Finite Group Databasebeta

← #30648466

Subgroups — #30648466

Order 16256.

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

normal subgroups only

This group has 446 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
2C2×#16861240C211yesmaximal, characteristic, complemented
3C2×#16861240C211yesmaximal, characteristic, complemented
4#22550642C211yesmaximal, characteristic, complemented
5#22550150C214yesmaximal, complemented
6#22550142C214yesmaximal, complemented
7#22549924C211yesmaximal, characteristic, complemented
8#22549858C211yesmaximal, characteristic, complemented
9C2×#101×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C2×#14523248C2211yescharacteristic
12#16861242—21no—
13#16861242C2212yes—
14#16861240—22no—
15#16861240—22no—
16#16861240—22no—
17#16861240C2214yes—
18#16861237—21no—
19#16861237C2212yes—
20#16861231—22no—
21C2×#14523238—21no—
22C2×#14523238C2212yescomplemented
23C2×#14523235—21no—
24C2×#14523235C2212yescomplemented
25#16861217C2214yes—
26#16861215—22no—
27C2×#14523232C2211yescharacteristic
28C2×#14523232C2211yescharacteristic
29M16×D127—22no—
30#16861144C2214yescomplemented
31#16861127C2212yescomplemented
32#16861125C2212yescomplemented
33C22×Q8×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic
35C2×M16×C127C2211yescharacteristic
36#101×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#14523238—24no—
55#14523238—24no—
56#14523237—22no—
57#14523237—22no—
58#14523235—24no—
59#14523235—24no—
60D1016—22no—
61#14523232—21no—
62#14523232D412yes—
63#14523232D412yes—
64#14523232C2312yes—
65#14523231—24no—
66C2×C127⋊C8—21no—
67C2×C127⋊C8C2312yes—
68C8×D127—24no—
69C2×Q8×C127—21no—
70C2×Q8×C127—21no—
71C2×Q8×C127—21no—
72C2×Q8×C127—21no—
73C2×Q8×C127—21no—
74C2×Q8×C127C2312yes—
75C22×C4×C127—21no—
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#4111yescharacteristic
106Q8×C127—22no—
107Q8×C127—22no—
108Q8×C127—41no—
109Q8×C127—41no—
110C2×C4×C127—22no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127#4111yescharacteristic
118C2×C4×C127C2411yescharacteristic
119C8×C127—22no—
120C8×C127—22no—
121C22×C127—41no—
122C22×C127#12111yescharacteristic
123C22×C127#12111yescharacteristic
124C22×C127C2×#4111yescharacteristic
125C2×D127—22no—
126C2×D127—41no—
127C2×D127—81no—
128C2×D127—81no—
129C4×C127—42no—
130C4×C127#12111yescharacteristic
131C4×C127#12111yescharacteristic
132C4×C127C22×D411yescharacteristic
133C4×C127C22×D411yescharacteristic
134Dic127—22no—
135Dic127—41no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—21no—
139C2×C127C2×#12111yescharacteristic
140D127—42no—
141D127—81no—
142#3069—1271nomaximal, Sylow
143C127#306911yescharacteristic, complemented, Sylow
144#575—1271no—
145C2×#137—1271no—
146C2×#137—1271no—
147#568—1271no—
148#473—1274no—
149#471—1274no—
150#430—1271no—
151#413—1271no—
152C2×#101D12711yescharacteristic, complemented
153C2×#41—1271no—
154C2×#41—1271no—
155C22×Q8C2×D12711yescharacteristic
156#137—1274no—
157#137—1272no—
158#137—2541no—
159#137—2542no—
160#137—2542no—
161#137—2542no—
162AGL(1,ℤ/8)—1272no—
163AGL(1,ℤ/8)—2541no—
164#135—2542no—
165C2×Q16—1272no—
166C2×Q16—2541no—
167C2×SD16—1272no—
168C2×SD16—2541no—
169#131—1274no—
170#131—2542no—
171C2×M16—1271no—
172C2×M16—1271no—
173C2×M16—2542no—
174C2×M16C2×D12711yescharacteristic
175C2×M16C2×D12711yescharacteristic
176#108—1274no—
177#101—1272no—
178#101C2×D12714yescomplemented
179#100—1272no—
180#41—1272no—
181#41—1272no—
182#41—2541no—
183#41—2541no—
184#41—2541no—
185#41—2541no—
186#41—2541no—
187#41—2541no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—1271no—
194C2×Q8—1271no—
195C2×Q8C22×D12712yes—
196C2×D4—1271no—
197C2×D4—1271no—
198C2×D4—2541no—
199C2×D4—2541no—
200C22×C4—21no—
201C22×C4—2541no—
202C22×C4—2541no—
203C22×C4C22×D12711yescharacteristic
204Q16—2542no—
205Q16—2544no—
206Q16—2544no—
207SD16—2542no—
208SD16—2542no—
209SD16—2544no—
210SD16—2544no—
211D8—2542no—
212M16—21no—
213M16—21no—
214M16—1272no—
215M16—1272no—
216M16—1272no—
217M16—2544no—
218M16—2541no—
219M16C22×D12712yes—
220M16C22×D12712yes—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—1272no—
224C2×C8—2544no—
225C2×C8—2541no—
226C23—2541no—
227C23—2541no—
228C23#1452324811yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—41no—
232Q8—41no—
233Q8—1272no—
234Q8—2541no—
235Q8—2541no—
236Q8—2541no—
237D4—1272no—
238D4—5081no—
239D4—5081no—
240D4—2542no—
241D4—2542no—
242D4—2541no—
243D4—2541no—
244D4—2541no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—5081no—
249C2×C4—5081no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4D4×D12711yescharacteristic
255C2×C4D4×D12711yescharacteristic
256C2×C4D4×D12711yescharacteristic
257C2×C4D4×D12711yescharacteristic
258C2×C4#1452324811yescharacteristic
259C2×C4C23×D12711yescharacteristic
260C8—22no—
261C8—22no—
262C8—1274no—
263C8—2542no—
264C22—41no—
265C22—10161no—
266C22—10161no—
267C22—5081no—
268C22—2542no—
269C22#1686118211yescharacteristic
270C22#1686118211yescharacteristic
271C22C2×#1452324811yescharacteristic
272C4—42no—
273C4—5081no—
274C4—2542no—
275C4#1686118211yescharacteristic
276C4#1686118211yescharacteristic
277C4C2×D4×D12711yescharacteristic
278C4C2×D4×D12711yescharacteristic
279C2—21no—
280C2—21no—
281C2—21no—
282C2—10161no—
283C2—5082no—
284C2C2×#1686118211yescharacteristic
subgroupG/H
centerC2C2×#16861182
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#101×C127C2
socleC2×C127C2×#121
prime radical#22550676C2
hypercenterC2×#101D127
hypocenterC127#3069
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. C2×#16861240C2semidirect
4. #22550642C2semidirect
5. #22550150C2semidirect
6. #22550142C2semidirect
7. #22549924C2semidirect
8. #22549858C2semidirect
9. C2×#101×C127C2semidirect
22. C2×#14523238C22semidirect
24. C2×#14523235C22semidirect
30. #16861144C22semidirect
31. #16861127C22semidirect
32. #16861125C22semidirect
36. #101×C127C22semidirect
143. C127#3069semidirect
152. C2×#101D127semidirect
178. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
C2×#168612402
#225506422
#225501502
#225501422
#225499242
#225498582
C2×#101×C1272
#3069127