Finite Group Databasebeta

← #27170217

Subgroups — #27170217

Order 12928.

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

normal subgroups only

This group has 240 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#20265892C211yesmaximal, characteristic, complemented
2#20265550C211yesmaximal, characteristic, complemented
3#20265373C211yesmaximal, characteristic, complemented
4#20265372C211yesmaximal, characteristic, complemented
5#20264925C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×#13108063—22no—
9C2×D808—21no—
10C2×#13108059—21no—
11#15692255C2211yescharacteristic
12#15692252C2211yescharacteristic
13#15692241—22no—
14#15692229—21no—
15#15692226—21no—
16C4×D404C2211yescharacteristic
17#15692214—21no—
18#15692213—21no—
19C4⋊#11717579C2211yescharacteristic
20#15692177—21no—
21#15692175—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202D412yes—
26C2×D404—21no—
27C2×D404—21no—
28C2×D404—21no—
29C2×D404D411yescharacteristic
30C2×C22⋊D101—22no—
31C2×C4×D101—21no—
32#13108063—44no—
33D808—42no—
34#13108059—42no—
35C2×#11717579—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101—21no—
38C4⋊Dic101D411yescharacteristic
39C22⋊Dic101—22no—
40#13108051—22no—
41#13108051—21no—
42C4×Dic101—22no—
43C2×D4×C101D411yescharacteristic
44C22×C4×C101—21no—
45C2×C8×C101—21no—
46C2×C8×C101—21no—
47C4⋊C4×C101D411yescharacteristic
48C22⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—21no—
51C22×D101—21no—
52C22×D101—41no—
53D4×C101—21no—
54D4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—41no—
57C2×C4×C101—41no—
58C2×C4×C101C2×D411yescharacteristic
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×D411yescharacteristic
61C22⋊D101—82no—
62C2×Dic101—21no—
63C2×Dic101—22no—
64D404—21no—
65D404—21no—
66D404—42no—
67D404—41no—
68D404—41no—
69D404—41no—
70D404—41no—
71C4×D101—41no—
72C4×D101—41no—
73Dic202—42no—
74C8×C101—22no—
75C8×C101—41no—
76#11717579—41no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101—41no—
80C22×C101#12011yescharacteristic
81C2×D101—41no—
82C2×D101—41no—
83C2×D101—41no—
84C2×D101—81no—
85C2×D101—81no—
86C2×D101—81no—
87C4×C101—21no—
88C4×C101—41no—
89C4×C101#13511yescharacteristic
90C4×C101#13511yescharacteristic
91C4×C101AGL(1,ℤ/8)11yescharacteristic
92C4×C101AGL(1,ℤ/8)11yescharacteristic
93Dic101—41no—
94Dic101—42no—
95C2×C101—81no—
96C2×C101#44911yescharacteristic
97C2×C101#44911yescharacteristic
98C2×C101#45311yescharacteristic
99D101—81no—
100D101—161no—
101#1624—1011nomaximal, Sylow
102C101#162411yescharacteristic, complemented, Sylow
103#555—1011no—
104#492—1011no—
105#459—1011no—
106#459—1011no—
107#333—1011no—
108D4⋊C8—1011no—
109D4⋊C8D10111yescharacteristic, complemented
110C2×SD16—2022no—
111C2×D8—2021no—
112C2×D8—2021no—
113C4⋊Q8—1011no—
114#127—1011no—
115#124—2022no—
116#121—2021no—
117#121—2021no—
118C4×D4—1011no—
119C4×D4C2×D10111yescharacteristic
120#117—2021no—
121#117—2021no—
122C4⋊C8—1011no—
123C4⋊C8C2×D10111yescharacteristic
124#102—2021no—
125#102—2021no—
126C4×C8C2×D10111yescharacteristic
127C2×Q8—1012no—
128C2×D4—1011no—
129C2×D4—2022no—
130C2×D4—2021no—
131C2×D4—2021no—
132C2×D4—2021no—
133C2×D4D40411yescharacteristic
134C22×C4—21no—
135C22×C4—2021no—
136SD16—4044no—
137D8—4042no—
138D8—4042no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—2021no—
142C4⋊C4—1011no—
143C4⋊C4—2021no—
144C4⋊C4—2021no—
145C4⋊C4D40411yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—2022no—
148C22⋊C4—2022no—
149C22⋊C4—2021no—
150C42—2022no—
151C42C22×D10111yescharacteristic
152C23—21no—
153C23—4041no—
154C23—2021no—
155Q8—4042no—
156D4—21no—
157D4—21no—
158D4—8082no—
159D4—4042no—
160D4—4041no—
161D4—4041no—
162D4—4041no—
163D4—4041no—
164D4—2021no—
165D4—2021no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—4041no—
170C2×C4—4041no—
171C2×C4—2021no—
172C2×C4—2022no—
173C2×C4D4×D10111yescharacteristic
174C2×C4D4×D10111yescharacteristic
175C2×C4C2×D40411yescharacteristic
176C8—22no—
177C8—41no—
178C8—4041no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—8081no—
183C22—8081no—
184C22—8081no—
185C22—4041no—
186C22—4041no—
187C22—4041no—
188C22#1569222511yescharacteristic
189C4—21no—
190C4—41no—
191C4—4041no—
192C4—4042no—
193C4#1569227611yescharacteristic
194C4#1569227711yescharacteristic
195C4#1569227911yescharacteristic
196C4#1569228311yescharacteristic
197C2—81no—
198C2—8081no—
199C2—16161no—
200C2#2026532311yescharacteristic
201C2#2026532411yescharacteristic
202C2#2026534611yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
hypercenterD4⋊C8D101
hypocenterC101#1624
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. #20265892C2semidirect
2. #20265550C2semidirect
3. #20265373C2semidirect
4. #20265372C2semidirect
5. #20264925C2semidirect
6. #20264906C2semidirect
7. D4⋊C8×C101C2semidirect
102. C101#1624semidirect
109. D4⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202658922
#202655502
#202653732
#202653722
#202649252
#202649062
D4⋊C8×C1012
#1624101