Finite Group Databasebeta

← C2×#22550047 #30648064

Subgroups — C2×#22550047 #30648064

Order 16256.

728 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861256C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C2×#16861196C211yesmaximal, characteristic, complemented
4#22550047C218yesmaximal, direct product factor
5C2×#16861136C211yesmaximal, characteristic, complemented
6C2×#16861135C211yesmaximal, characteristic, complemented
7C2×#16861127C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
9#16861256—22no—
10#16861256—22no—
11#16861256—22no—
12#16861256C2214yescomplemented
13C2×#14523248—21no—
14C2×#14523248C2211yescharacteristic
15C2×#14523247—22no—
16C2×#14523247—21no—
17C22×Dic254—21no—
18C22×Dic254C2211yescharacteristic
19C2×Q8×D127—21no—
20#16861235—22no—
21#16861235C2214yescomplemented
22C2×#14523236—21no—
23C2×#14523233—21no—
24C2×#14523232C2211yescharacteristic
25#16861196—22no—
26#16861196C2214yescomplemented
27C2×C22⋊Dic127—22no—
28C2×C4×Dic127C2211yescharacteristic
29#16861136C2214yescomplemented
30#16861135C2214yescomplemented
31#16861127C2214yescomplemented
32C2×#41×C127C2211yescharacteristic
33C22×D4×C127C2211yescharacteristic
34AGL(1,ℤ/8)×C127—22no—
35AGL(1,ℤ/8)×C127C2214yescomplemented
36C2×SD16×C127—21no—
37C2×D8×C127—21no—
38C2×M16×C127C2211yescharacteristic
39#14523248—24no—
40#14523248—24no—
41#14523248—21no—
42#14523248—21no—
43#14523248D412yes—
44#14523248C2312yes—
45#14523247—28no—
46#14523247—24no—
47#14523247—28no—
48#14523247—24no—
49C2×Dic254—22no—
50C2×Dic254—24no—
51C2×Dic254—24no—
52C2×Dic254—21no—
53C2×Dic254—21no—
54C2×Dic254—21no—
55C2×Dic254—21no—
56C2×Dic254D411yescharacteristic
57C2×Dic254D412yes—
58C2×Dic254C2312yes—
59Q8×D127—24no—
60Q8×D127—24no—
61C2×D508D411yescharacteristic
62C2×C22⋊D127—22no—
63C2×C22⋊D127—21no—
64C22×Dic127—22no—
65C22×Dic127—21no—
66C22×Dic127D412yes—
67C2×C4×D127—22no—
68C2×C4×D127—21no—
69#14523236—24no—
70#14523233—24no—
71#14523232—21no—
72#14523232C2312yes—
73C2×C127⋊C8—21no—
74C22⋊Dic127—28no—
75C4×Dic127—21no—
76C4×Dic127C2312yes—
77C24×C127—21no—
78#41×C127—21no—
79#41×C127—21no—
80#41×C127D412yes—
81#41×C127C2312yes—
82C2×Q8×C127D411yescharacteristic
83C2×D4×C127—21no—
84C2×D4×C127—21no—
85C2×D4×C127—21no—
86C2×D4×C127—21no—
87C2×D4×C127—21no—
88C2×D4×C127—21no—
89C2×D4×C127D411yescharacteristic
90C2×D4×C127C2312yes—
91C22×C4×C127—21no—
92C22×C4×C127C2311yescharacteristic
93SD16×C127—24no—
94D8×C127—24no—
95M16×C127—21no—
96M16×C127C2312yes—
97C2×C8×C127—21no—
98C22×D127—21no—
99Dic254—28no—
100Dic254—22no—
101Dic254—21no—
102Dic254—22no—
103Dic254—22no—
104Dic254—22no—
105Dic254C2×D412yes—
106D508—21no—
107D508C2×D412yes—
108C22⋊D127—28no—
109C22⋊D127—22no—
110C22⋊D127—22no—
111C2×Dic127—28no—
112C2×Dic127—21no—
113C2×Dic127—22no—
114C2×Dic127—22no—
115C2×Dic127—22no—
116C2×Dic127—42no—
117C2×Dic127C2×D414yes—
118C4×D127—28no—
119C4×D127—22no—
120C4×D127—22no—
121C127⋊C8—22no—
122C23×C127—21no—
123C23×C127—22no—
124C23×C127—44no—
125C23×C127—41no—
126C23×C127—41no—
127C23×C127C2×D411yescharacteristic
128Q8×C127—21no—
129Q8×C127C2×D412yes—
130D4×C127—22no—
131D4×C127—22no—
132D4×C127—21no—
133D4×C127—22no—
134D4×C127—22no—
135D4×C127—41no—
136D4×C127—41no—
137D4×C127C2×D412yes—
138C2×C4×C127—21no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2411yescharacteristic
147C8×C127—22no—
148C22×C127—22no—
149C22×C127—22no—
150C22×C127—44no—
151C22×C127—42no—
152C22×C127—42no—
153C22×C127—82no—
154C22×C127—82no—
155C22×C127#12011yescharacteristic
156C22×C127#12011yescharacteristic
157C22×C127C22×D411yescharacteristic
158C2×D127—22no—
159C2×D127—42no—
160C4×C127—22no—
161C4×C127#12011yescharacteristic
162C4×C127#12011yescharacteristic
163C4×C127C22×D411yescharacteristic
164C4×C127C22×D411yescharacteristic
165Dic127—22no—
166Dic127—24no—
167Dic127—42no—
168C2×C127—21no—
169C2×C127—21no—
170C2×C127—42no—
171C2×C127—82no—
172C2×C127#45412yescomplemented
173C2×C127C2×#12011yescharacteristic
174D127—42no—
175C2×#454—1271nomaximal, Sylow
176C127C2×#45411yescharacteristic, complemented, Sylow
177C2×2−1+4—1271no—
178C2×AGL(1,ℤ/8)—1271no—
179C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
180C2×#124—1271no—
181#454—1278no—
182C2×#104—1271no—
183C2×#104—1271no—
184C2×#101—1271no—
1852−1+4—1274no—
1862−1+4—2542no—
1872−1+4—2542no—
1882−1+4—2542no—
189C2×#41—1271no—
190C2×#41—2541no—
191C2×#41—2542no—
192C2×#41—2541no—
193C2×#41C2×D12711yescharacteristic
194C22×Q8—1271no—
195C22×Q8—2541no—
196C22×Q8—2541no—
197C22×D4C2×D12711yescharacteristic
198AGL(1,ℤ/8)—22no—
199AGL(1,ℤ/8)—1274no—
200AGL(1,ℤ/8)—2542no—
201AGL(1,ℤ/8)C2×D12714yescomplemented
202C2×SD16—21no—
203C2×SD16—2541no—
204C2×D8—21no—
205C2×D8—2541no—
206C2×M16—1271no—
207C2×M16C2×D12711yescharacteristic
208#124—1274no—
209#124—2542no—
210C2×C22⋊C4—2542no—
211C2×C42—1271no—
212#104—1274no—
213#104—1274no—
214#101—1274no—
215C24—21no—
216#41—21no—
217#41—21no—
218#41—1272no—
219#41—1272no—
220#41—2544no—
221#41—2544no—
222#41—2548no—
223#41—2548no—
224#41—2544no—
225#41—2544no—
226#41—2541no—
227#41—2541no—
228#41C22⋊D12712yes—
229#41C22×D12712yes—
230C2×Q8—1272no—
231C2×Q8—1272no—
232C2×Q8—1271no—
233C2×Q8—2544no—
234C2×Q8—2542no—
235C2×Q8—2544no—
236C2×Q8—2544no—
237C2×Q8—2544no—
238C2×Q8—2541no—
239C2×Q8—2541no—
240C2×Q8—2541no—
241C2×Q8—2541no—
242C2×Q8C22⋊D12711yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—1271no—
250C2×D4—2542no—
251C2×D4—2541no—
252C2×D4C22⋊D12711yescharacteristic
253C2×D4C22×D12712yes—
254C22×C4—21no—
255C22×C4—1272no—
256C22×C4—2542no—
257C22×C4—2541no—
258C22×C4—2542no—
259C22×C4—2541no—
260C22×C4C22×D12711yescharacteristic
261SD16—24no—
262SD16—2544no—
263D8—24no—
264D8—2544no—
265M16—21no—
266M16—1272no—
267M16—2541no—
268M16C22×D12712yes—
269C2×C8—21no—
270C2×C8—2541no—
271C22⋊C4—2548no—
272C42—1272no—
273C42—2541no—
274C23—21no—
275C23—22no—
276C23—44no—
277C23—41no—
278C23—41no—
279C23—2541no—
280C23D4×D12711yescharacteristic
281Q8—21no—
282Q8—1272no—
283Q8—2548no—
284Q8—2542no—
285Q8—2542no—
286Q8—2542no—
287Q8—2542no—
288Q8—2541no—
289Q8C2×C22⋊D12712yes—
290D4—22no—
291D4—22no—
292D4—22no—
293D4—22no—
294D4—21no—
295D4—41no—
296D4—41no—
297D4—1272no—
298D4—2548no—
299D4—2542no—
300D4—2542no—
301D4—2541no—
302D4C2×C22⋊D12712yes—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—1274no—
307C2×C4—5082no—
308C2×C4—2548no—
309C2×C4—2542no—
310C2×C4—2548no—
311C2×C4—2542no—
312C2×C4—2542no—
313C2×C4—2541no—
314C2×C4—2542no—
315C2×C4—2542no—
316C2×C4D4×D12711yescharacteristic
317C2×C4D4×D12711yescharacteristic
318C2×C4D4×D12711yescharacteristic
319C2×C4C2×C22⋊D12711yescharacteristic
320C2×C4C2×C22⋊D12711yescharacteristic
321C2×C4C23×D12711yescharacteristic
322C8—22no—
323C8—2542no—
324C22—22no—
325C22—22no—
326C22—44no—
327C22—42no—
328C22—42no—
329C22—82no—
330C22—82no—
331C22—5082no—
332C22—2542no—
333C22#1686117711yescharacteristic
334C22#1686117711yescharacteristic
335C22C2×D4×D12711yescharacteristic
336C4—22no—
337C4—5082no—
338C4—2544no—
339C4—2542no—
340C4#1686117711yescharacteristic
341C4#1686117711yescharacteristic
342C4C2×D4×D12711yescharacteristic
343C4C22×C22⋊D12711yescharacteristic
344C2—21no—
345C2—21no—
346C2—42no—
347C2—82no—
348C2—5082no—
349C2#2255004712yesdirect product factor
350C2C2×#1686117711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#120
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127C2×#454
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×#16861256C2semidirect
2. C2×#16861235C2semidirect
3. C2×#16861196C2semidirect
4. #22550047C2direct (N × H)
5. C2×#16861136C2semidirect
6. C2×#16861135C2semidirect
7. C2×#16861127C2semidirect
8. C2×AGL(1,ℤ/8)×C127C2semidirect
12. #16861256C22semidirect
21. #16861235C22semidirect
26. #16861196C22semidirect
29. #16861136C22semidirect
30. #16861135C22semidirect
31. #16861127C22semidirect
35. AGL(1,ℤ/8)×C127C22semidirect
172. C2×C127#454semidirect
176. C127C2×#454semidirect
179. C2×AGL(1,ℤ/8)D127semidirect
201. AGL(1,ℤ/8)C2×D127semidirect
349. C2#22550047direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#168612352
C2×#168611962
#225500472
C2×#168611362
C2×#168611352
C2×#168611272
C2×AGL(1,ℤ/8)×C1272
C2×#454127