Finite Group Databasebeta

← #30648460

Subgroups — #30648460

Order 16256.

466 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×#16861234C211yesmaximal, characteristic, complemented
3#22550643C211yesmaximal, characteristic, complemented
4#22550144C214yesmaximal, complemented
5#22550141C214yesmaximal, complemented
6#22549924C211yesmaximal, characteristic, complemented
7#22549860C211yesmaximal, characteristic, complemented
8C2×#16861125C211yesmaximal, characteristic, complemented
9#575×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11#16861234—22no—
12#16861234—22no—
13#16861234—22no—
14#16861234C2214yes—
15C2×#14523235—21no—
16C2×#14523235C2212yescomplemented
17C2×#14523233—21no—
18C2×#14523233C2212yescomplemented
19#16861216—22no—
20#16861216C2214yescomplemented
21C2×#14523232—22no—
22C2×#14523232C2211yescharacteristic
23C2×#14523232C2211yescharacteristic
24C2×#14523232C2211yescharacteristic
25#16861144C2214yescomplemented
26#16861128C2212yescomplemented
27#16861126C2212yescomplemented
28#16861125C2214yescomplemented
29C2×#41×C127C2211yescharacteristic
30C2×#41×C127C2211yescharacteristic
31#137×C127—21no—
32#137×C127C2212yescomplemented
33AGL(1,ℤ/8)×C127—21no—
34AGL(1,ℤ/8)×C127C2212yescomplemented
35#135×C127—22no—
36C2×M16×C127C2211yescharacteristic
37C23×D127—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508—21no—
42C2×D508—21no—
43C2×D508C2312yes—
44#14523235—24no—
45#14523235—24no—
46#14523233—24no—
47#14523233—24no—
48#14523232—24no—
49#14523232—21no—
50#14523232—21no—
51#14523232D412yes—
52#14523232D412yes—
53#14523232C2312yes—
54#14523232C2312yes—
55C2×C127⋊C8—24no—
56C2×C127⋊C8—21no—
57C2×C127⋊C8—21no—
58C2×C127⋊C8C2312yes—
59#41×C127—21no—
60#41×C127—21no—
61#41×C127—21no—
62#41×C127—21no—
63#41×C127—21no—
64#41×C127—21no—
65#41×C127D412yes—
66#41×C127C2312yes—
67C2×Q8×C127D411yescharacteristic
68C2×Q8×C127C2311yescharacteristic
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127D411yescharacteristic
72C2×D4×C127C2311yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76Q16×C127—22no—
77SD16×C127—22no—
78SD16×C127—22no—
79D8×C127—22no—
80M16×C127—21no—
81M16×C127C2312yes—
82C2×C8×C127—21no—
83C22×D127—22no—
84C22×D127—41no—
85C22×D127—41no—
86C22×D127—82no—
87D508—22no—
88D508—22no—
89D508—41no—
90D508—41no—
91C127⋊C8—22no—
92C127⋊C8—22no—
93C127⋊C8C2×D414yes—
94C23×C127—21no—
95C23×C127—21no—
96C23×C127#4111yescharacteristic
97Q8×C127—21no—
98Q8×C127—21no—
99Q8×C127—21no—
100Q8×C127C2×D412yes—
101D4×C127—22no—
102D4×C127—22no—
103D4×C127—21no—
104D4×C127—21no—
105D4×C127—21no—
106D4×C127—41no—
107D4×C127—41no—
108D4×C127C2×D412yes—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—21no—
113C2×C4×C127—41no—
114C2×C4×C127—41no—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127#4111yescharacteristic
120C2×C4×C127C2411yescharacteristic
121C8×C127—22no—
122C22×C127—22no—
123C22×C127—41no—
124C22×C127—41no—
125C22×C127—81no—
126C22×C127—81no—
127C22×C127#12111yescharacteristic
128C22×C127#12111yescharacteristic
129C22×C127C2×#4111yescharacteristic
130C2×D127—42no—
131C2×D127—82no—
132C2×D127—82no—
133C2×D127—82no—
134C4×C127—22no—
135C4×C127—41no—
136C4×C127#12111yescharacteristic
137C4×C127#12111yescharacteristic
138C4×C127C22×D411yescharacteristic
139C4×C127C22×D411yescharacteristic
140C2×C127—21no—
141C2×C127—21no—
142C2×C127—21no—
143C2×C127—42no—
144C2×C127—81no—
145C2×C127C2×#12111yescharacteristic
146D127—82no—
147#3068—1271nomaximal, Sylow
148C127#306811yescharacteristic, complemented, Sylow
149#575D12711yescharacteristic, complemented
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×#100—1271no—
158C2×#41C2×D12711yescharacteristic
159C2×#41C2×D12711yescharacteristic
160C22×D4—1271no—
161#137—21no—
162#137C2×D12712yescomplemented
163AGL(1,ℤ/8)—21no—
164AGL(1,ℤ/8)—1274no—
165AGL(1,ℤ/8)—2542no—
166AGL(1,ℤ/8)—2542no—
167AGL(1,ℤ/8)—2542no—
168AGL(1,ℤ/8)C2×D12712yescomplemented
169#135—22no—
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—1271no—
179C2×M16—2542no—
180C2×M16C2×D12711yescharacteristic
181#108—1274no—
182#101—1272no—
183#100—1272no—
184#100—1274no—
185C24—2541no—
186#41—21no—
187#41—21no—
188#41—21no—
189#41—21no—
190#41—21no—
191#41—21no—
192#41C22⋊D12712yes—
193#41C22×D12712yes—
194C2×Q8C22⋊D12711yescharacteristic
195C2×Q8C22×D12711yescharacteristic
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—1272no—
199C2×D4—2541no—
200C2×D4—2541no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2541no—
204C2×D4C22⋊D12711yescharacteristic
205C2×D4C22×D12711yescharacteristic
206C22×C4—21no—
207C22×C4—21no—
208C22×C4C22×D12711yescharacteristic
209Q16—22no—
210SD16—22no—
211SD16—22no—
212SD16—2544no—
213SD16—2544no—
214D8—22no—
215D8—2544no—
216D8—2544no—
217M16—21no—
218M16—1272no—
219M16—1272no—
220M16—1272no—
221M16—1272no—
222M16—2544no—
223M16—2541no—
224M16—2541no—
225M16C22×D12712yes—
226C2×C8—21no—
227C2×C8—1272no—
228C2×C8—2544no—
229C2×C8—2541no—
230C2×C8—2541no—
231C23—21no—
232C23—21no—
233C23—10162no—
234C23—5081no—
235C23—5081no—
236C23—2542no—
237C23#1452324711yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—21no—
241Q8C2×C22⋊D12712yes—
242D4—22no—
243D4—22no—
244D4—21no—
245D4—21no—
246D4—21no—
247D4—41no—
248D4—41no—
249D4—5081no—
250D4—5081no—
251D4—2542no—
252D4—2542no—
253D4C2×C22⋊D12712yes—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4D4×D12711yescharacteristic
261C2×C4D4×D12711yescharacteristic
262C2×C4C2×C22⋊D12711yescharacteristic
263C2×C4C2×C22⋊D12711yescharacteristic
264C2×C4#1452324711yescharacteristic
265C2×C4C23×D12711yescharacteristic
266C8—22no—
267C8—1274no—
268C8—2542no—
269C8—2542no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22—81no—
275C22—10162no—
276C22—10162no—
277C22—10162no—
278C22—5082no—
279C22#1686118111yescharacteristic
280C22#1686118111yescharacteristic
281C22C2×#1452324711yescharacteristic
282C4—22no—
283C4—41no—
284C4#1686118111yescharacteristic
285C4#1686118111yescharacteristic
286C4C2×D4×D12711yescharacteristic
287C4C22×C22⋊D12711yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—42no—
292C2—81no—
293C2—10162no—
294C2C2×#1686118111yescharacteristic
subgroupG/H
centerC2C2×#16861181
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#575×C127C2
socleC2×C127C2×#121
hypercenter#575D127
hypocenterC127#3068
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861234C2semidirect
3. #22550643C2semidirect
4. #22550144C2semidirect
5. #22550141C2semidirect
6. #22549924C2semidirect
7. #22549860C2semidirect
8. C2×#16861125C2semidirect
9. #575×C127C2semidirect
16. C2×#14523235C22semidirect
18. C2×#14523233C22semidirect
20. #16861216C22semidirect
25. #16861144C22semidirect
26. #16861128C22semidirect
27. #16861126C22semidirect
28. #16861125C22semidirect
32. #137×C127C22semidirect
34. AGL(1,ℤ/8)×C127C22semidirect
148. C127#3068semidirect
149. #575D127semidirect
162. #137C2×D127semidirect
168. AGL(1,ℤ/8)C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
#225506432
#225501442
#225501412
#225499242
#225498602
C2×#168611252
#575×C1272
#3068127