Finite Group Databasebeta

← C2×#22550137 #30648427

Subgroups — C2×#22550137 #30648427

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×#16861239C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3C2×#16861216C211yesmaximal, characteristic, complemented
4#22550137C218yesmaximal, direct product factor
5C2×#16861144C211yesmaximal, characteristic, complemented
6C2×#16861127C211yesmaximal, characteristic, complemented
7C2×#16861126C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10#16861239—22no—
11#16861239C2214yescomplemented
12C2×#14523238—21no—
13C2×#14523236—21no—
14C2×#14523236—21no—
15C2×#14523236—22no—
16C2×#14523236—21no—
17C2×#14523236C2214yescomplemented
18#16861216—22no—
19#16861216C2214yescomplemented
20C2×#14523232—21no—
21C2×#14523232C2211yescharacteristic
22C2×#14523232C2211yescharacteristic
23C22×C127⋊C8—21no—
24C22×C127⋊C8C2211yescharacteristic
25#16861144C2214yescomplemented
26#16861127C2214yescomplemented
27#16861126C2214yescomplemented
28C2×#41×C127C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic
30AGL(1,ℤ/8)×C127—22no—
31AGL(1,ℤ/8)×C127C2214yes—
32C2×SD16×C127—21no—
33C2×D8×C127—21no—
34C2×M16×C127C2211yescharacteristic
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—21no—
38C2×Dic254—21no—
39C2×Dic254—21no—
40C2×Dic254C2312yes—
41C22×Dic127—21no—
42#14523238—24no—
43#14523236—24no—
44#14523236—24no—
45#14523236—24no—
46#14523232—24no—
47#14523232—21no—
48#14523232—21no—
49#14523232C2312yes—
50#14523232C2312yes—
51C2×C127⋊C8—24no—
52C2×C127⋊C8—21no—
53C2×C127⋊C8—21no—
54C2×C127⋊C8D411yescharacteristic
55C2×C127⋊C8D411yescharacteristic
56C2×C127⋊C8D412yes—
57C2×C127⋊C8C2312yes—
58C24×C127—21no—
59#41×C127—21no—
60#41×C127—21no—
61#41×C127D412yes—
62#41×C127C2312yes—
63C2×Q8×C127D411yescharacteristic
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127D411yescharacteristic
71C2×D4×C127C2312yes—
72C22×C4×C127—21no—
73C22×C4×C127C2311yescharacteristic
74SD16×C127—24no—
75D8×C127—24no—
76M16×C127—21no—
77M16×C127C2312yes—
78C2×C8×C127—21no—
79Dic254—22no—
80Dic254—22no—
81Dic254—41no—
82Dic254—41no—
83C2×Dic127—22no—
84C2×Dic127—41no—
85C2×Dic127—41no—
86C127⋊C8—22no—
87C127⋊C8—22no—
88C127⋊C8C2×D412yes—
89C127⋊C8C2×D412yes—
90C23×C127—21no—
91C23×C127—22no—
92C23×C127—41no—
93C23×C127—41no—
94C23×C127—82no—
95C23×C127#4111yescharacteristic
96Q8×C127—21no—
97Q8×C127C2×D412yes—
98D4×C127—22no—
99D4×C127—22no—
100D4×C127—21no—
101D4×C127—22no—
102D4×C127—22no—
103D4×C127—41no—
104D4×C127—41no—
105D4×C127C2×D412yes—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127#4111yescharacteristic
114C2×C4×C127C2411yescharacteristic
115C8×C127—22no—
116C22×C127—22no—
117C22×C127—22no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—82no—
121C22×C127—82no—
122C22×C127—82no—
123C22×C127#12111yescharacteristic
124C22×C127#12111yescharacteristic
125C22×C127C2×#4111yescharacteristic
126C4×C127—22no—
127C4×C127#12111yescharacteristic
128C4×C127#12111yescharacteristic
129C4×C127C22×D411yescharacteristic
130C4×C127C22×D411yescharacteristic
131Dic127—42no—
132C2×C127—21no—
133C2×C127—21no—
134C2×C127—42no—
135C2×C127—82no—
136C2×C127#47112yescomplemented
137C2×C127C2×#12111yescharacteristic
138C2×#471—1271nomaximal, Sylow
139C127C2×#47111yescharacteristic, complemented, Sylow
140C2×#137—1271no—
141C2×AGL(1,ℤ/8)D12711yescharacteristic
142C22×SD16—1271no—
143C2×#131—1271no—
144#471—1278no—
145C2×#108—1271no—
146C2×#101—1271no—
147C2×#100—1271no—
148C2×#41C2×D12711yescharacteristic
149C22×Q8—1271no—
150C22×D4C2×D12711yescharacteristic
151#137—1274no—
152#137—2542no—
153AGL(1,ℤ/8)—22no—
154AGL(1,ℤ/8)C2×D12714yes—
155C2×Q16—2541no—
156C2×SD16—21no—
157C2×SD16—1274no—
158C2×SD16—2541no—
159C2×SD16—2541no—
160C2×SD16—2542no—
161C2×SD16—2541no—
162C2×D8—21no—
163#131—1274no—
164#131—2542no—
165C2×M16—1271no—
166C2×M16—1271no—
167C2×M16—2541no—
168C2×M16C2×D12711yescharacteristic
169C22×C8—1271no—
170C22×C8—2541no—
171#108—1274no—
172#101—1274no—
173#100—1274no—
174C24—21no—
175#41—21no—
176#41—21no—
177#41C22⋊D12712yes—
178#41C22×D12712yes—
179C2×Q8—1272no—
180C2×Q8—2541no—
181C2×Q8—2541no—
182C2×Q8—2541no—
183C2×Q8—2541no—
184C2×Q8—2541no—
185C2×Q8C22⋊D12711yescharacteristic
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4C22⋊D12711yescharacteristic
193C2×D4C22×D12712yes—
194C22×C4—21no—
195C22×C4—2541no—
196C22×C4C22×D12711yescharacteristic
197Q16—2544no—
198SD16—24no—
199SD16—2544no—
200SD16—2544no—
201SD16—2544no—
202D8—24no—
203M16—21no—
204M16—1272no—
205M16—1272no—
206M16—2544no—
207M16—2541no—
208M16—2541no—
209M16C22×D12712yes—
210C2×C8—21no—
211C2×C8—1271no—
212C2×C8—1271no—
213C2×C8—1272no—
214C2×C8—1272no—
215C2×C8—2544no—
216C2×C8—2541no—
217C2×C8—2541no—
218C23—21no—
219C23—22no—
220C23—41no—
221C23—41no—
222C23—82no—
223C23#1452324711yescharacteristic
224Q8—21no—
225Q8—5081no—
226Q8—5081no—
227Q8—2542no—
228Q8—2542no—
229Q8C2×C22⋊D12712yes—
230D4—22no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—21no—
235D4—41no—
236D4—41no—
237D4C2×C22⋊D12712yes—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—5081no—
242C2×C4—5081no—
243C2×C4—2542no—
244C2×C4D4×D12711yescharacteristic
245C2×C4D4×D12711yescharacteristic
246C2×C4C2×C22⋊D12711yescharacteristic
247C2×C4C2×C22⋊D12711yescharacteristic
248C2×C4#1452324711yescharacteristic
249C2×C4C23×D12711yescharacteristic
250C8—22no—
251C8—1272no—
252C8—1272no—
253C8—2542no—
254C8—2542no—
255C22—22no—
256C22—22no—
257C22—42no—
258C22—42no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22#1686118111yescharacteristic
263C22#1686118111yescharacteristic
264C22C2×#1452324711yescharacteristic
265C4—22no—
266C4—5082no—
267C4#1686118111yescharacteristic
268C4#1686118111yescharacteristic
269C4C2×D4×D12711yescharacteristic
270C4C22×C22⋊D12711yescharacteristic
271C2—21no—
272C2—21no—
273C2—42no—
274C2—82no—
275C2#2255013712yesdirect product factor
276C2C2×#1686118111yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#121
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127C2×#471
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C22×#14523236C2semidirect
3. C2×#16861216C2semidirect
4. #22550137C2direct (N × H)
5. C2×#16861144C2semidirect
6. C2×#16861127C2semidirect
7. C2×#16861126C2semidirect
11. #16861239C22semidirect
17. C2×#14523236C22semidirect
19. #16861216C22semidirect
25. #16861144C22semidirect
26. #16861127C22semidirect
27. #16861126C22semidirect
136. C2×C127#471semidirect
139. C127C2×#471semidirect
275. C2#22550137direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232362
C2×#168612162
#225501372
C2×#168611442
C2×#168611272
C2×#168611262
C2×AGL(1,ℤ/8)×C1272
C2×#471127