Finite Group Databasebeta

← #30648191

Subgroups — #30648191

Order 16256.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16861235C211yesmaximal, characteristic, complemented
2#22550422C211yesmaximal, characteristic, complemented
3C2×D4×Dic127C211yesmaximal, characteristic, complemented
4#22550093C212yesmaximal, complemented
5#22550089C212yesmaximal, complemented
6#22550079C212yesmaximal, complemented
7#22550069C212yesmaximal, complemented
8#22549899C211yesmaximal, characteristic, complemented
9#22549878C211yesmaximal, characteristic, complemented
10#22549877C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C23×Dic127—21no—
14#16861235—24no—
15C2×#14523236C2212yes—
16C2×#14523233C2212yes—
17C2×#14523232C2211yescharacteristic
18#16861207C2211yescharacteristic, complemented
19#16861205C2211yescharacteristic, complemented
20#16861198C2211yescharacteristic, complemented
21#16861196C2211yescharacteristic, complemented
22#16861181—21no—
23#16861181—21no—
24D4×Dic127—21no—
25D4×Dic127—21no—
26D4×Dic127C2212yes—
27D4×Dic127C2212yes—
28C4×C22⋊D127—21no—
29C4×C22⋊D127—21no—
30C2×C4⋊Dic127C2211yescharacteristic
31C2×C22⋊Dic127—21no—
32C2×C4×Dic127C2211yescharacteristic
33Dic127⋊C8C2212yescomplemented
34Dic127⋊C8C2212yescomplemented
35#16861134C2212yescomplemented
36#16861133C2212yescomplemented
37#16861131C2212yescomplemented
38#16861130C2212yescomplemented
39C2×#41×C127C2211yescharacteristic
40C22×D4×C127C2211yescharacteristic
41AGL(1,ℤ/8)×C127—24no—
42C2×SD16×C127C2212yes—
43C2×D8×C127C2212yes—
44C2×M16×C127C2211yescharacteristic
45#14523248—22no—
46#14523248—22no—
47C2×Dic254C2311yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—21no—
50C2×C22⋊D127—21no—
51C2×C22⋊D127—21no—
52C22×Dic127—22no—
53C22×Dic127—22no—
54C22×Dic127—21no—
55C22×Dic127—44no—
56C22×Dic127D411yescharacteristic
57C22×Dic127D411yescharacteristic
58C2×C4×D127—21no—
59#14523236—24no—
60#14523233—24no—
61#14523232—22no—
62C2×C127⋊C8C2312yes—
63C4⋊Dic127—21no—
64C4⋊Dic127C2312yes—
65Dic127⋊C4—21no—
66Dic127⋊C4—21no—
67C22⋊Dic127—21no—
68C22⋊Dic127—21no—
69C22⋊Dic127—22no—
70C22⋊Dic127—22no—
71(C2×D127)⋊C4—21no—
72(C2×D127)⋊C4—21no—
73C4×Dic127D411yescharacteristic
74C4×Dic127D411yescharacteristic
75C4×Dic127C2311yescharacteristic
76C4×Dic127C2311yescharacteristic
77C24×C127—21no—
78#41×C127—22no—
79#41×C127—22no—
80C2×Q8×C127C2311yescharacteristic
81C2×D4×C127—21no—
82C2×D4×C127—21no—
83C2×D4×C127—22no—
84C2×D4×C127D414yescomplemented
85C2×D4×C127C2311yescharacteristic
86C2×D4×C127C2312yes—
87C22×C4×C127—21no—
88C22×C4×C127C2311yescharacteristic
89SD16×C127—24no—
90D8×C127—24no—
91M16×C127—22no—
92C2×C8×C127C2312yes—
93C22×D127—21no—
94Dic254—21no—
95Dic254—21no—
96D508—21no—
97D508—21no—
98C22⋊D127—42no—
99C22⋊D127—42no—
100C22⋊D127—42no—
101C2×Dic127—22no—
102C2×Dic127—21no—
103C2×Dic127—22no—
104C2×Dic127—41no—
105C2×Dic127—44no—
106C2×Dic127—42no—
107C2×Dic127—44no—
108C2×Dic127C2×D411yescharacteristic
109C2×Dic127C2×D411yescharacteristic
110C2×Dic127C2×D411yescharacteristic
111C2×Dic127C2×D411yescharacteristic
112C4×D127—42no—
113C127⋊C8—22no—
114C23×C127—24no—
115C23×C127—22no—
116C23×C127—21no—
117C23×C127—42no—
118C23×C127—42no—
119C23×C127C2×D411yescharacteristic
120Q8×C127—21no—
121Q8×C127—21no—
122D4×C127—24no—
123D4×C127—21no—
124D4×C127—22no—
125D4×C127—21no—
126D4×C127—42no—
127D4×C127C2×D414yes—
128C2×C4×C127—21no—
129C2×C4×C127—42no—
130C2×C4×C127C2×D411yescharacteristic
131C2×C4×C127C2×D412yes—
132C2×C4×C127#4112yes—
133C2×C4×C127C2411yescharacteristic
134C8×C127—22no—
135C22×C127—21no—
136C22×C127—21no—
137C22×C127—24no—
138C22×C127—48no—
139C22×C127—42no—
140C22×C127—42no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127—41no—
144C22×C127#12112yes—
145C22×C127C22×D411yescharacteristic
146C2×D127—41no—
147C2×D127—41no—
148C2×D127—41no—
149C4×C127—41no—
150C4×C127#12112yes—
151C4×C127C22×D411yescharacteristic
152C4×C127C2×#4111yescharacteristic
153Dic127—21no—
154Dic127—21no—
155Dic127—21no—
156Dic127—41no—
157Dic127—42no—
158Dic127AGL(1,ℤ/8)12yescomplemented
159C2×C127—22no—
160C2×C127—44no—
161C2×C127—81no—
162C2×C127C2×#12111yescharacteristic
163C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C127#57511yescharacteristic
165D127—81no—
166#3036—1271nomaximal, Sylow
167C127#303611yescharacteristic, complemented, Sylow
168C2×AGL(1,ℤ/8)—1271no—
169C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
170#532—1271no—
171C2×C4×D4—1271no—
172#463—1272no—
173#463—1272no—
174#461—1272no—
175#459—1272no—
176#423—1271no—
177#417—1271no—
178#417—1271no—
179C2×#41—1271no—
180C2×#41C2×D12711yescharacteristic
181C22×D4C2×D12711yescharacteristic
182C23×C4—2541no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—2544no—
185C2×SD16—1272no—
186C2×SD16C2×D12712yes—
187C2×D8—1272no—
188C2×D8C2×D12712yes—
189C2×M16—1271no—
190C2×M16C2×D12711yescharacteristic
191C4⋊Q8—1271no—
192#127—1271no—
193#124—1271no—
194#124—1271no—
195#121—2541no—
196#121—2541no—
197C4×D4—1272no—
198C4×D4—1272no—
199C4×D4—2541no—
200C4×D4—2541no—
201C4×D4—2541no—
202C4×D4—2541no—
203C2×C4⋊C4—1271no—
204C2×C22⋊C4—2541no—
205C2×C42—1271no—
206C4⋊C8—1272no—
207C4⋊C8—1272no—
208#103—1272no—
209#103—1272no—
210#102—1272no—
211#102—1272no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—2542no—
216#41—2542no—
217C2×Q8—1271no—
218C2×Q8C22×D12711yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—1271no—
223C2×D4—2541no—
224C2×D4—2541no—
225C2×D4—2541no—
226C2×D4C22⋊D12714yescomplemented
227C2×D4C22×D12711yescharacteristic
228C2×D4C22×D12712yes—
229C22×C4—21no—
230C22×C4—1271no—
231C22×C4—1271no—
232C22×C4—5084no—
233C22×C4—2542no—
234C22×C4—2542no—
235C22×C4—2541no—
236C22×C4—2541no—
237C22×C4C22×D12711yescharacteristic
238SD16—24no—
239SD16—2544no—
240D8—24no—
241D8—2544no—
242M16—22no—
243M16—2542no—
244C2×C8—1272no—
245C2×C8C22×D12712yes—
246C4⋊C4—1272no—
247C4⋊C4—2541no—
248C4⋊C4—2541no—
249C4⋊C4—2541no—
250C22⋊C4—2541no—
251C22⋊C4—2541no—
252C22⋊C4—2542no—
253C22⋊C4—2542no—
254C22⋊C4—2541no—
255C22⋊C4—2541no—
256C42—1271no—
257C42—1271no—
258C42—1271no—
259C42—1271no—
260C23—22no—
261C23—24no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—2541no—
266C23D4×D12711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2541no—
270Q8—2541no—
271D4—21no—
272D4—22no—
273D4—21no—
274D4—24no—
275D4—42no—
276D4—5082no—
277D4—5082no—
278D4—5082no—
279D4—2541no—
280D4—2541no—
281D4C2×C22⋊D12714yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—1271no—
285C2×C4—1271no—
286C2×C4—1271no—
287C2×C4—1271no—
288C2×C4—5084no—
289C2×C4—5082no—
290C2×C4—5084no—
291C2×C4—5082no—
292C2×C4—5081no—
293C2×C4—2542no—
294C2×C4—2542no—
295C2×C4—2541no—
296C2×C4D4×D12711yescharacteristic
297C2×C4C2×C22⋊D12712yes—
298C2×C4#1452324712yes—
299C2×C4C23×D12711yescharacteristic
300C8—22no—
301C8—2542no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—5081no—
312C22—5081no—
313C22—5081no—
314C22#1686118112yes—
315C22C2×D4×D12711yescharacteristic
316C4—41no—
317C4—1272no—
318C4—5081no—
319C4—5082no—
320C4—2541no—
321C4—2541no—
322C4—2541no—
323C4#1686118112yes—
324C4C22×C22⋊D12711yescharacteristic
325C4C2×#1452324711yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—10161no—
330C2C2×#1686118111yescharacteristic
331C2AGL(1,ℤ/8)×D12711yescharacteristic
332C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3036
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×#16861235C2semidirect
2. #22550422C2semidirect
3. C2×D4×Dic127C2semidirect
4. #22550093C2semidirect
5. #22550089C2semidirect
6. #22550079C2semidirect
7. #22550069C2semidirect
8. #22549899C2semidirect
9. #22549878C2semidirect
10. #22549877C2semidirect
11. C2×AGL(1,ℤ/8)×C127C2semidirect
18. #16861207C22semidirect
19. #16861205C22semidirect
20. #16861198C22semidirect
21. #16861196C22semidirect
33. Dic127⋊C8C22semidirect
34. Dic127⋊C8C22semidirect
35. #16861134C22semidirect
36. #16861133C22semidirect
37. #16861131C22semidirect
38. #16861130C22semidirect
84. C2×D4×C127D4semidirect
158. Dic127AGL(1,ℤ/8)semidirect
167. C127#3036semidirect
169. C2×AGL(1,ℤ/8)D127semidirect
226. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225504222
C2×D4×Dic1272
#225500932
#225500892
#225500792
#225500692
#225498992
#225498782
#225498772
C2×AGL(1,ℤ/8)×C1272
#3036127