Finite Group Databasebeta

← C2×#20265376 #27180563

Subgroups — C2×#20265376 #27180563

Order 12928.

584 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 157 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
3C2×#15692251C211yesmaximal, characteristic
4C2×#15692220C211yesmaximal, characteristic, complemented
5#20265376C218yesmaximal, direct product factor
6C2×#15692184C211yesmaximal, characteristic, complemented
7C2×#15692175C211yesmaximal, characteristic, complemented
8C2×#102×C101C211yesmaximal, characteristic, complemented
9C22×Dic202C2211yescharacteristic
10C22×D404C2211yescharacteristic
11C22×C22⋊D101—21no—
12C22×C22⋊D101—21no—
13C2×#13108063—24no—
14C2×#13108063C2214yescomplemented
15C2×#13108062—24no—
16C2×#13108062C2214yescomplemented
17C22×#11717579C2211yescharacteristic
18#15692251—22no—
19#15692251C2214yes—
20#15692220—22no—
21#15692220C2214yescomplemented
22C2×#13108053—21no—
23C2×C4×Dic101—21no—
24C2×C4×Dic101C2211yescharacteristic
25#15692184C2214yescomplemented
26#15692175C2214yescomplemented
27C22×D4×C101C2211yescharacteristic
28C22×C8×C101C2211yescharacteristic
29C2×C4⋊C4×C101C2211yescharacteristic
30#102×C101C2214yescomplemented
31C23×D101—21no—
32C2×Dic202—21no—
33C2×Dic202—22no—
34C2×Dic202D414yescomplemented
35C2×Dic202C2312yes—
36C2×D404—22no—
37C2×D404—22no—
38C2×D404—21no—
39C2×D404C2312yes—
40C2×C22⋊D101—24no—
41C2×C22⋊D101—24no—
42C2×C22⋊D101—44no—
43C2×C22⋊D101—44no—
44C22×Dic101—21no—
45C22×Dic101D411yescharacteristic
46C22×Dic101D411yescharacteristic
47#13108063—28no—
48#13108062—28no—
49C2×#11717579—22no—
50C2×#11717579C2312yes—
51#13108053—24no—
52C4×Dic101—24no—
53C4×Dic101D412yes—
54C4×Dic101C2312yes—
55C24×C101—21no—
56C2×D4×C101—22no—
57C2×D4×C101—21no—
58C2×D4×C101—22no—
59C2×D4×C101C2312yes—
60C22×C4×C101—21no—
61C22×C4×C101C2311yescharacteristic
62C2×C8×C101—22no—
63C2×C8×C101C2312yes—
64C4⋊C4×C101—21no—
65C4⋊C4×C101C2312yes—
66C23×C101—22no—
67C23×C101—42no—
68C23×C101—42no—
69C23×C101—42no—
70C23×C101C2×D411yescharacteristic
71C22×D101—22no—
72C22×D101—42no—
73C22×D101—42no—
74C22×D101—42no—
75D4×C101—22no—
76D4×C101—22no—
77D4×C101—42no—
78C2×C4×C101—22no—
79C2×C4×C101—42no—
80C2×C4×C101C2×D411yescharacteristic
81C2×C4×C101C2×D412yes—
82C2×C4×C101#4112yes—
83C2×C4×C101C2411yescharacteristic
84C22⋊D101—48no—
85C22⋊D101—48no—
86C2×Dic101—24no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101SD1614yescomplemented
90C2×Dic101C2×D412yes—
91C2×Dic101C2×D412yes—
92D404—22no—
93D404—22no—
94D404—42no—
95Dic202—24no—
96Dic202—22no—
97Dic202C2×D414yescomplemented
98C8×C101—22no—
99#11717579—22no—
100C22×C101—42no—
101C22×C101—42no—
102C22×C101—42no—
103C22×C101—84no—
104C22×C101#12112yes—
105C22×C101C2×SD1612yes—
106C22×C101AGL(1,ℤ/8)12yes—
107C22×C101C22×D411yescharacteristic
108C2×D101—42no—
109C2×D101—42no—
110C2×D101—42no—
111C2×D101—84no—
112C4×C101—42no—
113C4×C101#12112yes—
114C4×C101C22×D411yescharacteristic
115C4×C101C2×#4111yescharacteristic
116Dic101—24no—
117Dic101—22no—
118Dic101C2×SD1614yescomplemented
119C2×C101—82no—
120C2×C101#46014yescomplemented
121C2×C101C2×#12111yescharacteristic
122C2×C101C22×SD1611yescharacteristic
123C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
124D101—82no—
125C2×#460—1011nomaximal, Sylow
126C101C2×#46011yescharacteristic, complemented, Sylow
127C22×SD16—1011no—
128C22×SD16—1011no—
129C2×#127—1011no—
130C2×C4×Q8—1011no—
131#460—1018no—
132C2×C4⋊C8—1011no—
133C2×#102—1011no—
134C2×#102D10111yescharacteristic, complemented
135C22×Q8—1011no—
136C22×D4—1011no—
137C22×D4—2021no—
138C22×D4—2021no—
139C22×D4C2×D10111yescharacteristic
140C2×SD16—1014no—
141C2×SD16—1014no—
142C2×SD16—2024no—
143C2×SD16—2024no—
144C22×C8—1011no—
145C22×C8C2×D10111yescharacteristic
146#127—1014no—
147#127—2022no—
148C4×Q8—1014no—
149C4×Q8—2022no—
150C2×C4⋊C4—2021no—
151C2×C4⋊C4C2×D10111yescharacteristic
152C2×C42—1011no—
153C2×C42—2021no—
154C4⋊C8—1014no—
155#102—1014no—
156#102C2×D10114yescomplemented
157C24—21no—
158C24—2021no—
159C2×Q8—1014no—
160C2×Q8—1012no—
161C2×Q8—2021no—
162C2×Q8—2022no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—1012no—
167C2×D4—4044no—
168C2×D4—4044no—
169C2×D4—2024no—
170C2×D4—2024no—
171C2×D4—2021no—
172C2×D4—2022no—
173C2×D4—2022no—
174C2×D4C22×D10112yes—
175C22×C4—21no—
176C22×C4—1011no—
177C22×C4—1011no—
178C22×C4—2021no—
179C22×C4C22×D10111yescharacteristic
180SD16—2028no—
181SD16—2028no—
182C2×C8—22no—
183C2×C8—1012no—
184C2×C8—2022no—
185C2×C8C22×D10112yes—
186C4⋊C4—21no—
187C4⋊C4—2024no—
188C4⋊C4C22×D10112yes—
189C42—1012no—
190C42—1012no—
191C42—2024no—
192C23—22no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—4042no—
197C23—4042no—
198C23—4042no—
199C23—2022no—
200C23D4×D10111yescharacteristic
201Q8—1014no—
202Q8—2022no—
203Q8—2024no—
204D4—22no—
205D4—22no—
206D4—42no—
207D4—4048no—
208D4—4048no—
209D4—4042no—
210D4—2022no—
211D4—2022no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—1012no—
215C2×C4—1014no—
216C2×C4—1012no—
217C2×C4—2022no—
218C2×C4—2024no—
219C2×C4—2022no—
220C2×C4D4×D10112yes—
221C2×C4D4×D10111yescharacteristic
222C2×C4#1310807412yes—
223C2×C4C23×D10111yescharacteristic
224C8—22no—
225C8—2022no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22—8084no—
231C22—4042no—
232C22—4042no—
233C22—4042no—
234C22#1569222812yes—
235C22SD16×D10112yes—
236C22#1569227912yes—
237C22C2×D4×D10111yescharacteristic
238C4—42no—
239C4—1014no—
240C4—2024no—
241C4—2022no—
242C4#1569222812yes—
243C4C2×D4×D10111yescharacteristic
244C4C2×#1310807411yescharacteristic
245C2—82no—
246C2—8082no—
247C2#2026537614yesdirect product factor
248C2C2×#1569222811yescharacteristic
249C2C2×SD16×D10111yescharacteristic
250C2C2×#1569227911yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#102×C101C2
socleC23×C101C2×D4
prime radicalC2×#15692251C2
hypercenterC2×#102D101
hypocenterC101C2×#460
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#13108063C2semidirect
2. C22×#13108062C2semidirect
4. C2×#15692220C2semidirect
5. #20265376C2direct (N × H)
6. C2×#15692184C2semidirect
7. C2×#15692175C2semidirect
8. C2×#102×C101C2semidirect
14. C2×#13108063C22semidirect
16. C2×#13108062C22semidirect
21. #15692220C22semidirect
25. #15692184C22semidirect
26. #15692175C22semidirect
30. #102×C101C22semidirect
34. C2×Dic202D4semidirect
89. C2×Dic101SD16semidirect
97. Dic202C2×D4semidirect
118. Dic101C2×SD16semidirect
120. C2×C101#460semidirect
126. C101C2×#460semidirect
134. C2×#102D101semidirect
156. #102C2×D101semidirect
247. C2#20265376direct (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
C2×#156922512
C2×#156922202
#202653762
C2×#156921842
C2×#156921752
C2×#102×C1012
C2×#460101