Finite Group Databasebeta

← #30650484

Subgroups — #30650484

Order 16256.

504 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16861232C211yesmaximal, characteristic, complemented
2#22550545C212yesmaximal, complemented
3#22550251C211yesmaximal, characteristic, complemented
4#22550112C211yesmaximal, characteristic, complemented
5#22550109C211yesmaximal, characteristic, complemented
6#22550074C212yesmaximal, complemented
7#22550021C212yesmaximal, complemented
8#22550020C212yesmaximal, complemented
9Q16×Dic127C211yesmaximal, characteristic, complemented
10C8×C22⋊D127C211yesmaximal, characteristic, complemented
11C2×#135×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13#16861232—24no—
14C2×#14523235C2212yes—
15C2×D1016C2211yescharacteristic
16C2×#14523233C2212yes—
17C2×C8×D127C2211yescharacteristic
18#16861205—22no—
19#16861205C2212yes—
20#16861183—22no—
21#16861183C2212yes—
22#16861180—22no—
23#16861180C2212yes—
24Q8×Dic127C2212yes—
25C4×C22⋊D127—22no—
26C4×C22⋊D127C2211yescharacteristic
27#16861143C2211yescharacteristic
28Dic127⋊C8C2211yescharacteristic
29#16861133C2212yes—
30#16861131C2212yes—
31#16861130C2212yes—
32C22⋊(C127⋊C8)C2211yescharacteristic
33(C2×D127)⋊C8C2211yescharacteristic
34C8×Dic127C2211yescharacteristic
35C2×#41×C127C2212yes—
36#135×C127—24no—
37C2×Q16×C127C2211yescharacteristic
38C2×SD16×C127C2212yes—
39C2×D8×C127C2211yescharacteristic
40C22×C8×C127C2211yescharacteristic
41#14523249—24no—
42#14523249—42no—
43C2×D508—22no—
44C2×D508C2312yes—
45C2×C22⋊D127—22no—
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—22no—
49C2×C22⋊D127D411yescharacteristic
50C2×C4×D127—22no—
51C2×C4×D127C2311yescharacteristic
52#14523235—24no—
53D1016—22no—
54#14523233—42no—
55C2×C127⋊C8C2311yescharacteristic
56C2×C127⋊C8C2311yescharacteristic
57C8×D127—22no—
58C4⋊Dic127—22no—
59C4⋊Dic127C2312yes—
60Dic127⋊C4D411yescharacteristic
61C22⋊Dic127—22no—
62C22⋊Dic127D411yescharacteristic
63(C2×D127)⋊C4—22no—
64(C2×D127)⋊C4D411yescharacteristic
65C4×Dic127—22no—
66C4×Dic127C2311yescharacteristic
67#41×C127—24no—
68#41×C127—42no—
69C2×Q8×C127C2312yes—
70C2×D4×C127—22no—
71C2×D4×C127C2312yes—
72C22×C4×C127—22no—
73C22×C4×C127C2311yescharacteristic
74Q16×C127D414yescomplemented
75SD16×C127—24no—
76D8×C127—22no—
77C2×C8×C127—22no—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C22×D127—22no—
81C22×D127C2×D411yescharacteristic
82D508—22no—
83D508—22no—
84D508—44no—
85C22⋊D127—22no—
86C22⋊D127—44no—
87C22⋊D127—82no—
88C22⋊D127—82no—
89C22⋊D127—82no—
90C2×Dic127—22no—
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C2×Dic127C2×D411yescharacteristic
94C4×D127—21no—
95C4×D127—21no—
96C4×D127—44no—
97C127⋊C8—21no—
98C127⋊C8—21no—
99C23×C127—22no—
100C23×C127C2×D411yescharacteristic
101Q8×C127—22no—
102Q8×C127C2×D414yes—
103D4×C127—22no—
104D4×C127—22no—
105D4×C127—44no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—44no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C8×C127—21no—
113C8×C127C2×D412yes—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127—21no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127C22×D411yescharacteristic
121C2×D127—21no—
122C2×D127—21no—
123C2×D127—21no—
124C2×D127—42no—
125C2×D127—42no—
126C2×D127—42no—
127C4×C127—21no—
128C4×C127—24no—
129C4×C127C22×D411yescharacteristic
130C4×C127H(5,𝔽2)11yescharacteristic
131Dic127—21no—
132Dic127—21no—
133Dic127—42no—
134Dic127#13512yescomplemented
135C2×C127—41no—
136C2×C127—82no—
137C2×C127D4211yescharacteristic
138C2×C127C2×#13511yescharacteristic
139C2×C127#57611yescharacteristic
140D127—41no—
141D127—82no—
142#3287—1271nomaximal, Sylow
143C127#328711yescharacteristic, complemented, Sylow
144C2×#135—1271no—
145C2×#135D12711yescharacteristic, complemented
146#550—1272no—
147#493—1271no—
148#466—1271no—
149#466—1271no—
150#460—1272no—
151#449—1272no—
152#449—1272no—
153C4×Q16—1271no—
154C8×D4—1271no—
155C2×#41—1272no—
156C2×#41C2×D12712yes—
157#135—24no—
158#135—2544no—
159C2×Q16C2×D12711yescharacteristic
160C2×SD16—1272no—
161C2×SD16C2×D12712yes—
162C2×D8—1272no—
163C2×D8—1271no—
164C2×D8C2×D12711yescharacteristic
165C22×C8—1271no—
166C22×C8C2×D12711yescharacteristic
167#127—1272no—
168#127—2542no—
169#121—1272no—
170#121—1272no—
171#121—2542no—
172#121—2542no—
173C4×Q8—1272no—
174C4×D4—1271no—
175C4×D4—2542no—
176#107—1271no—
177C4⋊C8—1271no—
178#103—1272no—
179#102—1272no—
180#102—1272no—
181C22⋊C8—1271no—
182C22⋊C8—1271no—
183C4×C8—1271no—
184#41—24no—
185#41—42no—
186#41—5082no—
187#41—2544no—
188C2×Q8C22×D12712yes—
189C2×D4—22no—
190C2×D4—1271no—
191C2×D4—1272no—
192C2×D4—2542no—
193C2×D4—2542no—
194C2×D4—2542no—
195C2×D4—2542no—
196C2×D4—2542no—
197C2×D4C22×D12712yes—
198C22×C4—22no—
199C22×C4—1271no—
200C22×C4—2542no—
201C22×C4C22×D12711yescharacteristic
202Q16C22⋊D12714yescomplemented
203SD16—24no—
204SD16—2544no—
205D8—22no—
206D8—5082no—
207D8—2542no—
208C2×C8—22no—
209C2×C8—1271no—
210C2×C8—1271no—
211C2×C8—2542no—
212C2×C8C22×D12711yescharacteristic
213C2×C8C22×D12711yescharacteristic
214C4⋊C4—1271no—
215C4⋊C4—1272no—
216C4⋊C4—2542no—
217C22⋊C4—1271no—
218C22⋊C4—1271no—
219C22⋊C4—2542no—
220C22⋊C4—2542no—
221C42—1271no—
222C42—2542no—
223C23—22no—
224C23—1271no—
225C23—2542no—
226C23D4×D12711yescharacteristic
227Q8—22no—
228Q8C2×C22⋊D12714yes—
229D4—22no—
230D4—22no—
231D4—44no—
232D4—10162no—
233D4—10162no—
234D4—10162no—
235D4—5084no—
236D4—5084no—
237D4—2542no—
238D4—2542no—
239D4—2542no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—44no—
244C2×C4—1271no—
245C2×C4—1271no—
246C2×C4—1271no—
247C2×C4—5084no—
248C2×C4—2542no—
249C2×C4—2541no—
250C2×C4—2541no—
251C2×C4D4×D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—21no—
254C8—2541no—
255C8—2541no—
256C8C2×C22⋊D12712yes—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—5082no—
264C22—5082no—
265C22—5082no—
266C22—2541no—
267C22—2541no—
268C22—2541no—
269C22C2×D4×D12711yescharacteristic
270C4—21no—
271C4—24no—
272C4—1272no—
273C4—5082no—
274C4—2541no—
275C4—2541no—
276C4C22×C22⋊D12711yescharacteristic
277C4#1686125411yescharacteristic
278C2—41no—
279C2—82no—
280C2—10162no—
281C2—5081no—
282C2D4×C22⋊D12711yescharacteristic
283C2#135×D12711yescharacteristic
284C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#135×C127C2
socleC22×C127C22×D4
hypercenterC2×#135D127
hypocenterC127#3287
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. #22550545C2semidirect
3. #22550251C2semidirect
4. #22550112C2semidirect
5. #22550109C2semidirect
6. #22550074C2semidirect
7. #22550021C2semidirect
8. #22550020C2semidirect
9. Q16×Dic127C2semidirect
10. C8×C22⋊D127C2semidirect
11. C2×#135×C127C2semidirect
74. Q16×C127D4semidirect
134. Dic127#135semidirect
143. C127#3287semidirect
145. C2×#135D127semidirect
202. Q16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612322
#225505452
#225502512
#225501122
#225501092
#225500742
#225500212
#225500202
Q16×Dic1272
C8×C22⋊D1272
C2×#135×C1272
#3287127