Finite Group Databasebeta

← #27170223

Subgroups — #27170223

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#20265831C211yesmaximal, characteristic, complemented
2#20265524C211yesmaximal, characteristic, complemented
3#20265373C211yesmaximal, characteristic, complemented
4#20265372C211yesmaximal, characteristic, complemented
5#20264922C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×#13108073—21no—
9C2×D808—21no—
10C2×#13108059—21no—
11#15692255C2211yescharacteristic
12#15692252C2211yescharacteristic
13#15692239—22no—
14#15692235—21no—
15#15692229—21no—
16C4×D404C2211yescharacteristic
17C2×C4⋊Dic101—21no—
18C4⋊#11717579C2211yescharacteristic
19#15692177—22no—
20#15692177—21no—
21#15692175—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25#13108073—42no—
26#13108073—42no—
27C2×Dic202—21no—
28C2×D404—21no—
29C2×D404—21no—
30C2×D404—21no—
31C2×D404D411yescharacteristic
32C2×C22⋊D101—22no—
33C22×Dic101—22no—
34C2×C4×D101—21no—
35D808—42no—
36#13108059—42no—
37C2×#11717579—21no—
38C4⋊Dic101—22no—
39C4⋊Dic101—21no—
40C4⋊Dic101D411yescharacteristic
41C4⋊Dic101D412yes—
42#13108051—22no—
43#13108051—21no—
44C2×D4×C101D411yescharacteristic
45C22×C4×C101—21no—
46C2×C8×C101—21no—
47C2×C8×C101—21no—
48C4⋊C4×C101D411yescharacteristic
49C22⋊C4×C101—21no—
50C42×C101C2311yescharacteristic
51C23×C101—21no—
52C22×D101—21no—
53C22×D101—41no—
54D4×C101—21no—
55D4×C101D812yescomplemented
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×D411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C22⋊D101—44no—
63C2×Dic101—21no—
64C2×Dic101—22no—
65C2×Dic101—44no—
66D404—21no—
67D404—21no—
68D404—42no—
69D404—41no—
70D404—41no—
71D404—41no—
72D404—41no—
73C4×D101—41no—
74C4×D101—41no—
75Dic202—41no—
76Dic202—41no—
77C8×C101—41no—
78C8×C101—41no—
79#11717579—41no—
80C22×C101—22no—
81C22×C101—41no—
82C22×C101—41no—
83C22×C101#12011yescharacteristic
84C2×D101—41no—
85C2×D101—41no—
86C2×D101—41no—
87C2×D101—81no—
88C2×D101—81no—
89C2×D101—81no—
90C4×C101—21no—
91C4×C101—41no—
92C4×C101C2×D811yescharacteristic
93C4×C101#13511yescharacteristic
94C4×C101AGL(1,ℤ/8)11yescharacteristic
95C4×C101AGL(1,ℤ/8)11yescharacteristic
96Dic101—42no—
97Dic101—41no—
98C2×C101—42no—
99C2×C101#44711yescharacteristic
100C2×C101#44911yescharacteristic
101C2×C101#45411yescharacteristic
102D101—81no—
103D101—161no—
104#1625—1011nomaximal, Sylow
105C101#162511yescharacteristic, complemented, Sylow
106#547—1011no—
107#486—1011no—
108#459—1011no—
109#459—1011no—
110#332—1011no—
111D4⋊C8—1011no—
112D4⋊C8D10111yescharacteristic, complemented
113C2×#41—2021no—
114C2×D8—2021no—
115C2×D8—2021no—
116C4⋊Q8—1011no—
117#127—1011no—
118#123—2022no—
119C22⋊Q8—2021no—
120#121—2021no—
121C4×D4—1011no—
122C4×D4C2×D10111yescharacteristic
123C2×C4⋊C4—2021no—
124C4⋊C8—1011no—
125C4⋊C8C2×D10111yescharacteristic
126#102—2021no—
127#102—2022no—
128#102—2021no—
129C4×C8C2×D10111yescharacteristic
130#41—4042no—
131#41—4042no—
132C2×Q8—2021no—
133C2×D4—1011no—
134C2×D4—2022no—
135C2×D4—2021no—
136C2×D4—2021no—
137C2×D4—2021no—
138C2×D4D40411yescharacteristic
139C22×C4—21no—
140C22×C4—2022no—
141C22×C4—2021no—
142D8—4042no—
143D8—4042no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—2021no—
147C4⋊C4—1011no—
148C4⋊C4—1012no—
149C4⋊C4—2022no—
150C4⋊C4—2021no—
151C4⋊C4D40411yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—2022no—
154C22⋊C4—2021no—
155C42C22×D10111yescharacteristic
156C23—21no—
157C23—4041no—
158C23—2021no—
159Q8—4041no—
160Q8—4041no—
161D4—21no—
162D4—4042no—
163D4—4044no—
164D4—4041no—
165D4—4041no—
166D4—4041no—
167D4—4041no—
168D4—2021no—
169D4—2021no—
170D4D80812yescomplemented
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—4044no—
175C2×C4—4041no—
176C2×C4—4041no—
177C2×C4—2021no—
178C2×C4—2022no—
179C2×C4D4×D10111yescharacteristic
180C2×C4D4×D10111yescharacteristic
181C2×C4C2×D40411yescharacteristic
182C8—41no—
183C8—41no—
184C8—4041no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—8081no—
189C22—8081no—
190C22—8081no—
191C22—4041no—
192C22—4041no—
193C22—4041no—
194C22#1569222511yescharacteristic
195C4—21no—
196C4—41no—
197C4—4041no—
198C4—4042no—
199C4C2×D80811yescharacteristic
200C4#1569227611yescharacteristic
201C4#1569227911yescharacteristic
202C4#1569228311yescharacteristic
203C2—42no—
204C2—8081no—
205C2—16161no—
206C2#2026531311yescharacteristic
207C2#2026532311yescharacteristic
208C2#2026535111yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
hypercenterD4⋊C8D101
hypocenterC101#1625
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. #20265831C2semidirect
2. #20265524C2semidirect
3. #20265373C2semidirect
4. #20265372C2semidirect
5. #20264922C2semidirect
6. #20264906C2semidirect
7. D4⋊C8×C101C2semidirect
55. D4×C101D8semidirect
105. C101#1625semidirect
112. D4⋊C8D101semidirect
170. D4D808semidirect

Maximal subgroups

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

subgroupindex
#202658312
#202655242
#202653732
#202653722
#202649222
#202649062
D4⋊C8×C1012
#1625101