Finite Group Databasebeta

← C2×#20265370 #27180542

Subgroups — C2×#20265370 #27180542

Order 12928.

648 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13108059C211yesmaximal, characteristic, complemented
2C2×#15692251C211yesmaximal, characteristic, complemented
3C2×D4×Dic101C211yesmaximal, characteristic, complemented
4#20265370C218yesmaximal, direct product factor
5C2×#15692184C211yesmaximal, characteristic, complemented
6C2×#15692177C211yesmaximal, characteristic, complemented
7C2×#15692176C211yesmaximal, characteristic, complemented
8C22×D8×C101C211yesmaximal, characteristic, complemented
9C22×D404C2211yescharacteristic
10C22×C22⋊D101—21no—
11C22×C22⋊D101—21no—
12C23×Dic101—21no—
13C2×#13108059—24no—
14C2×#13108059C2214yescomplemented
15C22×#11717579C2211yescharacteristic
16#15692251—22no—
17#15692251C2214yescomplemented
18D4×Dic101—22no—
19D4×Dic101C2214yescomplemented
20C2×C4⋊Dic101C2211yescharacteristic
21C2×C22⋊Dic101—21no—
22C2×C4×Dic101C2211yescharacteristic
23#15692184C2214yescomplemented
24#15692177C2214yescomplemented
25#15692176C2214yescomplemented
26C22×D4×C101C2211yescharacteristic
27C22×D4×C101C2211yescharacteristic
28C2×D8×C101—24no—
29C2×D8×C101C2214yescomplemented
30C22×C8×C101C2211yescharacteristic
31C23×D101—21no—
32C2×D404—22no—
33C2×D404—22no—
34C2×D404—21no—
35C2×D404C2312yes—
36C2×C22⋊D101—24no—
37C2×C22⋊D101—24no—
38C2×C22⋊D101—44no—
39C2×C22⋊D101—44no—
40C22×Dic101—21no—
41C22×Dic101—24no—
42C22×Dic101—44no—
43C22×Dic101D411yescharacteristic
44C22×Dic101D411yescharacteristic
45#13108059—28no—
46C2×#11717579—22no—
47C2×#11717579C2312yes—
48C4⋊Dic101—21no—
49C4⋊Dic101C2312yes—
50C22⋊Dic101—24no—
51C4×Dic101D412yes—
52C4×Dic101C2312yes—
53C24×C101—21no—
54C24×C101—21no—
55C2×D4×C101—21no—
56C2×D4×C101—22no—
57C2×D4×C101—21no—
58C2×D4×C101—22no—
59C2×D4×C101—22no—
60C2×D4×C101D414yescomplemented
61C2×D4×C101C2312yes—
62C2×D4×C101C2312yes—
63C22×C4×C101C2311yescharacteristic
64D8×C101—28no—
65C2×C8×C101—22no—
66C2×C8×C101C2312yes—
67C23×C101—24no—
68C23×C101—22no—
69C23×C101—22no—
70C23×C101—42no—
71C23×C101—42no—
72C23×C101—42no—
73C23×C101—42no—
74C23×C101—42no—
75C23×C101C2×D411yescharacteristic
76C22×D101—22no—
77C22×D101—42no—
78C22×D101—42no—
79C22×D101—42no—
80D4×C101—24no—
81D4×C101—22no—
82D4×C101—22no—
83D4×C101—22no—
84D4×C101—42no—
85D4×C101C2×D414yescomplemented
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101C2×D412yes—
88C2×C4×C101#4112yes—
89C2×C4×C101C2411yescharacteristic
90C22⋊D101—48no—
91C22⋊D101—48no—
92C2×Dic101—22no—
93C2×Dic101—22no—
94C2×Dic101—42no—
95C2×Dic101—48no—
96C2×Dic101D814yescomplemented
97C2×Dic101C2×D412yes—
98C2×Dic101C2×D412yes—
99D404—22no—
100D404—22no—
101D404—42no—
102C8×C101—22no—
103#11717579—22no—
104C22×C101—24no—
105C22×C101—42no—
106C22×C101—42no—
107C22×C101—48no—
108C22×C101—42no—
109C22×C101—42no—
110C22×C101—42no—
111C22×C101—84no—
112C22×C101#12112yes—
113C22×C101C2×D812yes—
114C22×C101AGL(1,ℤ/8)12yes—
115C22×C101C22×D411yescharacteristic
116C2×D101—42no—
117C2×D101—42no—
118C2×D101—42no—
119C2×D101—84no—
120C4×C101#12112yes—
121C4×C101C22×D411yescharacteristic
122C4×C101C2×#4111yescharacteristic
123Dic101—22no—
124Dic101—42no—
125Dic101C2×D814yescomplemented
126C2×C101—44no—
127C2×C101—82no—
128C2×C101#45914yescomplemented
129C2×C101C2×#12111yescharacteristic
130C2×C101C22×D811yescharacteristic
131C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
132D101—82no—
133C2×#459—1011nomaximal, Sylow
134C101C2×#45911yescharacteristic, complemented, Sylow
135C22×D8—1011no—
136C22×D8D10111yescharacteristic, complemented
137C2×#127—1011no—
138C2×C4×D4—1011no—
139#459—1018no—
140C2×C4⋊C8—1011no—
141C2×#102—1011no—
142C2×#102—1011no—
143C22×D4—1011no—
144C22×D4—2021no—
145C22×D4—2021no—
146C22×D4C2×D10111yescharacteristic
147C22×D4C2×D10111yescharacteristic
148C23×C4—2021no—
149C2×D8—24no—
150C2×D8—1014no—
151C2×D8—2024no—
152C2×D8C2×D10114yescomplemented
153C22×C8—1011no—
154C22×C8C2×D10111yescharacteristic
155#127—1014no—
156#127—2022no—
157C4×D4—1014no—
158C4×D4—2022no—
159C2×C4⋊C4—1011no—
160C2×C22⋊C4—2021no—
161C2×C42—1011no—
162C4⋊C8—1014no—
163#102—1014no—
164#102—1014no—
165C24—21no—
166C24—21no—
167C24—2021no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—1012no—
174C2×D4—4044no—
175C2×D4—4044no—
176C2×D4—2024no—
177C2×D4—2024no—
178C2×D4—2021no—
179C2×D4—2022no—
180C2×D4—2022no—
181C2×D4C22⋊D10114yescomplemented
182C2×D4C22×D10112yes—
183C2×D4C22×D10112yes—
184C22×C4—1011no—
185C22×C4—1011no—
186C22×C4—4044no—
187C22×C4—2024no—
188C22×C4—2021no—
189C22×C4C22×D10111yescharacteristic
190D8—28no—
191D8—2028no—
192C2×C8—22no—
193C2×C8—1012no—
194C2×C8—2022no—
195C2×C8C22×D10112yes—
196C4⋊C4—1012no—
197C4⋊C4—2021no—
198C22⋊C4—2024no—
199C42—1012no—
200C42—1012no—
201C23—22no—
202C23—24no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—4042no—
210C23—4042no—
211C23—4042no—
212C23—2022no—
213C23D4×D10111yescharacteristic
214D4—22no—
215D4—22no—
216D4—22no—
217D4—24no—
218D4—42no—
219D4—4048no—
220D4—4048no—
221D4—4042no—
222D4—2022no—
223D4—2022no—
224D4C2×C22⋊D10114yescomplemented
225C2×C4—1012no—
226C2×C4—1014no—
227C2×C4—1012no—
228C2×C4—4048no—
229C2×C4—4042no—
230C2×C4—2022no—
231C2×C4—2022no—
232C2×C4D4×D10111yescharacteristic
233C2×C4C2×C22⋊D10112yes—
234C2×C4#1310807312yes—
235C2×C4C23×D10111yescharacteristic
236C8—22no—
237C8—2022no—
238C22—24no—
239C22—48no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—84no—
246C22—8084no—
247C22—4042no—
248C22—4042no—
249C22—4042no—
250C22#1569222712yes—
251C22D8×D10112yes—
252C22#1569227912yes—
253C22C2×D4×D10111yescharacteristic
254C4—1014no—
255C4—4042no—
256C4—2022no—
257C4#1569222712yes—
258C4C22×C22⋊D10111yescharacteristic
259C4C2×#1310807311yescharacteristic
260C2—44no—
261C2—82no—
262C2—8082no—
263C2#2026537014yesdirect product factor
264C2C2×#1569222711yescharacteristic
265C2C2×D8×D10111yescharacteristic
266C2C2×#1569227911yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×D8×C101C2
socleC23×C101C2×D4
hypercenterC22×D8D101
hypocenterC101C2×#459
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#13108059C2semidirect
2. C2×#15692251C2semidirect
3. C2×D4×Dic101C2semidirect
4. #20265370C2direct (N × H)
5. C2×#15692184C2semidirect
6. C2×#15692177C2semidirect
7. C2×#15692176C2semidirect
8. C22×D8×C101C2semidirect
14. C2×#13108059C22semidirect
17. #15692251C22semidirect
19. D4×Dic101C22semidirect
23. #15692184C22semidirect
24. #15692177C22semidirect
25. #15692176C22semidirect
29. C2×D8×C101C22semidirect
60. C2×D4×C101D4semidirect
85. D4×C101C2×D4semidirect
96. C2×Dic101D8semidirect
125. Dic101C2×D8semidirect
128. C2×C101#459semidirect
134. C101C2×#459semidirect
136. C22×D8D101semidirect
152. C2×D8C2×D101semidirect
181. C2×D4C22⋊D101semidirect
224. D4C2×C22⋊D101semidirect
263. C2#20265370direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080592
C2×#156922512
C2×D4×Dic1012
#202653702
C2×#156921842
C2×#156921772
C2×#156921762
C22×D8×C1012
C2×#459101