Finite Group Databasebeta

← #27170309

Subgroups — #27170309

Order 12928.

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

normal subgroups only

This group has 308 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#20265525C211yesmaximal, characteristic, complemented
3#20265454C211yesmaximal, characteristic, complemented
4#20264920C211yesmaximal, characteristic, complemented
5D4⋊#11717579C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7#461×C101C211yesmaximal, characteristic, complemented
8C22×D404—21no—
9C2×D4×D101—21no—
10#15692252C2211yescharacteristic
11#15692230—21no—
12#15692229—21no—
13#15692225—22no—
14C4×D404C2211yescharacteristic
15#15692188—21no—
16C4⋊#11717579C2211yescharacteristic, complemented
17#15692175—22no—
18#15692175—21no—
19C4×#11717579C2211yescharacteristic, complemented
20C2×SD16×C101—21no—
21C4⋊Q8×C101C2211yescharacteristic, complemented
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic, complemented
24#103×C101—21no—
25C23×D101—22no—
26C2×D404—22no—
27C2×D404—21no—
28C2×D404—21no—
29C2×D404—42no—
30C2×D404—42no—
31C2×D404D411yescharacteristic
32C2×D404D412yes—
33C2×C22⋊D101—22no—
34D4×D101—42no—
35D4×D101—42no—
36C2×C4×D101—21no—
37C2×#11717579—21no—
38C2×#11717579—21no—
39C4⋊Dic101D411yescharacteristic
40#13108051—22no—
41#13108051—21no—
42C2×Q8×C101—21no—
43C2×D4×C101D411yescharacteristic
44C22×C4×C101—21no—
45SD16×C101—42no—
46C2×C8×C101—21no—
47C4⋊C4×C101—21no—
48C4⋊C4×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—
58Q8×C101—41no—
59Q8×C101—41no—
60D4×C101—21no—
61D4×C101D812yes—
62C2×C4×C101—21no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101C2×D411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C22⋊D101—44no—
70C2×Dic101—21no—
71D404—21no—
72D404—42no—
73D404—44no—
74D404—41no—
75D404—41no—
76D404—41no—
77D404—41no—
78D404SD1612yescomplemented
79C4×D101—41no—
80C4×D101—41no—
81C8×C101—41no—
82#11717579—41no—
83#11717579—41no—
84C22×C101—22no—
85C22×C101—41no—
86C22×C101—41no—
87C22×C101#12011yescharacteristic
88C2×D101—22no—
89C2×D101—41no—
90C2×D101—41no—
91C2×D101—42no—
92C2×D101—42no—
93C2×D101—42no—
94C2×D101—88no—
95C4×C101—21no—
96C4×C101—41no—
97C4×C101—81no—
98C4×C101C2×D811yescharacteristic
99C4×C101C2×SD1611yescharacteristic
100C4×C101AGL(1,ℤ/8)11yescharacteristic
101C4×C101AGL(1,ℤ/8)11yescharacteristic
102Dic101—41no—
103C2×C101—42no—
104C2×C101#44711yescharacteristic
105C2×C101#45011yescharacteristic
106C2×C101#45511yescharacteristic
107D101—42no—
108D101—82no—
109#1639—1011nomaximal, Sylow
110C101#163911yescharacteristic, complemented, Sylow
111D42—1011no—
112#486—1011no—
113#474—1011no—
114#461D10111yescharacteristic, complemented
115#331—1011no—
116D4⋊C8—1011no—
117D4⋊C8—1011no—
118C22×D4—2021no—
119C22×D4—2021no—
120C2×SD16—21no—
121C4⋊Q8C2×D10111yescharacteristic, complemented
122#127—1011no—
123#121—2021no—
124#121—2021no—
125#120—2022no—
126C4×D4—1011no—
127C4×D4C2×D10111yescharacteristic
128#107—2021no—
129C4⋊C8—1011no—
130C4⋊C8C2×D10111yescharacteristic, complemented
131#103—21no—
132#102—2021no—
133#102—2022no—
134C4×C8—1011no—
135C24—2022no—
136C2×Q8—21no—
137C2×D4—1011no—
138C2×D4—1012no—
139C2×D4—4042no—
140C2×D4—4042no—
141C2×D4—4042no—
142C2×D4—4042no—
143C2×D4—2022no—
144C2×D4—2022no—
145C2×D4—2021no—
146C2×D4—2021no—
147C2×D4C22⋊D10111yescharacteristic
148C22×C4—21no—
149C22×C4—2021no—
150SD16—42no—
151C2×C8—21no—
152C2×C8—2021no—
153C2×C8—2021no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1011no—
157C4⋊C4C22⋊D10111yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—2022no—
160C22⋊C4—2021no—
161C42C22×D10111yescharacteristic
162C23—21no—
163C23—8082no—
164C23—4044no—
165C23—4044no—
166C23—2022no—
167C23—2021no—
168Q8—41no—
169Q8—41no—
170D4—21no—
171D4—1012no—
172D4—4042no—
173D4—4044no—
174D4—4044no—
175D4—4041no—
176D4—4041no—
177D4—4041no—
178D4—4041no—
179D4—2021no—
180D4#1310805912yes—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—4041no—
186C2×C4—4041no—
187C2×C4—2021no—
188C2×C4D4×D10111yescharacteristic
189C2×C4D4×D10111yescharacteristic
190C2×C4C2×C22⋊D10111yescharacteristic
191C8—41no—
192C8—4041no—
193C8—4041no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—8088no—
198C22—4042no—
199C22—4042no—
200C22—4042no—
201C22—4041no—
202C22—4041no—
203C22—2022no—
204C22#1569222311yescharacteristic
205C4—21no—
206C4—41no—
207C4—81no—
208C4—4041no—
209C4C2×#1310805911yescharacteristic
210C4SD16×D10111yescharacteristic
211C4#1569228011yescharacteristic
212C4#1569228211yescharacteristic
213C2—42no—
214C2—8082no—
215C2—4042no—
216C2#2026531011yescharacteristic
217C2#2026532611yescharacteristic
218C2#2026535311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#461×C101C2
socleC22×C101#120
prime radicalD4×D404C2
hypercenter#461D101
hypocenterC101#1639
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #20265525C2semidirect
3. #20265454C2semidirect
4. #20264920C2semidirect
5. D4⋊#11717579C2semidirect
6. #20264904C2semidirect
7. #461×C101C2semidirect
16. C4⋊#11717579C22semidirect
19. C4×#11717579C22semidirect
21. C4⋊Q8×C101C22semidirect
23. C4⋊C8×C101C22semidirect
78. D404SD16semidirect
110. C101#1639semidirect
114. #461D101semidirect
121. C4⋊Q8C2×D101semidirect
130. C4⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
D4×D4042
#202655252
#202654542
#202649202
D4⋊#117175792
#202649042
#461×C1012
#1639101