Finite Group Databasebeta

← C2×#20265548 #27181537

Subgroups — C2×#20265548 #27181537

Order 12928.

656 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×#13108063C211yesmaximal, characteristic, complemented
2C22×#13108062C211yesmaximal, characteristic, complemented
3C22×#13108061C211yesmaximal, characteristic, complemented
4C2×#15692253C211yesmaximal, characteristic, complemented
5C2×#15692251C211yesmaximal, characteristic
6#20265548C218yesmaximal, direct product factor
7C2×C8×Dic101C211yesmaximal, characteristic, complemented
8C22×SD16×C101C211yesmaximal, characteristic, complemented
9C22×Dic202C2211yescharacteristic
10C22×D404C2211yescharacteristic
11C22×C22⋊D101—22no—
12C2×#13108063—24no—
13C2×#13108063C2214yescomplemented
14C2×#13108062—24no—
15C2×#13108062C2214yescomplemented
16C2×#13108061—24no—
17C2×#13108061C2214yescomplemented
18C22×#11717579C2211yescharacteristic
19#15692253—22no—
20#15692253C2214yescomplemented
21#15692251—22no—
22#15692251C2214yes—
23C2×#13108053—22no—
24C2×C4×Dic101C2211yescharacteristic
25C8×Dic101C2214yescomplemented
26C22×Q8×C101C2211yescharacteristic
27C22×D4×C101C2211yescharacteristic
28C2×SD16×C101—24no—
29C2×SD16×C101C2214yescomplemented
30C22×C8×C101C2211yescharacteristic
31C23×D101—21no—
32C2×Dic202—22no—
33C2×Dic202—22no—
34C2×Dic202—21no—
35C2×Dic202C2312yes—
36C2×D404—22no—
37C2×D404—22no—
38C2×D404—21no—
39C2×D404C2312yes—
40C2×C22⋊D101—28no—
41C2×C22⋊D101—48no—
42C22×Dic101—21no—
43C22×Dic101D412yes—
44#13108063—28no—
45#13108062—28no—
46#13108061—28no—
47C2×#11717579D414yescomplemented
48C2×#11717579C2312yes—
49#13108053—28no—
50C4×Dic101D412yes—
51C4×Dic101C2312yes—
52C24×C101—21no—
53C2×Q8×C101—21no—
54C2×Q8×C101—22no—
55C2×Q8×C101—22no—
56C2×Q8×C101C2312yes—
57C2×D4×C101—22no—
58C2×D4×C101—22no—
59C2×D4×C101—21no—
60C2×D4×C101C2312yes—
61C22×C4×C101—21no—
62C22×C4×C101C2311yescharacteristic
63SD16×C101—28no—
64C2×C8×C101D414yescomplemented
65C2×C8×C101C2312yes—
66C23×C101—22no—
67C23×C101—42no—
68C23×C101—44no—
69C23×C101C2×D411yescharacteristic
70C22×D101—22no—
71C22×D101—42no—
72C22×D101—44no—
73Q8×C101—22no—
74Q8×C101—22no—
75Q8×C101—42no—
76D4×C101—22no—
77D4×C101—22no—
78D4×C101—42no—
79C2×C4×C101—22no—
80C2×C4×C101—42no—
81C2×C4×C101C2×D411yescharacteristic
82C2×C4×C101C2×D412yes—
83C2×C4×C101C2×D412yes—
84C2×C4×C101C2411yescharacteristic
85C22⋊D101—416no—
86C2×Dic101—22no—
87C2×Dic101—42no—
88C2×Dic101SD1618yescomplemented
89C2×Dic101C2×D414yes—
90D404—22no—
91D404—22no—
92D404—42no—
93Dic202—22no—
94Dic202—22no—
95Dic202—42no—
96C8×C101C2×D414yescomplemented
97#11717579C2×D414yescomplemented
98C22×C101—42no—
99C22×C101—44no—
100C22×C101—84no—
101C22×C101#12712yes—
102C22×C101C2×SD1614yes—
103C22×C101C22×D411yescharacteristic
104C2×D101—42no—
105C2×D101—44no—
106C2×D101—84no—
107C4×C101—42no—
108C4×C101#12712yes—
109C4×C101C22×D411yescharacteristic
110C4×C101C22×D411yescharacteristic
111Dic101—42no—
112Dic101C2×SD1618yescomplemented
113C2×C101—82no—
114C2×C101#49214yescomplemented
115C2×C101C2×#12711yescharacteristic
116C2×C101C22×SD1612yes—
117D101—82no—
118C2×#492—1011nomaximal, Sylow
119C101C2×#49211yescharacteristic, complemented, Sylow
120C22×SD16—1011no—
121C22×SD16—1011no—
122C22×SD16—1011no—
123C22×SD16D10111yescharacteristic, complemented
124C2×C4⋊Q8—1011no—
125C2×#127—1011no—
126#492—1018no—
127C2×C4×C8—1011no—
128C22×Q8—1011no—
129C22×Q8C2×D10111yescharacteristic
130C22×D4—1011no—
131C22×D4—2022no—
132C22×D4C2×D10111yescharacteristic
133C2×SD16—24no—
134C2×SD16—1014no—
135C2×SD16—1014no—
136C2×SD16—1014no—
137C2×SD16—2024no—
138C2×SD16—2024no—
139C2×SD16—2024no—
140C2×SD16C2×D10114yescomplemented
141C22×C8—1011no—
142C22×C8C2×D10111yescharacteristic
143C4⋊Q8—1014no—
144C4⋊Q8—2022no—
145#127—1014no—
146#127—2022no—
147C2×C4⋊C4—2022no—
148C2×C42—1011no—
149C4×C8—1014no—
150C24—21no—
151C24—2021no—
152C2×Q8—21no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—1012no—
156C2×Q8—2022no—
157C2×Q8—2022no—
158C2×Q8—2021no—
159C2×Q8C22×D10112yes—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—1012no—
164C2×D4—4048no—
165C2×D4—2028no—
166C2×D4—2021no—
167C2×D4—2022no—
168C2×D4—2022no—
169C2×D4C22×D10112yes—
170C22×C4—21no—
171C22×C4—1012no—
172C22×C4—2021no—
173C22×C4C22×D10111yescharacteristic
174SD16—28no—
175SD16—2028no—
176SD16—2028no—
177SD16—2028no—
178C2×C8—1014no—
179C2×C8—1012no—
180C2×C8C22⋊D10114yescomplemented
181C2×C8C22×D10112yes—
182C4⋊C4—2028no—
183C42—1012no—
184C42—1012no—
185C23—22no—
186C23—42no—
187C23—44no—
188C23—4042no—
189C23—4044no—
190C23—2022no—
191C23D4×D10111yescharacteristic
192Q8—22no—
193Q8—22no—
194Q8—42no—
195Q8—4042no—
196Q8—2022no—
197Q8—2022no—
198D4—22no—
199D4—22no—
200D4—42no—
201D4—40416no—
202D4—4042no—
203D4—2022no—
204D4—2022no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—1014no—
208C2×C4—1018no—
209C2×C4—4042no—
210C2×C4—2022no—
211C2×C4D4×D10112yes—
212C2×C4D4×D10111yescharacteristic
213C2×C4C2×C22⋊D10112yes—
214C2×C4C23×D10111yescharacteristic
215C8—1014no—
216C8C2×C22⋊D10114yescomplemented
217C22—44no—
218C22—42no—
219C22—84no—
220C22—8084no—
221C22—4044no—
222C22—4042no—
223C22#1569225112yes—
224C22SD16×D10114yes—
225C22C2×D4×D10111yescharacteristic
226C4—42no—
227C4—1018no—
228C4—4042no—
229C4#1569225112yes—
230C4C2×D4×D10111yescharacteristic
231C4C22×C22⋊D10111yescharacteristic
232C2—82no—
233C2—8082no—
234C2#2026554814yesdirect product factor
235C2C2×#1569225111yescharacteristic
236C2C2×SD16×D10112yes—
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×SD16×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692251C2
hypercenterC22×SD16D101
hypocenterC101C2×#492
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13108063C2semidirect
2. C22×#13108062C2semidirect
3. C22×#13108061C2semidirect
4. C2×#15692253C2semidirect
6. #20265548C2direct (N × H)
7. C2×C8×Dic101C2semidirect
8. C22×SD16×C101C2semidirect
13. C2×#13108063C22semidirect
15. C2×#13108062C22semidirect
17. C2×#13108061C22semidirect
20. #15692253C22semidirect
25. C8×Dic101C22semidirect
29. C2×SD16×C101C22semidirect
47. C2×#11717579D4semidirect
64. C2×C8×C101D4semidirect
88. C2×Dic101SD16semidirect
96. C8×C101C2×D4semidirect
97. #11717579C2×D4semidirect
112. Dic101C2×SD16semidirect
114. C2×C101#492semidirect
119. C101C2×#492semidirect
123. C22×SD16D101semidirect
140. C2×SD16C2×D101semidirect
180. C2×C8C22⋊D101semidirect
216. C8C2×C22⋊D101semidirect
234. C2#20265548direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080632
C22×#131080622
C22×#131080612
C2×#156922532
C2×#156922512
#202655482
C2×C8×Dic1012
C22×SD16×C1012
C2×#492101