Finite Group Databasebeta

← #30648464

Subgroups — #30648464

Order 16256.

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

normal subgroups only

This group has 446 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
1#22550674C211yesmaximal, characteristic
2C2×#16861240C211yesmaximal, characteristic, complemented
3#22550643C211yesmaximal, characteristic, complemented
4#22550149C214yesmaximal, complemented
5#22550141C214yesmaximal, complemented
6#22549924C211yesmaximal, characteristic, complemented
7#22549858C211yesmaximal, characteristic, complemented
8C2×#16861128C211yesmaximal, characteristic, complemented
9C2×#137×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11#16861240—22no—
12#16861240—22no—
13#16861240—21no—
14#16861239C2212yes—
15#16861235—21no—
16#16861234C2212yes—
17#16861229—22no—
18C2×#14523238C2212yescomplemented
19C2×#14523235C2212yescomplemented
20#16861216—22no—
21#16861216C2214yes—
22C2×#14523232—22no—
23C2×#14523232C2211yescharacteristic
24C2×#14523232C2211yescharacteristic
25C2×#14523232C2211yescharacteristic
26#16861144C2214yescomplemented
27#16861128C2214yescomplemented
28#16861127C2212yescomplemented
29#16861125C2212yescomplemented
30C2×#41×C127C2211yescharacteristic
31C22×Q8×C127C2211yescharacteristic
32#137×C127—22no—
33#137×C127C2214yes—
34C2×Q16×C127—21no—
35C2×SD16×C127—21no—
36C2×M16×C127C2211yescharacteristic
37#14523248—21no—
38#14523248—21no—
39#14523248—21no—
40#14523248—21no—
41C2×Dic254C2311yescharacteristic
42C2×D508C2311yescharacteristic
43C2×C22⋊D127—21no—
44C2×C4×D127—21no—
45#14523238—24no—
46#14523238—22no—
47#14523236—22no—
48#14523235—24no—
49#14523235—22no—
50#14523233—22no—
51#14523232—24no—
52#14523232—21no—
53#14523232—21no—
54#14523232D412yes—
55#14523232D412yes—
56#14523232C2312yes—
57#14523232C2312yes—
58C2×C127⋊C8—24no—
59C2×C127⋊C8—21no—
60C2×C127⋊C8—21no—
61C2×C127⋊C8C2312yes—
62#41×C127—21no—
63#41×C127—21no—
64#41×C127D412yes—
65#41×C127C2312yes—
66C2×Q8×C127—21no—
67C2×Q8×C127—21no—
68C2×Q8×C127—21no—
69C2×Q8×C127—21no—
70C2×Q8×C127—21no—
71C2×Q8×C127D411yescharacteristic
72C2×Q8×C127C2312yes—
73C2×D4×C127—21no—
74C2×D4×C127D411yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78Q16×C127—24no—
79SD16×C127—24no—
80M16×C127—21no—
81M16×C127C2312yes—
82C2×C8×C127—21no—
83C22×D127—21no—
84Dic254—21no—
85Dic254—21no—
86D508—21no—
87D508—21no—
88C22⋊D127—41no—
89C22⋊D127—41no—
90C2×Dic127—21no—
91C4×D127—41no—
92C4×D127—41no—
93C127⋊C8—22no—
94C127⋊C8—22no—
95C127⋊C8C2×D414yes—
96C23×C127—21no—
97C23×C127#4111yescharacteristic
98Q8×C127—21no—
99Q8×C127—22no—
100Q8×C127—22no—
101Q8×C127—41no—
102Q8×C127—41no—
103Q8×C127C2×D412yes—
104D4×C127—22no—
105D4×C127—22no—
106D4×C127—21no—
107D4×C127C2×D412yes—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—41no—
113C2×C4×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—
121C22×C127—22no—
122C22×C127—41no—
123C22×C127—81no—
124C22×C127#12111yescharacteristic
125C22×C127#12111yescharacteristic
126C22×C127C2×#4111yescharacteristic
127C2×D127—41no—
128C2×D127—81no—
129C4×C127—22no—
130C4×C127—42no—
131C4×C127#12111yescharacteristic
132C4×C127#12111yescharacteristic
133C4×C127C22×D411yescharacteristic
134C4×C127C22×D411yescharacteristic
135Dic127—41no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—21no—
139C2×C127—42no—
140C2×C127C2×#12111yescharacteristic
141D127—81no—
142#3069—1271nomaximal, Sylow
143C127#306911yescharacteristic, complemented, Sylow
144#575—1271no—
145C2×#137—1271no—
146C2×#137D12711yescharacteristic, complemented
147#568—1271no—
148#473—1274no—
149#471—1274no—
150#430—1271no—
151#413—1271no—
152C2×#101—1271no—
153C2×#41—1271no—
154C2×#41C2×D12711yescharacteristic
155C22×Q8C2×D12711yescharacteristic
156#137—22no—
157#137—1272no—
158#137—2542no—
159#137—2542no—
160#137—2541no—
161#137C2×D12714yes—
162AGL(1,ℤ/8)—1272no—
163AGL(1,ℤ/8)—2541no—
164#135—2542no—
165C2×Q16—21no—
166C2×Q16—1272no—
167C2×SD16—21no—
168C2×SD16—1272no—
169#131—1274no—
170#131—2542no—
171C2×M16—1271no—
172C2×M16—1271no—
173C2×M16—1271no—
174C2×M16—2542no—
175C2×M16C2×D12711yescharacteristic
176#108—1274no—
177#101—1274no—
178#101—1272no—
179#100—1272no—
180#41—21no—
181#41—21no—
182#41—2541no—
183#41—2541no—
184#41—2541no—
185#41—2541no—
186#41C22⋊D12712yes—
187#41C22×D12712yes—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—1271no—
194C2×Q8C22⋊D12711yescharacteristic
195C2×Q8C22×D12712yes—
196C2×D4—21no—
197C2×D4—1271no—
198C2×D4—2541no—
199C2×D4C22⋊D12711yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—2541no—
203C22×C4C22×D12711yescharacteristic
204Q16—24no—
205Q16—2544no—
206Q16—2542no—
207SD16—24no—
208SD16—2544no—
209SD16—2542no—
210SD16—2542no—
211D8—2542no—
212M16—21no—
213M16—1272no—
214M16—1272no—
215M16—1272no—
216M16—1272no—
217M16—2544no—
218M16—2541no—
219M16—2541no—
220M16C22×D12712yes—
221C2×C8—21no—
222C2×C8—1272no—
223C2×C8—2544no—
224C2×C8—2541no—
225C2×C8—2541no—
226C23—21no—
227C23—2541no—
228C23#1452324711yescharacteristic
229Q8—21no—
230Q8—22no—
231Q8—22no—
232Q8—41no—
233Q8—41no—
234Q8—2541no—
235Q8—2541no—
236Q8C2×C22⋊D12712yes—
237D4—22no—
238D4—22no—
239D4—21no—
240D4—5081no—
241D4—5081no—
242D4—2541no—
243D4—2541no—
244D4C2×C22⋊D12712yes—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—5081no—
252C2×C4—5081no—
253C2×C4—2541no—
254C2×C4D4×D12711yescharacteristic
255C2×C4D4×D12711yescharacteristic
256C2×C4C2×C22⋊D12711yescharacteristic
257C2×C4C2×C22⋊D12711yescharacteristic
258C2×C4#1452324711yescharacteristic
259C2×C4C23×D12711yescharacteristic
260C8—22no—
261C8—1274no—
262C8—2542no—
263C8—2542no—
264C22—22no—
265C22—41no—
266C22—81no—
267C22—10161no—
268C22—5081no—
269C22#1686118111yescharacteristic
270C22#1686118111yescharacteristic
271C22C2×#1452324711yescharacteristic
272C4—22no—
273C4—42no—
274C4—5081no—
275C4#1686118111yescharacteristic
276C4#1686118111yescharacteristic
277C4C2×D4×D12711yescharacteristic
278C4C22×C22⋊D12711yescharacteristic
279C2—21no—
280C2—21no—
281C2—21no—
282C2—42no—
283C2—10161no—
284C2C2×#1686118111yescharacteristic
subgroupG/H
centerC2C2×#16861181
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC2×C127C2×#121
prime radical#22550674C2
hypercenterC2×#137D127
hypocenterC127#3069
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22550643C2semidirect
4. #22550149C2semidirect
5. #22550141C2semidirect
6. #22549924C2semidirect
7. #22549858C2semidirect
8. C2×#16861128C2semidirect
9. C2×#137×C127C2semidirect
18. C2×#14523238C22semidirect
19. C2×#14523235C22semidirect
26. #16861144C22semidirect
27. #16861128C22semidirect
28. #16861127C22semidirect
29. #16861125C22semidirect
143. C127#3069semidirect
146. C2×#137D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612402
#225506432
#225501492
#225501412
#225499242
#225498582
C2×#168611282
C2×#137×C1272
#3069127