Finite Group Databasebeta

← #27180495

Subgroups — #27180495

Order 12928.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#15692303C211yesmaximal, characteristic, complemented
2#20265975C212yesmaximal, complemented
3#20265359C214yesmaximal, complemented
4#20265349C214yesmaximal, complemented
5#20265195C212yesmaximal, complemented
6C2×#15692174C211yesmaximal, characteristic, complemented
7#536×C101C211yesmaximal, characteristic, complemented
8#15692303—24no—
9#15692303—22no—
10#15692303C2214yes—
11C2×#13108075—22no—
12C2×#13108074—22no—
13C2×#13108074C2212yes—
14C2×Q8×D101—22no—
15#15692286C2214yescomplemented
16#15692285—22no—
17#15692281C2214yescomplemented
18#15692280—22no—
19#15692275—24no—
20C2×#13108058C2212yes—
21#15692183C2218yescomplemented
22#15692174C2214yescomplemented
23C2×#41×C101C2211yescharacteristic
24C22×Q8×C101C2211yescharacteristic
25C4⋊Q8×C101C2212yescomplemented
26#124×C101C2212yescomplemented
27#123×C101—22no—
28C22⋊Q8×C101—22no—
29#117×C101C2211yescharacteristic
30#13108075—28no—
31#13108075—28no—
32#13108074—28no—
33#13108074—28no—
34#13108074—22no—
35#13108074—22no—
36#13108074D414yes—
37#13108074C2314yes—
38C2×Dic202—22no—
39C2×Dic202D412yes—
40Q8×D101—28no—
41Q8×D101—28no—
42C2×D404—22no—
43C2×D404D412yes—
44C2×C22⋊D101—22no—
45C2×C4×D101—24no—
46C2×C4×D101—22no—
47#13108064—24no—
48#13108062—24no—
49#13108061—24no—
50#13108059—24no—
51#13108058—22no—
52#13108058C2314yes—
53C2×#11717579—22no—
54#41×C101—22no—
55#41×C101—22no—
56C2×Q8×C101—22no—
57C2×Q8×C101—22no—
58C2×Q8×C101D412yes—
59C2×Q8×C101C2311yescharacteristic
60C2×Q8×C101C2312yes—
61C2×D4×C101—21no—
62C2×D4×C101C2311yescharacteristic
63C22×C4×C101—21no—
64C22×C4×C101D412yes—
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—21no—
67C4⋊C4×C101—24no—
68C22⋊C4×C101—21no—
69C22⋊C4×C101—24no—
70C42×C101C2312yes—
71C23×C101—21no—
72C23×C101C2×D411yescharacteristic
73C22×D101—22no—
74Q8×C101—24no—
75Q8×C101—24no—
76Q8×C101—22no—
77D4×C101—22no—
78D4×C101—41no—
79D4×C101—41no—
80C2×C4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101—21no—
84C2×C4×C101—42no—
85C2×C4×C101C2×D412yes—
86C2×C4×C101C2×D412yes—
87C2×C4×C101C2×D414yes—
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101C2411yescharacteristic
90C22⋊D101—24no—
91C22⋊D101—24no—
92C2×Dic101—22no—
93D404—28no—
94D404—22no—
95D404C2×D414yes—
96C4×D101—216no—
97C4×D101—24no—
98C4×D101—24no—
99Dic202—28no—
100Dic202—22no—
101Dic202C2×D414yes—
102#11717579—24no—
103C22×C101—41no—
104C22×C101—42no—
105C22×C101—41no—
106C22×C101#12011yescharacteristic
107C22×C101#12011yescharacteristic
108C22×C101C22×D411yescharacteristic
109C2×D101—24no—
110C2×D101—44no—
111C4×C101—24no—
112C4×C101—42no—
113C4×C101—41no—
114C4×C101#12012yes—
115C4×C101C22×D412yes—
116Dic101—24no—
117C2×C101—21no—
118C2×C101—21no—
119C2×C101—21no—
120C2×C101—81no—
121C2×C101C2×#12011yescharacteristic
122D101—44no—
123#3020—1011nomaximal, Sylow
124C101#302011yescharacteristic, complemented, Sylow
125C2×2−1+4—1011no—
126#575—1012no—
127#536D10111yescharacteristic, complemented
128#456—1014no—
129#454—1014no—
130#421—1012no—
131C2×#101—1011no—
1322−1+4—1014no—
1332−1+4—2024no—
1342−1+4—2022no—
135C2×#41—1012no—
136C2×#41—2022no—
137C2×#41—2022no—
138C2×#41C2×D10111yescharacteristic
139C22×Q8—2022no—
140C22×Q8C2×D10111yescharacteristic
141#137—1014no—
142#137—2022no—
143AGL(1,ℤ/8)—1014no—
144AGL(1,ℤ/8)—2022no—
145#135—2024no—
146C2×M16—1012no—
147C4⋊Q8C2×D10112yescomplemented
148#124C2×D10112yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D10111yescharacteristic
152#104—1018no—
153#101—1014no—
154#41—22no—
155#41—22no—
156#41—1014no—
157#41—1014no—
158#41—2028no—
159#41—2028no—
160#41—2028no—
161#41—2022no—
162#41—2022no—
163#41—2028no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—1012no—
167C2×Q8—2022no—
168C2×Q8—2028no—
169C2×Q8—2028no—
170C2×Q8C22⋊D10112yes—
171C2×Q8C22×D10111yescharacteristic
172C2×Q8C22×D10112yes—
173C2×D4—21no—
174C2×D4—1012no—
175C2×D4—2022no—
176C2×D4—2022no—
177C2×D4C22×D10111yescharacteristic
178C22×C4—21no—
179C22×C4—2024no—
180C22×C4—2022no—
181C22×C4C22⋊D10112yes—
182C22×C4C22×D10111yescharacteristic
183Q16—2024no—
184SD16—2024no—
185SD16—2024no—
186D8—2024no—
187M16—1014no—
188M16—2022no—
189C2×C8—2022no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D10112yes—
195C23—21no—
196C23—2022no—
197C23C2×C22⋊D10111yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—1014no—
202Q8—2028no—
203Q8—2022no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—1014no—
208D4—2028no—
209D4—2024no—
210D4—2024no—
211D4—2022no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—20216no—
218C2×C4—2022no—
219C2×C4—2024no—
220C2×C4—2024no—
221C2×C4D4×D10112yes—
222C2×C4D4×D10112yes—
223C2×C4C2×C22⋊D10114yes—
224C2×C4C2×C22⋊D10111yescharacteristic
225C2×C4C23×D10111yescharacteristic
226C8—2024no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—4044no—
231C22—2024no—
232C22#1569222311yescharacteristic
233C22#1569222311yescharacteristic
234C22C22×C22⋊D10111yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—2024no—
239C4#1569222312yes—
240C4C2×D4×D10112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—4044no—
246C2C2×#1569222311yescharacteristic
subgroupG/H
centerC2C2×#15692223
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#536×C101C2
socleC2×C101C2×#120
hypercenter#536D101
hypocenterC101#3020
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#15692303C2semidirect
2. #20265975C2semidirect
3. #20265359C2semidirect
4. #20265349C2semidirect
5. #20265195C2semidirect
6. C2×#15692174C2semidirect
7. #536×C101C2semidirect
15. #15692286C22semidirect
17. #15692281C22semidirect
21. #15692183C22semidirect
22. #15692174C22semidirect
25. C4⋊Q8×C101C22semidirect
26. #124×C101C22semidirect
124. C101#3020semidirect
127. #536D101semidirect
147. C4⋊Q8C2×D101semidirect
148. #124C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156923032
#202659752
#202653592
#202653492
#202651952
C2×#156921742
#536×C1012
#3020101