Finite Group Databasebeta

← #27170250

Subgroups — #27170250

Order 12928.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D404C211yesmaximal, characteristic
2#20265550C211yesmaximal, characteristic, complemented
3#20265468C211yesmaximal, characteristic, complemented
4#20265467C211yesmaximal, characteristic, complemented
5#20264920C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic, complemented
8C22×D404—21no—
9C2×D4×D101—21no—
10C2×#13108063—22no—
11#15692255C2211yescharacteristic, complemented
12#15692252C2211yescharacteristic
13#15692230—21no—
14#15692229—21no—
15#15692225—22no—
16C4×D404C2211yescharacteristic
17#15692187—21no—
18#15692186—21no—
19C4⋊#11717579C2211yescharacteristic, complemented
20#15692177—21no—
21#15692176—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic, complemented
24C4×C8×C101C2211yescharacteristic, complemented
25C23×D101—22no—
26C2×Dic202—21no—
27C2×D404—22no—
28C2×D404—21no—
29C2×D404—21no—
30C2×D404—42no—
31C2×D404—42no—
32C2×D404D411yescharacteristic
33C2×D404D412yes—
34C2×C22⋊D101—22no—
35D4×D101—42no—
36D4×D101—42no—
37C2×C4×D101—21no—
38#13108063—44no—
39C2×#11717579—21no—
40C4⋊Dic101—21no—
41C4⋊Dic101—21no—
42C4⋊Dic101D411yescharacteristic
43#13108051—22no—
44#13108051—21no—
45C2×D4×C101D411yescharacteristic
46C22×C4×C101—21no—
47C2×C8×C101—21no—
48C2×C8×C101—21no—
49C4⋊C4×C101D411yescharacteristic
50C22⋊C4×C101—21no—
51C42×C101C2311yescharacteristic
52C23×C101—21no—
53C22×D101—21no—
54C22×D101—22no—
55C22×D101—44no—
56C22×D101—44no—
57C22×D101—82no—
58D4×C101—21no—
59D4×C101SD1612yescomplemented
60C2×C4×C101—21no—
61C2×C4×C101—41no—
62C2×C4×C101—41no—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C22⋊D101—44no—
67C2×Dic101—21no—
68C2×Dic101—41no—
69D404—21no—
70D404—42no—
71D404—44no—
72D404—41no—
73D404—41no—
74D404—41no—
75D404—41no—
76D404SD1612yescomplemented
77C4×D101—41no—
78C4×D101—41no—
79Dic202—42no—
80C8×C101—22no—
81C8×C101—41no—
82#11717579—41no—
83C22×C101—22no—
84C22×C101—41no—
85C22×C101—41no—
86C22×C101#12011yescharacteristic
87C2×D101—22no—
88C2×D101—41no—
89C2×D101—41no—
90C2×D101—42no—
91C2×D101—42no—
92C2×D101—42no—
93C2×D101—88no—
94C4×C101—21no—
95C4×C101—41no—
96C4×C101C2×SD1611yescharacteristic
97C4×C101C2×SD1611yescharacteristic
98C4×C101AGL(1,ℤ/8)11yescharacteristic
99C4×C101AGL(1,ℤ/8)11yescharacteristic
100Dic101—41no—
101Dic101—81no—
102C2×C101—42no—
103C2×C101#45011yescharacteristic
104C2×C101#45011yescharacteristic
105C2×C101#45311yescharacteristic
106D101—42no—
107D101—82no—
108#1629—1011nomaximal, Sylow
109C101#162911yescharacteristic, complemented, Sylow
110D42—1011no—
111#492—1011no—
112#476—1011no—
113#476—1011no—
114#331—1011no—
115D4⋊C8—1011no—
116D4⋊C8D10111yescharacteristic, complemented
117C22×D4—2021no—
118C22×D4—2021no—
119C2×SD16—2022no—
120C4⋊Q8—1011no—
121#127—1011no—
122#121—2021no—
123#121—2021no—
124#120—2022no—
125C4×D4—1011no—
126C4×D4C2×D10111yescharacteristic
127#106—2021no—
128#106—2021no—
129C4⋊C8—1011no—
130C4⋊C8C2×D10111yescharacteristic, complemented
131#102—2021no—
132#102—2021no—
133C4×C8C2×D10111yescharacteristic, complemented
134C24—2022no—
135C2×Q8—2021no—
136C2×D4—1011no—
137C2×D4—1012no—
138C2×D4—4042no—
139C2×D4—4042no—
140C2×D4—4042no—
141C2×D4—4042no—
142C2×D4—2022no—
143C2×D4—2022no—
144C2×D4—2021no—
145C2×D4—2021no—
146C2×D4D40411yescharacteristic
147C22×C4—21no—
148C22×C4—2021no—
149SD16—4044no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—2021no—
153C4⋊C4—1011no—
154C4⋊C4—2021no—
155C4⋊C4—2021no—
156C4⋊C4D40411yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—2022no—
159C22⋊C4—2021no—
160C42C22×D10111yescharacteristic
161C23—21no—
162C23—8082no—
163C23—4044no—
164C23—4044no—
165C23—2022no—
166C23—2021no—
167Q8—4042no—
168D4—21no—
169D4—1012no—
170D4—4042no—
171D4—4044no—
172D4—4044no—
173D4—4041no—
174D4—4041no—
175D4—4041no—
176D4—4041no—
177D4—2021no—
178D4#1310806312yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—4041no—
183C2×C4—4041no—
184C2×C4—4041no—
185C2×C4—2021no—
186C2×C4D4×D10111yescharacteristic
187C2×C4D4×D10111yescharacteristic
188C2×C4C2×D40411yescharacteristic
189C8—22no—
190C8—41no—
191C8—4041no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—8088no—
196C22—4042no—
197C22—4042no—
198C22—4042no—
199C22—4041no—
200C22—4041no—
201C22—2022no—
202C22#1569222511yescharacteristic
203C4—21no—
204C4—41no—
205C4—8081no—
206C4—4041no—
207C4SD16×D10111yescharacteristic
208C4C2×#1310806311yescharacteristic
209C4#1569227911yescharacteristic
210C4#1569228311yescharacteristic
211C2—42no—
212C2—8082no—
213C2—4042no—
214C2#2026532911yescharacteristic
215C2#2026533011yescharacteristic
216C2#2026534611yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
prime radicalD4×D404C2
hypercenterD4⋊C8D101
hypocenterC101#1629
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
2. #20265550C2semidirect
3. #20265468C2semidirect
4. #20265467C2semidirect
5. #20264920C2semidirect
6. #20264906C2semidirect
7. D4⋊C8×C101C2semidirect
11. #15692255C22semidirect
19. C4⋊#11717579C22semidirect
23. C4⋊C8×C101C22semidirect
24. C4×C8×C101C22semidirect
59. D4×C101SD16semidirect
76. D404SD16semidirect
109. C101#1629semidirect
116. D4⋊C8D101semidirect
130. C4⋊C8C2×D101semidirect
133. C4×C8C2×D101semidirect
178. D4#13108063semidirect

Maximal subgroups

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

subgroupindex
D4×D4042
#202655502
#202654682
#202654672
#202649202
#202649062
D4⋊C8×C1012
#1629101