Finite Group Databasebeta

← C2×#22550136 #30648426

Subgroups — C2×#22550136 #30648426

Order 16256.

468 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic
3C2×#16861215C211yesmaximal, characteristic, complemented
4#22550136C218yesmaximal, direct product factor
5C2×#16861145C211yesmaximal, characteristic, complemented
6C2×#16861128C211yesmaximal, characteristic, complemented
7C2×#16861126C211yesmaximal, characteristic, complemented
8C22×SD16×C127C211yesmaximal, characteristic, complemented
9C2×#14523248C2211yescharacteristic
10#16861240—22no—
11#16861240C2214yescomplemented
12#16861235—22no—
13#16861235C2214yes—
14C2×#14523238—21no—
15C2×#14523236—21no—
16C2×#14523235—21no—
17C2×#14523233—21no—
18#16861215—22no—
19#16861215C2214yescomplemented
20C2×#14523232C2211yescharacteristic
21C2×#14523232C2211yescharacteristic
22C2×#14523232C2211yescharacteristic
23C2×#14523231—21no—
24C2×C8×D127—21no—
25#16861145C2214yescomplemented
26#16861128C2214yescomplemented
27#16861126C2214yescomplemented
28C22×Q8×C127C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic
30C2×SD16×C127—21no—
31C2×SD16×C127—21no—
32C2×SD16×C127—22no—
33C2×SD16×C127C2214yescomplemented
34C22×C8×C127C2211yescharacteristic
35#14523248—21no—
36#14523248—21no—
37#14523248D412yes—
38#14523248C2312yes—
39C2×Dic254D411yescharacteristic
40C2×D508D411yescharacteristic
41C2×C22⋊D127—21no—
42C2×C4×D127—21no—
43#14523238—24no—
44#14523236—24no—
45#14523235—24no—
46#14523233—24no—
47#14523232—21no—
48#14523232—21no—
49#14523232—21no—
50#14523232C2312yes—
51#14523232C2312yes—
52#14523232C2312yes—
53#14523231—24no—
54C2×C127⋊C8—21no—
55C2×C127⋊C8—21no—
56C2×C127⋊C8—21no—
57C8×D127—24no—
58C24×C127—21no—
59C2×Q8×C127—21no—
60C2×Q8×C127—21no—
61C2×Q8×C127—21no—
62C2×Q8×C127—21no—
63C2×Q8×C127—21no—
64C2×Q8×C127C2312yes—
65C2×D4×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127C2312yes—
71C22×C4×C127—21no—
72C22×C4×C127C2311yescharacteristic
73SD16×C127—24no—
74SD16×C127—24no—
75C2×C8×C127D411yescharacteristic
76C2×C8×C127D411yescharacteristic
77C2×C8×C127D412yes—
78C2×C8×C127C2312yes—
79C22×D127—21no—
80Dic254—21no—
81Dic254C2×D412yes—
82D508—21no—
83D508C2×D412yes—
84C22⋊D127—22no—
85C22⋊D127—22no—
86C2×Dic127—21no—
87C4×D127—22no—
88C4×D127—22no—
89C127⋊C8—22no—
90C127⋊C8—22no—
91C127⋊C8—22no—
92C23×C127—22no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127—82no—
96C23×C127#4111yescharacteristic
97Q8×C127—22no—
98Q8×C127—22no—
99Q8×C127—41no—
100Q8×C127—41no—
101D4×C127—22no—
102D4×C127—22no—
103D4×C127—41no—
104D4×C127—41no—
105C2×C4×C127—22no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127#4111yescharacteristic
113C2×C4×C127C2411yescharacteristic
114C8×C127C2×D412yes—
115C8×C127C2×D412yes—
116C22×C127—22no—
117C22×C127—42no—
118C22×C127—82no—
119C22×C127—82no—
120C22×C127—82no—
121C22×C127#12111yescharacteristic
122C22×C127#12111yescharacteristic
123C22×C127C2×#4111yescharacteristic
124C2×D127—22no—
125C2×D127—42no—
126C4×C127—42no—
127C4×C127#12111yescharacteristic
128C4×C127#12111yescharacteristic
129C4×C127C22×D411yescharacteristic
130C4×C127C22×D411yescharacteristic
131Dic127—22no—
132C2×C127—21no—
133C2×C127—21no—
134C2×C127—82no—
135C2×C127#47112yescomplemented
136C2×C127C2×#12111yescharacteristic
137D127—42no—
138C2×#471—1271nomaximal, Sylow
139C127C2×#47111yescharacteristic, complemented, Sylow
140C2×#137—1271no—
141C2×AGL(1,ℤ/8)—1271no—
142C22×SD16D12711yescharacteristic, complemented
143C2×#131—1271no—
144#471—1278no—
145C2×#108—1271no—
146C2×#101—1271no—
147C2×#100—1271no—
148C2×#41—1271no—
149C22×Q8C2×D12711yescharacteristic
150C22×D4C2×D12711yescharacteristic
151#137—1274no—
152#137—2542no—
153AGL(1,ℤ/8)—1274no—
154AGL(1,ℤ/8)—2542no—
155C2×Q16—2541no—
156C2×SD16—21no—
157C2×SD16—21no—
158C2×SD16—22no—
159C2×SD16—2541no—
160C2×SD16—2541no—
161C2×SD16C2×D12714yescomplemented
162C2×D8—2541no—
163#131—1274no—
164#131—2542no—
165C2×M16—1271no—
166C2×M16—1271no—
167C2×M16—1271no—
168C2×M16—2541no—
169C22×C8—2541no—
170C22×C8C2×D12711yescharacteristic
171#108—1274no—
172#101—1274no—
173#100—1274no—
174C24—21no—
175#41—1272no—
176#41—1272no—
177#41—2541no—
178#41—2541no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—1271no—
185C2×Q8C22×D12712yes—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—1271no—
192C2×D4—2541no—
193C2×D4C22×D12712yes—
194C22×C4—21no—
195C22×C4—2541no—
196C22×C4C22×D12711yescharacteristic
197Q16—2544no—
198SD16—24no—
199SD16—24no—
200SD16—2544no—
201SD16—2544no—
202D8—2544no—
203M16—1272no—
204M16—1272no—
205M16—1272no—
206M16—2541no—
207M16—2541no—
208M16—2541no—
209M16—2544no—
210C2×C8—2541no—
211C2×C8—2541no—
212C2×C8—2541no—
213C2×C8—2544no—
214C2×C8C22⋊D12711yescharacteristic
215C2×C8C22⋊D12711yescharacteristic
216C2×C8C22⋊D12712yes—
217C2×C8C22×D12712yes—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—82no—
222C23—2541no—
223C23#1452324711yescharacteristic
224Q8—22no—
225Q8—22no—
226Q8—41no—
227Q8—41no—
228Q8—1272no—
229Q8—2541no—
230D4—22no—
231D4—22no—
232D4—41no—
233D4—41no—
234D4—1272no—
235D4—2542no—
236D4—2542no—
237D4—2541no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—2541no—
242C2×C4—2542no—
243C2×C4—2542no—
244C2×C4D4×D12711yescharacteristic
245C2×C4D4×D12711yescharacteristic
246C2×C4C2×C22⋊D12711yescharacteristic
247C2×C4C2×C22⋊D12711yescharacteristic
248C2×C4#1452324711yescharacteristic
249C2×C4C23×D12711yescharacteristic
250C8—2542no—
251C8—2542no—
252C8—2542no—
253C8C2×C22⋊D12712yes—
254C8C2×C22⋊D12712yes—
255C22—22no—
256C22—42no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—5082no—
261C22—2542no—
262C22#1686118011yescharacteristic
263C22#1686118011yescharacteristic
264C22C2×#1452324711yescharacteristic
265C4—42no—
266C4—2542no—
267C4#1686118011yescharacteristic
268C4#1686118011yescharacteristic
269C4C2×D4×D12711yescharacteristic
270C4C22×C22⋊D12711yescharacteristic
271C2—21no—
272C2—21no—
273C2—82no—
274C2—5082no—
275C2#2255013612yesdirect product factor
276C2C2×#1686118011yescharacteristic
subgroupG/H
centerC22#16861180
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC22×C127#121
prime radicalC2×#16861235C2
hypercenterC22×SD16D127
hypocenterC127C2×#471
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. C2×#16861215C2semidirect
4. #22550136C2direct (N × H)
5. C2×#16861145C2semidirect
6. C2×#16861128C2semidirect
7. C2×#16861126C2semidirect
8. C22×SD16×C127C2semidirect
11. #16861240C22semidirect
19. #16861215C22semidirect
25. #16861145C22semidirect
26. #16861128C22semidirect
27. #16861126C22semidirect
33. C2×SD16×C127C22semidirect
135. C2×C127#471semidirect
139. C127C2×#471semidirect
142. C22×SD16D127semidirect
161. C2×SD16C2×D127semidirect
275. C2#22550136direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612352
C2×#168612152
#225501362
C2×#168611452
C2×#168611282
C2×#168611262
C22×SD16×C1272
C2×#471127