Finite Group Databasebeta

← #27181384

Subgroups — #27181384

Order 12928.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265789C211yesmaximal, characteristic, complemented
2#20265518C212yesmaximal, complemented
3#20265477C212yesmaximal, complemented
4#20265382C212yesmaximal, complemented
5#20265328C212yesmaximal
6#20265277C212yesmaximal, complemented
7C4×#13108062C212yesmaximal, complemented
8#20265238C211yesmaximal, characteristic, complemented
9#518×C101C211yesmaximal, characteristic
10C2×#13108062C2212yes—
11C2×#13108062C2212yes—
12#15692255C2211yescharacteristic, complemented
13#15692248—22no—
14#15692247C2211yescharacteristic, complemented
15#15692245—22no—
16#15692234C2212yes—
17C4×Dic202C2212yescomplemented
18#15692188C2212yescomplemented
19#15692186C2212yescomplemented
20C4⋊#11717579C2212yescomplemented
21#15692178C2212yescomplemented
22#15692178C2212yescomplemented
23#15692176C2212yes—
24#15692176C2212yes—
25C22⋊#11717579C2212yes—
26#15692166C2211yescharacteristic, complemented
27C4×#11717579C2211yescharacteristic, complemented
28C22×D4×C101C2211yescharacteristic
29C4×D4×C101—22no—
30C4×D4×C101C2212yes—
31C4×D4×C101C2212yes—
32#117×C101C2211yescharacteristic
33#117×C101C2211yescharacteristic
34C2×C22⋊C4×C101—22no—
35C2×Dic202C2312yes—
36#13108062—24no—
37#13108062—24no—
38C2×#11717579C2312yes—
39C2×#11717579C2312yes—
40C4⋊Dic101—22no—
41C4⋊Dic101C2312yes—
42#13108053—22no—
43#13108053—22no—
44#13108053—22no—
45C22⋊Dic101—22no—
46C4×Dic101—22no—
47C24×C101—21no—
48C2×D4×C101—24no—
49C2×D4×C101—21no—
50C2×D4×C101C2312yes—
51C22×C4×C101—22no—
52C22×C4×C101—22no—
53C22×C4×C101C2311yescharacteristic
54C4⋊C4×C101D412yes—
55C4⋊C4×C101C2312yes—
56C22⋊C4×C101—21no—
57C22⋊C4×C101—22no—
58C22⋊C4×C101—22no—
59C22⋊C4×C101D412yes—
60C42×C101—21no—
61C42×C101C2311yescharacteristic
62C42×C101C2311yescharacteristic
63C23×C101—24no—
64C23×C101—22no—
65C23×C101—82no—
66C23×C101C2×D411yescharacteristic
67D4×C101—24no—
68D4×C101—22no—
69D4×C101#4114yes—
70C2×C4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101—82no—
73C2×C4×C101—82no—
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101C2×D412yes—
76C2×C4×C101C2×D412yes—
77C2×C4×C101C2411yescharacteristic
78C2×Dic101—22no—
79C2×Dic101—22no—
80Dic202—22no—
81Dic202—22no—
82#11717579—22no—
83#11717579—22no—
84C22×C101—21no—
85C22×C101—24no—
86C22×C101—22no—
87C22×C101—42no—
88C22×C101—42no—
89C22×C101—84no—
90C22×C101C22×D411yescharacteristic
91C4×C101—21no—
92C4×C101—22no—
93C4×C101—22no—
94C4×C101—42no—
95C4×C101C2×#4112yes—
96Dic101—42no—
97Dic101—42no—
98C2×C101—41no—
99C2×C101—44no—
100C2×C101#52511yescharacteristic
101C2×C101#57712yes—
102#3123—1011nomaximal, Sylow
103C101#312311yescharacteristic, complemented, Sylow
104#541—1011no—
105#518D10111yescharacteristic
106#485—1012no—
107#478—1012no—
108#461—1012no—
109#450—1012no—
110#440—1012no—
111C4×SD16—1012no—
112#433—1011no—
113C22×D4C2×D10111yescharacteristic
114C2×SD16—1012no—
115C2×SD16—1012no—
116C4⋊Q8—1011no—
117#126—2022no—
118#125—1011no—
119#125—2022no—
120C22⋊Q8—1012no—
121C4×Q8—1012no—
122C4×D4—22no—
123C4×D4C2×D10112yes—
124C4×D4C2×D10112yes—
125#117C2×D10111yescharacteristic
126#117C2×D10111yescharacteristic
127C2×C22⋊C4—22no—
128#107—1012no—
129#106—1012no—
130C4⋊C8—1012no—
131#103—1012no—
132#103—1012no—
133#102—1012no—
134#102—1012no—
135C22⋊C8—1012no—
136#97—1011no—
137C4×C8—1011no—
138C24—21no—
139C2×Q8—1012no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D10112yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D10111yescharacteristic
146SD16—2024no—
147SD16—2024no—
148C2×C8—1012no—
149C2×C8—1012no—
150C4⋊C4—1012no—
151C4⋊C4—2022no—
152C4⋊C4—2022no—
153C4⋊C4—2022no—
154C4⋊C4—2022no—
155C4⋊C4C22⋊D10112yes—
156C4⋊C4C22×D10112yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—2022no—
161C22⋊C4C22⋊D10112yes—
162C42—21no—
163C42—2022no—
164C42C22×D10111yescharacteristic
165C42C22×D10111yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D10111yescharacteristic
170Q8—2022no—
171Q8—2022no—
172D4—24no—
173D4—22no—
174D4#1310807414yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—2022no—
180C2×C4—2022no—
181C2×C4C2×C22⋊D10111yescharacteristic
182C2×C4C2×C22⋊D10112yes—
183C2×C4C2×C22⋊D10112yes—
184C2×C4C23×D10111yescharacteristic
185C8—2022no—
186C8—2022no—
187C22—22no—
188C22—21no—
189C22—24no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D10111yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—42no—
198C4—4042no—
199C4—4042no—
200C4C2×#1310807412yes—
201C2—41no—
202C2—44no—
203C2#2026568811yescharacteristic
204C2#2026598812yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#518×C101C2
socleC22×C101C22×D4
prime radicalC22×D4×C101C22
hypercenter#518D101
hypocenterC101#3123
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20265789C2semidirect
2. #20265518C2semidirect
3. #20265477C2semidirect
4. #20265382C2semidirect
6. #20265277C2semidirect
7. C4×#13108062C2semidirect
8. #20265238C2semidirect
12. #15692255C22semidirect
14. #15692247C22semidirect
17. C4×Dic202C22semidirect
18. #15692188C22semidirect
19. #15692186C22semidirect
20. C4⋊#11717579C22semidirect
21. #15692178C22semidirect
22. #15692178C22semidirect
26. #15692166C22semidirect
27. C4×#11717579C22semidirect
103. C101#3123semidirect

Maximal subgroups

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

subgroupindex
#202657892
#202655182
#202654772
#202653822
#202653282
#202652772
C4×#131080622
#202652382
#518×C1012
#3123101