Finite Group Databasebeta

← #27170256

Subgroups — #27170256

Order 12928.

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

normal subgroups only

This group has 256 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#20265469C211yesmaximal, characteristic, complemented
3#20265464C211yesmaximal, characteristic, complemented
4#20264921C211yesmaximal, characteristic, complemented
5#20264909C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7#492×C101C211yesmaximal, characteristic, complemented
8C2×#13108073—21no—
9#15692251—21no—
10#15692227—22no—
11#15692226—21no—
12C4×Dic202C2211yescharacteristic
13C4×D404C2211yescharacteristic
14D4×Dic101—21no—
15#15692188—21no—
16#15692186—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692178—21no—
20#15692175—21no—
21C2×SD16×C101—22no—
22C4⋊Q8×C101C2211yescharacteristic
23#127×C101C2211yescharacteristic
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—
33C2×#11717579—21no—
34C2×#11717579—21no—
35C4⋊Dic101D411yescharacteristic
36C4⋊Dic101D411yescharacteristic
37#13108053—21no—
38C22⋊Dic101—22no—
39#13108051—21no—
40C4×Dic101—21no—
41C2×Q8×C101—21no—
42C2×D4×C101—21no—
43C2×D4×C101—21no—
44C2×D4×C101D412yes—
45SD16×C101—44no—
46C2×C8×C101—21no—
47C4⋊C4×C101—21no—
48C4⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—22no—
51C22×D101—21no—
52Q8×C101—42no—
53D4×C101—42no—
54D4×C101—41no—
55D4×C101—41no—
56D4×C101—41no—
57D4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C22⋊D101—44no—
63C22⋊D101—82no—
64C2×Dic101—21no—
65C2×Dic101—21no—
66C2×Dic101—21no—
67C2×Dic101—44no—
68D404—21no—
69D404—21no—
70C4×D101—41no—
71C4×D101—41no—
72Dic202—21no—
73Dic202SD1612yescomplemented
74C8×C101—22no—
75#11717579—41no—
76#11717579—41no—
77C22×C101—42no—
78C22×C101—42no—
79C22×C101—42no—
80C22×C101#12011yescharacteristic
81C2×D101—41no—
82C2×D101—41no—
83C2×D101—41no—
84C4×C101—21no—
85C4×C101—81no—
86C4×C101C2×SD1611yescharacteristic
87C4×C101#13511yescharacteristic
88C4×C101AGL(1,ℤ/8)11yescharacteristic
89C4×C101#13711yescharacteristic
90Dic101—22no—
91Dic101—41no—
92Dic101—41no—
93C2×C101—82no—
94C2×C101#45011yescharacteristic
95C2×C101#45211yescharacteristic
96C2×C101#45311yescharacteristic
97D101—81no—
98#1631—1011nomaximal, Sylow
99C101#163111yescharacteristic, complemented, Sylow
100#550—1011no—
101#492D10111yescharacteristic, complemented
102#477—1011no—
103#476—1011no—
104#331—1011no—
105Q8⋊C8—1011no—
106D4⋊C8—1011no—
107C2×#41—2021no—
108C2×SD16—22no—
109C4⋊Q8C2×D10111yescharacteristic
110#127—2021no—
111#127C2×D10111yescharacteristic
112#121—2021no—
113#121—2022no—
114C4×Q8—1011no—
115C4×D4—1011no—
116C4×D4—2021no—
117#107—2021no—
118#106—2021no—
119C4⋊C8—1011no—
120C4⋊C8—1011no—
121#103—2021no—
122#102—2021no—
123C4×C8C2×D10111yescharacteristic
124#41—4042no—
125#41—4042no—
126C2×Q8—21no—
127C2×Q8—1011no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—1011no—
131C2×D4—2022no—
132C2×D4—2022no—
133C2×D4C22⋊D10112yes—
134C22×C4—2022no—
135C22×C4—2021no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—2021no—
139C2×C8—2021no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—1011no—
143C4⋊C4—1011no—
144C4⋊C4—2021no—
145C22⋊C4—2022no—
146C22⋊C4—2021no—
147C42—2021no—
148C42C22×D10111yescharacteristic
149C23—22no—
150C23—2021no—
151Q8—42no—
152Q8—1012no—
153Q8—2021no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—8082no—
160D4—4044no—
161D4—2021no—
162D4—2021no—
163C2×C4—41no—
164C2×C4—4044no—
165C2×C4—4041no—
166C2×C4—4041no—
167C2×C4—2021no—
168C2×C4—2021no—
169C2×C4—2021no—
170C2×C4D4×D10111yescharacteristic
171C2×C4D4×D10111yescharacteristic
172C2×C4C2×C22⋊D10111yescharacteristic
173C8—22no—
174C8—4041no—
175C8—4041no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—4041no—
180C22—4041no—
181C22—4041no—
182C22#1569222311yescharacteristic
183C4—21no—
184C4—81no—
185C4—4041no—
186C4—4041no—
187C4—2022no—
188C4SD16×D10111yescharacteristic
189C4#1569227611yescharacteristic
190C4#1569228211yescharacteristic
191C4#1569228411yescharacteristic
192C2—82no—
193C2—8081no—
194C2#2026532611yescharacteristic
195C2#2026533611yescharacteristic
196C2#2026534411yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#492×C101C2
socleC22×C101#120
prime radical#20265847C2
hypercenter#492D101
hypocenterC101#1631
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #20265469C2semidirect
3. #20265464C2semidirect
4. #20264921C2semidirect
5. #20264909C2semidirect
6. #20264906C2semidirect
7. #492×C101C2semidirect
73. Dic202SD16semidirect
99. C101#1631semidirect
101. #492D101semidirect

Maximal subgroups

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

subgroupindex
#202658472
#202654692
#202654642
#202649212
#202649092
#202649062
#492×C1012
#1631101