C2×Q8—21no—
127C2×Q8C22⋊D10111yescharacteristic
128C2×D4—1012no—
129C2×D4—1011no—
130C2×D4—2022no—
131C2×D4—2022no—
132C2×D4—2021no—
133C2×D4—2021no—
134C22×C4—2022no—
135C22×C4—2021no—
136SD16—4044no—
137C2×C8—21no—
138C2×C8—2021no—
139C2×C8—2021no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1011no—
144C4⋊C4C22⋊D10111yescharacteristic
145C22⋊C4—2022no—
146C22⋊C4—2021no—
147C42—21no—
148C42C22×D10111yescharacteristic
149C23—2022no—
150C23—2021no—
151Q8—21no—
152Q8—42no—
153Q8#1310806112yescomplemented
154D4—8082no—
155D4—4042no—
156D4—4044no—
157D4—4041no—
158D4—4041no—
159D4—4041no—
160D4—4041no—
161D4—2021no—
162D4—2021no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—4044no—
167C2×C4—4041no—
168C2×C4—4041no—
169C2×C4—2021no—
170C2×C4D4×D10111yescharacteristic
171C2×C4D4×D10111yescharacteristic
172C2×C4C2×C22⋊D10111yescharacteristic
173C8—41no—
174C8—4041no—
175C8—2022no—
176C22—4042no—
177C22—4041no—
178C22—4042no—
179C22—4041no—
180C22—4041no—
181C22—4042no—
182C22#1569222311yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—81no—
187C4—4041no—
188C4C2×#1310806111yescharacteristic
189C4#1569227811yescharacteristic
190C4#1569228011yescharacteristic
191C4#1569228711yescharacteristic
192C2—8082no—
193C2—8081no—
194C2#2026532711yescharacteristic
195C2#2026533811yescharacteristic
196C2#2026534311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#477×C101C2
socleC22×C101#120
prime radical#20265849C2
hypercenter#477D101
hypocenterC101#1631
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #20265551C2semidirect
3. #20265464C2semidirect
4. #20264920C2semidirect
5. Q8⋊#11717579C2semidirect
6. #20264904C2semidirect
7. #477×C101C2semidirect
54. Q8×C101SD16semidirect
99. C101#1631semidirect
102. #477D101semidirect
153. Q8#13108061semidirect

Maximal subgroups

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

subgroupindex
#202658492
#202655512
#202654642
#202649202
Q8⋊#117175792
#202649042
#477×C1012
#1631101