Finite Group Databasebeta

← #30650463

Subgroups — #30650463

Order 16256.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16861228C212yesmaximal, complemented
2#22550546C211yesmaximal, characteristic, complemented
3#22550530C211yesmaximal, characteristic, complemented
4#22550248C211yesmaximal, characteristic, complemented
5#22550104C212yesmaximal, complemented
6#22550087C211yesmaximal, characteristic, complemented
7#22550039C212yesmaximal, complemented
8#22550018C212yesmaximal, complemented
9C4×#14523236C211yesmaximal, characteristic, complemented
10#22549941C211yesmaximal, characteristic, complemented
11#459×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yes—
13C2×#14523247C2212yes—
14#16861228—28no—
15C2×Dic508C2212yes—
16C2×#14523236C2212yes—
17C2×#14523236C2212yes—
18C2×#14523236C2211yescharacteristic
19C2×C8×D127C2212yes—
20#16861209C2211yescharacteristic
21#16861205—21no—
22#16861193—22no—
23#16861189C2212yes—
24#16861183—21no—
25#16861181—22no—
26#16861180C2212yes—
27C4×Dic254C2211yescharacteristic
28C4×D508C2211yescharacteristic
29D4×Dic127—21no—
30C2×C4⋊Dic127—21no—
31#16861140C2211yescharacteristic
32#16861134C2212yes—
33#16861132C2211yescharacteristic
34#16861130C2211yescharacteristic
35(C2×D127)⋊C8C2212yes—
36C4×C127⋊C8C2211yescharacteristic
37C2×D8×C127C2212yes—
38#127×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40C4⋊C8×C127C2211yescharacteristic
41#102×C127C2212yes—
42#14523247—24no—
43#14523247—24no—
44#14523247—42no—
45#14523247—42no—
46C2×Dic254C2312yes—
47C2×Dic254C2311yescharacteristic
48C2×D508D411yescharacteristic
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—22no—
52C22×Dic127—22no—
53C22×Dic127—22no—
54C2×C4×D127C2312yes—
55Dic508—24no—
56#14523236—24no—
57#14523236—42no—
58#14523236D414yescomplemented
59C2×C127⋊C8C2311yescharacteristic
60C2×C127⋊C8C2311yescharacteristic
61C8×D127—24no—
62C4⋊Dic127—22no—
63C4⋊Dic127—21no—
64C4⋊Dic127—21no—
65C4⋊Dic127D411yescharacteristic
66C4⋊Dic127C2311yescharacteristic
67Dic127⋊C4—21no—
68C22⋊Dic127—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4D412yes—
71C4×Dic127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—21no—
74C2×D4×C127C2312yes—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77D8×C127—24no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127C2311yescharacteristic
80C22⋊C4×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127C2×D412yes—
83Dic254—21no—
84Dic254—22no—
85Dic254—22no—
86Dic254C2×D412yes—
87D508—22no—
88C22⋊D127—44no—
89C22⋊D127—44no—
90C22⋊D127—82no—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C2×Dic127—22no—
94C2×Dic127—44no—
95C2×Dic127—44no—
96C2×Dic127C2×D412yes—
97C4×D127—22no—
98C4×D127—22no—
99C127⋊C8—21no—
100C127⋊C8C2×D412yes—
101C23×C127—21no—
102C23×C127—22no—
103D4×C127—22no—
104D4×C127—21no—
105D4×C127—22no—
106D4×C127—42no—
107D4×C127—81no—
108D4×C127C2×D412yes—
109C2×C4×C127—21no—
110C2×C4×C127—42no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2411yescharacteristic
114C8×C127—22no—
115C22×C127—22no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127C22×D411yescharacteristic
122C2×D127—22no—
123C2×D127—22no—
124C2×D127—22no—
125C4×C127—21no—
126C4×C127—41no—
127C4×C127#13512yescomplemented
128C4×C127C22×D411yescharacteristic
129C4×C127H(5,𝔽2)11yescharacteristic
130Dic127—22no—
131Dic127—22no—
132Dic127—42no—
133Dic127—41no—
134C2×C127—42no—
135C2×C127—82no—
136C2×C127D4211yescharacteristic
137C2×C127C2×#13511yescharacteristic
138C2×C127#57711yescharacteristic
139D127—42no—
140#3284—1271nomaximal, Sylow
141C127#328411yescharacteristic, complemented, Sylow
142C2×#135—1272no—
143#550—1271no—
144#547—1271no—
145#492—1271no—
146#465—1272no—
147#462—1271no—
148#459D12711yescharacteristic, complemented
149#452—1272no—
150#449—1272no—
151C4×SD16—1271no—
152C8×D4—1271no—
153C2×#41—1272no—
154C2×#41—1272no—
155#135—2548no—
156C2×Q16—1272no—
157C2×SD16—1271no—
158C2×SD16—1272no—
159C2×SD16—1272no—
160C2×D8C2×D12712yes—
161C22×C8—1272no—
162C4⋊Q8—1271no—
163#127—2541no—
164#127C2×D12711yescharacteristic
165#123—2542no—
166C22⋊Q8—1272no—
167#121—1272no—
168#121—2542no—
169#121—2541no—
170C4×Q8—1271no—
171C4×D4—1271no—
172C4×D4—2541no—
173C4×D4C2×D12711yescharacteristic
174C2×C4⋊C4—2541no—
175#106—1271no—
176C4⋊C8C2×D12711yescharacteristic
177#103—1272no—
178#103—1271no—
179#102—1271no—
180#102C2×D12712yes—
181C22⋊C8—1272no—
182C4×C8—1271no—
183#41—5082no—
184#41—5082no—
185#41—2544no—
186#41—2544no—
187C2×Q8—1271no—
188C2×Q8—1272no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—1271no—
192C2×D4—2542no—
193C2×D4—2542no—
194C2×D4—2542no—
195C2×D4C22×D12712yes—
196C2×D4C22×D12711yescharacteristic
197C22×C4—21no—
198C22×C4—1272no—
199C22×C4—2542no—
200C22×C4—2542no—
201Q16—2544no—
202SD16—1274no—
203SD16—5082no—
204SD16—2544no—
205D8—24no—
206C2×C8—1271no—
207C2×C8—1271no—
208C2×C8—2544no—
209C2×C8C22×D12712yes—
210C4⋊C4—1271no—
211C4⋊C4—1271no—
212C4⋊C4—2542no—
213C4⋊C4—2541no—
214C4⋊C4—2541no—
215C4⋊C4—2541no—
216C4⋊C4C22×D12711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1272no—
219C22⋊C4—2542no—
220C22⋊C4—2542no—
221C42—2541no—
222C42C22×D12711yescharacteristic
223C23—21no—
224C23—22no—
225C23—1272no—
226Q8—1272no—
227Q8—2542no—
228Q8—2542no—
229Q8—2541no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—81no—
235D4—10162no—
236D4—5084no—
237D4—5084no—
238D4—2542no—
239D4D4×D12712yes—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—1272no—
243C2×C4—5084no—
244C2×C4—5084no—
245C2×C4—2541no—
246C2×C4—2542no—
247C2×C4—2541no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—22no—
254C8—1272no—
255C8—2541no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—2542no—
263C22—2542no—
264C22—2542no—
265C22C2×D4×D12711yescharacteristic
266C4—21no—
267C4—41no—
268C4—5081no—
269C4—5082no—
270C4—2542no—
271C4—2542no—
272C4#1686122812yescomplemented
273C4C2×D4×D12711yescharacteristic
274C4#1686125511yescharacteristic
275C2—42no—
276C2—82no—
277C2—5082no—
278C2#2255051011yescharacteristic
279C2C2×#1686122811yescharacteristic
280C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3284
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16861228C2semidirect
2. #22550546C2semidirect
3. #22550530C2semidirect
4. #22550248C2semidirect
5. #22550104C2semidirect
6. #22550087C2semidirect
7. #22550039C2semidirect
8. #22550018C2semidirect
9. C4×#14523236C2semidirect
10. #22549941C2semidirect
11. #459×C127C2semidirect
58. #14523236D4semidirect
127. C4×C127#135semidirect
141. C127#3284semidirect
148. #459D127semidirect
272. C4#16861228semidirect

Maximal subgroups

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

subgroupindex
C2×#168612282
#225505462
#225505302
#225502482
#225501042
#225500872
#225500392
#225500182
C4×#145232362
#225499412
#459×C1272
#3284127