Finite Group Databasebeta

← #27181596

Subgroups — #27181596

Order 12928.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#15692286C212yesmaximal, complemented
2C2×#15692281C212yesmaximal, complemented
3#20265727C211yesmaximal, characteristic, complemented
4#20265572C212yesmaximal, complemented
5#20265567C212yesmaximal, complemented
6#20265559C214yesmaximal, complemented
7C4×#13108058C211yesmaximal, characteristic, complemented
8C2×#124×C101C211yesmaximal, characteristic, complemented
9C2×#13108074C2212yes—
10#15692286—28no—
11#15692281—28no—
12C2×#13108064C2214yescomplemented
13C2×#13108062C2214yes—
14C2×#13108061C2214yescomplemented
15C2×#13108059C2214yes—
16C2×#13108058C2212yes—
17#15692255C2211yescharacteristic, complemented
18#15692252C2211yescharacteristic, complemented
19#15692243C2212yescomplemented
20#15692229—22no—
21C4×D404—22no—
22#15692166C2212yescomplemented
23C4×#11717579C2212yescomplemented
24C22×Q8×C101C2211yescharacteristic
25C22×D4×C101C2211yescharacteristic
26#124×C101—22no—
27#124×C101C2212yes—
28#124×C101C2212yes—
29C2×C22⋊C4×C101—22no—
30C2×C42×C101C2211yescharacteristic
31#13108074—24no—
32#13108074—24no—
33C2×Dic202C2312yes—
34C2×D404—22no—
35C2×D404C2312yes—
36C2×C22⋊D101—22no—
37C2×C4×D101—22no—
38#13108064—28no—
39#13108062—28no—
40#13108061—28no—
41#13108059—28no—
42#13108058D418yescomplemented
43C2×#11717579C2314yes—
44C4⋊Dic101—22no—
45#13108051—24no—
46C24×C101—21no—
47C2×Q8×C101—21no—
48C2×Q8×C101—24no—
49C2×Q8×C101C2312yes—
50C2×D4×C101—24no—
51C2×D4×C101—21no—
52C2×D4×C101C2312yes—
53C22×C4×C101—21no—
54C22×C4×C101D412yes—
55C22×C4×C101C2311yescharacteristic
56C22⋊C4×C101—24no—
57C22⋊C4×C101—24no—
58C42×C101D412yes—
59C42×C101C2311yescharacteristic
60C42×C101C2311yescharacteristic
61C23×C101—22no—
62C23×C101—44no—
63C23×C101—42no—
64C23×C101C2×D411yescharacteristic
65C22×D101—22no—
66Q8×C101—24no—
67Q8×C101—42no—
68D4×C101—24no—
69D4×C101—42no—
70C2×C4×C101—22no—
71C2×C4×C101—42no—
72C2×C4×C101—42no—
73C2×C4×C101C2×D414yes—
74C2×C4×C101C2×D412yes—
75C2×C4×C101C2×D412yes—
76C2×C4×C101C2×D411yescharacteristic
77C2×C4×C101C2411yescharacteristic
78C22⋊D101—44no—
79C2×Dic101—22no—
80D404—22no—
81D404—22no—
82D404—44no—
83C4×D101—44no—
84Dic202—22no—
85Dic202—22no—
86#11717579C2×D418yes—
87C22×C101—22no—
88C22×C101—42no—
89C22×C101—44no—
90C22×C101—84no—
91C22×C101#12712yes—
92C22×C101C22×D411yescharacteristic
93C2×D101—42no—
94C2×D101—44no—
95C4×C101—22no—
96C4×C101—22no—
97C4×C101—42no—
98C4×C101#12712yes—
99C4×C101C22×D412yes—
100Dic101—42no—
101C2×C101—22no—
102C2×C101—82no—
103C2×C101C2×#12711yescharacteristic
104C2×C101#57512yes—
105D101—82no—
106#3153—1011nomaximal, Sylow
107C101#315311yescharacteristic, complemented, Sylow
108C2×#137—1012no—
109C2×AGL(1,ℤ/8)—1012no—
110#532—1011no—
111C2×#124D10111yescharacteristic, complemented
112#497—1012no—
113#496—1012no—
114#495—1014no—
115C4×M16—1011no—
116C2×#41—1012no—
117C22×Q8C2×D10111yescharacteristic
118C22×D4C2×D10111yescharacteristic
119#137—2028no—
120AGL(1,ℤ/8)—2028no—
121C2×Q16—1014no—
122C2×SD16—1014no—
123C2×SD16—1014no—
124C2×D8—1014no—
125C2×M16—1012no—
126C4⋊Q8—1011no—
127#127—1011no—
128#124—22no—
129#124—1012no—
130#124C2×D10112yes—
131#124C2×D10112yes—
132#121—2022no—
133C4×D4—2022no—
134C2×C22⋊C4—22no—
135C2×C42C2×D10111yescharacteristic
136#97—1012no—
137C4×C8—1012no—
138C24—21no—
139#41—2024no—
140#41—2024no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—1012no—
144C2×Q8C22×D10112yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—1012no—
148C2×D4—2022no—
149C2×D4—2022no—
150C2×D4C22×D10112yes—
151C22×C4—21no—
152C22×C4—2022no—
153C22×C4C22⋊D10112yes—
154C22×C4C22×D10111yescharacteristic
155Q16—2028no—
156SD16—2028no—
157SD16—2028no—
158D8—2028no—
159M16—1018no—
160C2×C8—1014no—
161C4⋊C4—2022no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—2024no—
165C42C22⋊D10112yes—
166C42C22×D10111yescharacteristic
167C42C22×D10111yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—2022no—
172C23C2×C22⋊D10111yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—2022no—
176Q8—2022no—
177D4—24no—
178D4—42no—
179D4—4044no—
180D4—4044no—
181D4—2022no—
182D4—2022no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—4044no—
187C2×C4—2022no—
188C2×C4D4×D10114yes—
189C2×C4C2×C22⋊D10112yes—
190C2×C4C2×C22⋊D10112yes—
191C2×C4C2×C22⋊D10111yescharacteristic
192C2×C4C23×D10111yescharacteristic
193C8—1018no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—4042no—
199C22—4044no—
200C22#1569225112yes—
201C22C22×C22⋊D10111yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—4042no—
206C4#1569225112yes—
207C4C2×D4×D10112yes—
208C2—22no—
209C2—82no—
210C2—8082no—
211C2C2×#1569225111yescharacteristic
212C2#2026597512yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#124×C101C2
socleC22×C101C22×D4
hypercenterC2×#124D101
hypocenterC101#3153
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#15692286C2semidirect
2. C2×#15692281C2semidirect
3. #20265727C2semidirect
4. #20265572C2semidirect
5. #20265567C2semidirect
6. #20265559C2semidirect
7. C4×#13108058C2semidirect
8. C2×#124×C101C2semidirect
12. C2×#13108064C22semidirect
14. C2×#13108061C22semidirect
17. #15692255C22semidirect
18. #15692252C22semidirect
19. #15692243C22semidirect
22. #15692166C22semidirect
23. C4×#11717579C22semidirect
42. #13108058D4semidirect
107. C101#3153semidirect
111. C2×#124D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922862
C2×#156922812
#202657272
#202655722
#202655672
#202655592
C4×#131080582
C2×#124×C1012
#3153101