Finite Group Databasebeta

← #30648457

Subgroups — #30648457

Order 16256.

466 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#22550676C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C2×#16861235C211yesmaximal, characteristic, complemented
4#22550642C211yesmaximal, characteristic, complemented
5#22550145C214yesmaximal, complemented
6#22550138C214yesmaximal, complemented
7#22549924C211yesmaximal, characteristic, complemented
8#22549858C211yesmaximal, characteristic, complemented
9C2×#100×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C2×#14523248C2211yescharacteristic
12#16861242—21no—
13#16861242C2212yescomplemented
14#16861237—21no—
15#16861237C2212yescomplemented
16#16861235—22no—
17#16861235—22no—
18#16861235—22no—
19#16861235C2214yes—
20#16861231—22no—
21C2×#14523236—21no—
22C2×#14523236C2212yescomplemented
23C2×#14523233—21no—
24C2×#14523233C2212yescomplemented
25#16861217C2214yescomplemented
26#16861215—22no—
27C2×#14523232C2211yescharacteristic
28C2×#14523232C2211yescharacteristic
29M16×D127—22no—
30#16861144C2214yescomplemented
31#16861127C2212yescomplemented
32#16861125C2212yescomplemented
33C22×D4×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic
35C2×M16×C127C2211yescharacteristic
36#100×C127C2214yescomplemented
37#14523248—21no—
38#14523248—21no—
39#14523248—21no—
40#14523248—21no—
41#14523248—21no—
42#14523248—21no—
43#14523248D412yes—
44#14523248C2312yes—
45C2×Dic254D411yescharacteristic
46C2×Dic254C2311yescharacteristic
47C2×D508D411yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—21no—
50C2×C22⋊D127—21no—
51C2×C4×D127—21no—
52C2×C4×D127—21no—
53Dic508—22no—
54#14523237—22no—
55#14523237—22no—
56#14523236—24no—
57#14523236—24no—
58D1016—22no—
59#14523233—24no—
60#14523233—24no—
61#14523232—21no—
62#14523232D412yes—
63#14523232D412yes—
64#14523232C2312yes—
65#14523231—24no—
66C2×C127⋊C8—21no—
67C2×C127⋊C8C2312yes—
68C8×D127—24no—
69C24×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—21no—
74C2×D4×C127—21no—
75C2×D4×C127C2312yes—
76C22×C4×C127C2311yescharacteristic
77M16×C127—21no—
78M16×C127—21no—
79M16×C127C2312yes—
80M16×C127C2312yes—
81C2×C8×C127—21no—
82C2×C8×C127—21no—
83C22×D127—21no—
84C22×D127—21no—
85Dic254—21no—
86Dic254—21no—
87Dic254—21no—
88Dic254C2×D412yes—
89D508—21no—
90D508—21no—
91D508—21no—
92D508C2×D412yes—
93C22⋊D127—22no—
94C22⋊D127—22no—
95C22⋊D127—41no—
96C22⋊D127—41no—
97C2×Dic127—21no—
98C2×Dic127—21no—
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—41no—
102C4×D127—41no—
103C127⋊C8—22no—
104C127⋊C8C2×D414yes—
105C23×C127—22no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127—82no—
109C23×C127#4111yescharacteristic
110D4×C127—22no—
111D4×C127—22no—
112D4×C127—41no—
113D4×C127—41no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127#4111yescharacteristic
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C8×C127—22no—
122C22×C127—41no—
123C22×C127—42no—
124C22×C127—82no—
125C22×C127—82no—
126C22×C127—82no—
127C22×C127#12111yescharacteristic
128C22×C127#12111yescharacteristic
129C22×C127C2×#4111yescharacteristic
130C2×D127—22no—
131C2×D127—41no—
132C2×D127—81no—
133C2×D127—81no—
134C4×C127#12111yescharacteristic
135C4×C127#12111yescharacteristic
136C4×C127C22×D411yescharacteristic
137C4×C127C22×D411yescharacteristic
138Dic127—22no—
139Dic127—41no—
140C2×C127—21no—
141C2×C127—21no—
142C2×C127—21no—
143C2×C127—82no—
144C2×C127C2×#12111yescharacteristic
145D127—42no—
146D127—81no—
147#3068—1271nomaximal, Sylow
148C127#306811yescharacteristic, complemented, Sylow
149#575—1271no—
150C2×AGL(1,ℤ/8)—1271no—
151C2×AGL(1,ℤ/8)—1271no—
152#568—1271no—
153#472—1274no—
154#471—1274no—
155#430—1271no—
156#413—1271no—
157C2×#100D12711yescharacteristic, complemented
158C2×#41—1271no—
159C2×#41—1271no—
160C22×D4C2×D12711yescharacteristic
161#137—1272no—
162#137—2541no—
163AGL(1,ℤ/8)—1274no—
164AGL(1,ℤ/8)—1272no—
165AGL(1,ℤ/8)—2541no—
166AGL(1,ℤ/8)—2542no—
167AGL(1,ℤ/8)—2542no—
168AGL(1,ℤ/8)—2542no—
169#135—2542no—
170C2×SD16—1272no—
171C2×SD16—2541no—
172C2×D8—1272no—
173C2×D8—2541no—
174#131—1274no—
175#131—2542no—
176C2×M16—1271no—
177C2×M16—1271no—
178C2×M16—2542no—
179C2×M16C2×D12711yescharacteristic
180C2×M16C2×D12711yescharacteristic
181#108—1274no—
182#101—1272no—
183#100—1272no—
184#100C2×D12714yescomplemented
185C24—21no—
186#41—1272no—
187#41—1272no—
188#41—2541no—
189#41—2541no—
190#41—2541no—
191#41—2541no—
192#41—2541no—
193#41—2541no—
194C2×Q8—1271no—
195C2×Q8—1271no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1271no—
202C2×D4—1271no—
203C2×D4—2541no—
204C2×D4—2541no—
205C2×D4C22×D12712yes—
206C22×C4—2541no—
207C22×C4—2541no—
208C22×C4C22×D12711yescharacteristic
209Q16—2542no—
210SD16—2542no—
211SD16—2542no—
212SD16—2544no—
213SD16—2544no—
214D8—2542no—
215D8—2544no—
216D8—2544no—
217M16—21no—
218M16—21no—
219M16—1272no—
220M16—1272no—
221M16—1272no—
222M16—2544no—
223M16—2541no—
224M16C22×D12712yes—
225M16C22×D12712yes—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—1272no—
229C2×C8—2544no—
230C2×C8—2541no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—82no—
235C23—2541no—
236C23—2541no—
237C23#1452324811yescharacteristic
238Q8—1272no—
239Q8—2541no—
240Q8—2541no—
241Q8—2541no—
242D4—22no—
243D4—22no—
244D4—41no—
245D4—41no—
246D4—1272no—
247D4—5081no—
248D4—5081no—
249D4—2542no—
250D4—2542no—
251D4—2541no—
252D4—2541no—
253D4—2541no—
254C2×C4—5081no—
255C2×C4—5081no—
256C2×C4—2541no—
257C2×C4—2541no—
258C2×C4—2542no—
259C2×C4—2542no—
260C2×C4D4×D12711yescharacteristic
261C2×C4D4×D12711yescharacteristic
262C2×C4D4×D12711yescharacteristic
263C2×C4D4×D12711yescharacteristic
264C2×C4#1452324811yescharacteristic
265C2×C4C23×D12711yescharacteristic
266C8—22no—
267C8—22no—
268C8—1274no—
269C8—2542no—
270C22—41no—
271C22—42no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—10161no—
276C22—10161no—
277C22—5081no—
278C22—2542no—
279C22#1686118211yescharacteristic
280C22#1686118211yescharacteristic
281C22C2×#1452324811yescharacteristic
282C4—5081no—
283C4—2542no—
284C4#1686118211yescharacteristic
285C4#1686118211yescharacteristic
286C4C2×D4×D12711yescharacteristic
287C4C2×D4×D12711yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—82no—
292C2—10161no—
293C2—5082no—
294C2C2×#1686118211yescharacteristic
subgroupG/H
centerC2C2×#16861182
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#100×C127C2
socleC2×C127C2×#121
hypercenterC2×#100D127
hypocenterC127#3068
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. #22550676C2semidirect
2. C2×#16861235C2semidirect
3. C2×#16861235C2semidirect
4. #22550642C2semidirect
5. #22550145C2semidirect
6. #22550138C2semidirect
7. #22549924C2semidirect
8. #22549858C2semidirect
9. C2×#100×C127C2semidirect
13. #16861242C22semidirect
15. #16861237C22semidirect
22. C2×#14523236C22semidirect
24. C2×#14523233C22semidirect
25. #16861217C22semidirect
30. #16861144C22semidirect
31. #16861127C22semidirect
32. #16861125C22semidirect
36. #100×C127C22semidirect
148. C127#3068semidirect
157. C2×#100D127semidirect
184. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
C2×#168612352
#225506422
#225501452
#225501382
#225499242
#225498582
C2×#100×C1272
#3068127