Finite Group Databasebeta

← C2×#22550144 #30648435

Subgroups — C2×#22550144 #30648435

Order 16256.

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

normal subgroups only

This group has 488 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×#16861234C211yesmaximal, characteristic, complemented
2C22×#14523233C211yesmaximal, characteristic, complemented
3C2×#16861216C211yesmaximal, characteristic, complemented
4#22550144C218yesmaximal, direct product factor
5C2×#16861144C211yesmaximal, characteristic, complemented
6C2×#16861126C211yesmaximal, characteristic, complemented
7C2×#16861125C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10#16861234—22no—
11#16861234C2214yescomplemented
12C2×#14523235—21no—
13C2×#14523233—21no—
14C2×#14523233—21no—
15C2×#14523233—22no—
16C2×#14523233—21no—
17C2×#14523233C2214yescomplemented
18#16861216—22no—
19#16861216C2214yescomplemented
20C2×#14523232—21no—
21C2×#14523232C2211yescharacteristic
22C2×#14523232C2211yescharacteristic
23C22×C127⋊C8—21no—
24C22×C127⋊C8C2211yescharacteristic
25#16861144C2214yescomplemented
26#16861126C2214yescomplemented
27#16861125C2214yescomplemented
28C2×#41×C127C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic
30AGL(1,ℤ/8)×C127—22no—
31AGL(1,ℤ/8)×C127C2214yescomplemented
32C2×SD16×C127—21no—
33C2×D8×C127—21no—
34C2×M16×C127C2211yescharacteristic
35C23×D127—21no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508C2312yes—
42#14523235—24no—
43#14523233—24no—
44#14523233—24no—
45#14523233—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—
79C22×D127—22no—
80C22×D127—41no—
81C22×D127—41no—
82C22×D127—82no—
83D508—22no—
84D508—22no—
85D508—41no—
86D508—41no—
87C127⋊C8—22no—
88C127⋊C8—22no—
89C127⋊C8C2×D412yes—
90C127⋊C8C2×D412yes—
91C23×C127—21no—
92C23×C127—22no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127—82no—
96C23×C127#4111yescharacteristic
97Q8×C127—21no—
98Q8×C127C2×D412yes—
99D4×C127—22no—
100D4×C127—22no—
101D4×C127—21no—
102D4×C127—22no—
103D4×C127—22no—
104D4×C127—41no—
105D4×C127—41no—
106D4×C127C2×D412yes—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127#4111yescharacteristic
115C2×C4×C127C2411yescharacteristic
116C8×C127—22no—
117C22×C127—22no—
118C22×C127—22no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127—82no—
122C22×C127—82no—
123C22×C127—82no—
124C22×C127#12111yescharacteristic
125C22×C127#12111yescharacteristic
126C22×C127C2×#4111yescharacteristic
127C2×D127—42no—
128C2×D127—82no—
129C2×D127—82no—
130C2×D127—82no—
131C4×C127—22no—
132C4×C127#12111yescharacteristic
133C4×C127#12111yescharacteristic
134C4×C127C22×D411yescharacteristic
135C4×C127C22×D411yescharacteristic
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—42no—
139C2×C127—82no—
140C2×C127#47212yescomplemented
141C2×C127C2×#12111yescharacteristic
142D127—82no—
143C2×#472—1271nomaximal, Sylow
144C127C2×#47211yescharacteristic, complemented, Sylow
145C2×AGL(1,ℤ/8)—1271no—
146C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
147C22×D8—1271no—
148C2×#131—1271no—
149#472—1278no—
150C2×#108—1271no—
151C2×#100—1271no—
152C2×#100—1271no—
153C2×#41C2×D12711yescharacteristic
154C22×D4—1271no—
155C22×D4C2×D12711yescharacteristic
156AGL(1,ℤ/8)—22no—
157AGL(1,ℤ/8)—1274no—
158AGL(1,ℤ/8)—2542no—
159AGL(1,ℤ/8)C2×D12714yescomplemented
160C2×SD16—21no—
161C2×SD16—2541no—
162C2×D8—21no—
163C2×D8—1274no—
164C2×D8—2541no—
165C2×D8—2541no—
166C2×D8—2542no—
167C2×D8—2541no—
168#131—1274no—
169#131—2542no—
170C2×M16—1271no—
171C2×M16—1271no—
172C2×M16—2541no—
173C2×M16C2×D12711yescharacteristic
174C22×C8—1271no—
175C22×C8—2541no—
176#108—1274no—
177#100—1274no—
178#100—1274no—
179C24—21no—
180C24—2541no—
181#41—21no—
182#41—21no—
183#41C22⋊D12712yes—
184#41C22×D12712yes—
185C2×Q8C22⋊D12711yescharacteristic
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—1272no—
193C2×D4—2541no—
194C2×D4—2541no—
195C2×D4—2541no—
196C2×D4—2541no—
197C2×D4—2541no—
198C2×D4C22⋊D12711yescharacteristic
199C2×D4C22×D12712yes—
200C22×C4—21no—
201C22×C4C22×D12711yescharacteristic
202SD16—24no—
203SD16—2544no—
204D8—24no—
205D8—2544no—
206D8—2544no—
207D8—2544no—
208M16—21no—
209M16—1272no—
210M16—1272no—
211M16—2544no—
212M16—2541no—
213M16—2541no—
214M16C22×D12712yes—
215C2×C8—21no—
216C2×C8—1271no—
217C2×C8—1271no—
218C2×C8—1272no—
219C2×C8—1272no—
220C2×C8—2544no—
221C2×C8—2541no—
222C2×C8—2541no—
223C23—21no—
224C23—22no—
225C23—41no—
226C23—41no—
227C23—82no—
228C23—10162no—
229C23—5081no—
230C23—5081no—
231C23—2542no—
232C23#1452324711yescharacteristic
233Q8—21no—
234Q8C2×C22⋊D12712yes—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—22no—
239D4—21no—
240D4—41no—
241D4—41no—
242D4—5081no—
243D4—5081no—
244D4—2542no—
245D4—2542no—
246D4C2×C22⋊D12712yes—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4C2×C22⋊D12711yescharacteristic
253C2×C4C2×C22⋊D12711yescharacteristic
254C2×C4#1452324711yescharacteristic
255C2×C4C23×D12711yescharacteristic
256C8—22no—
257C8—1272no—
258C8—1272no—
259C8—2542no—
260C8—2542no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—10162no—
269C22—10162no—
270C22—10162no—
271C22—5082no—
272C22#1686118111yescharacteristic
273C22#1686118111yescharacteristic
274C22C2×#1452324711yescharacteristic
275C4—22no—
276C4#1686118111yescharacteristic
277C4#1686118111yescharacteristic
278C4C2×D4×D12711yescharacteristic
279C4C22×C22⋊D12711yescharacteristic
280C2—21no—
281C2—21no—
282C2—42no—
283C2—82no—
284C2—10162no—
285C2#2255014412yesdirect product factor
286C2C2×#1686118111yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#121
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127C2×#472
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×#16861234C2semidirect
2. C22×#14523233C2semidirect
3. C2×#16861216C2semidirect
4. #22550144C2direct (N × H)
5. C2×#16861144C2semidirect
6. C2×#16861126C2semidirect
7. C2×#16861125C2semidirect
8. C2×AGL(1,ℤ/8)×C127C2semidirect
11. #16861234C22semidirect
17. C2×#14523233C22semidirect
19. #16861216C22semidirect
25. #16861144C22semidirect
26. #16861126C22semidirect
27. #16861125C22semidirect
31. AGL(1,ℤ/8)×C127C22semidirect
140. C2×C127#472semidirect
144. C127C2×#472semidirect
146. C2×AGL(1,ℤ/8)D127semidirect
159. AGL(1,ℤ/8)C2×D127semidirect
285. C2#22550144direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×#145232332
C2×#168612162
#225501442
C2×#168611442
C2×#168611262
C2×#168611252
C2×AGL(1,ℤ/8)×C1272
C2×#472127