Finite Group Databasebeta

← #30649121

Subgroups — #30649121

Order 16256.

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

normal subgroups only

This group has 424 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#22550422C211yesmaximal, characteristic
2C2×#16861208C211yesmaximal, characteristic, complemented
3#22550237C212yesmaximal, complemented
4#22550234C212yesmaximal, complemented
5#22550225C212yesmaximal, complemented
6#22550221C212yesmaximal, complemented
7#22549878C211yesmaximal, characteristic, complemented
8#22549877C211yesmaximal, characteristic, complemented
9#22549875C211yesmaximal, characteristic, complemented
10M16×Dic127C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254—21no—
14C2×#14523232C2211yescharacteristic
15#16861208—21no—
16#16861208—21no—
17#16861208C2212yes—
18#16861208C2212yes—
19#16861207C2211yescharacteristic
20#16861205C2211yescharacteristic
21#16861198C2211yescharacteristic
22#16861196C2211yescharacteristic
23#16861181—21no—
24#16861181—21no—
25C4×C22⋊D127—21no—
26C4×C22⋊D127—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×Dic127⋊C4—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861134C2212yes—
31#16861133C2212yes—
32#16861132C2212yes—
33#16861131C2212yes—
34#16861130C2212yes—
35#16861129C2212yes—
36#16861119C2212yes—
37C8×Dic127C2212yes—
38C2×#41×C127C2211yescharacteristic
39C2×M16×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#103×C127C2212yes—
42#102×C127C2212yes—
43#14523248—22no—
44#14523248—22no—
45C2×Dic254—22no—
46C2×Dic254—22no—
47C2×Dic254—44no—
48C2×Dic254C2311yescharacteristic
49C2×D508C2311yescharacteristic
50C2×C22⋊D127—21no—
51C2×C22⋊D127—21no—
52C2×C22⋊D127—21no—
53C22×Dic127—21no—
54C22×Dic127D411yescharacteristic
55C22×Dic127D411yescharacteristic
56C2×C4×D127—21no—
57#14523232—22no—
58C2×C127⋊C8C2312yes—
59C4⋊Dic127—21no—
60C4⋊Dic127C2312yes—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63Dic127⋊C4—22no—
64Dic127⋊C4—22no—
65C22⋊Dic127—21no—
66C22⋊Dic127—21no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C4×Dic127C2311yescharacteristic
73#41×C127—22no—
74#41×C127—22no—
75C2×Q8×C127C2311yescharacteristic
76C2×D4×C127—21no—
77C2×D4×C127C2311yescharacteristic
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127C2311yescharacteristic
81M16×C127—22no—
82C2×C8×C127C2312yes—
83C4⋊C4×C127—21no—
84C4⋊C4×C127C2312yes—
85C22×D127—21no—
86Dic254—21no—
87Dic254—21no—
88Dic254—44no—
89Dic254—44no—
90D508—21no—
91D508—21no—
92C22⋊D127—42no—
93C22⋊D127—42no—
94C22⋊D127—42no—
95C2×Dic127—22no—
96C2×Dic127—22no—
97C2×Dic127—21no—
98C2×Dic127—22no—
99C2×Dic127—42no—
100C2×Dic127C2×D411yescharacteristic
101C2×Dic127C2×D411yescharacteristic
102C2×Dic127C2×D411yescharacteristic
103C2×Dic127C2×D411yescharacteristic
104C4×D127—42no—
105C127⋊C8—22no—
106C23×C127—21no—
107C23×C127C2×D411yescharacteristic
108Q8×C127—21no—
109Q8×C127—21no—
110D4×C127—21no—
111D4×C127—21no—
112D4×C127—42no—
113C2×C4×C127—22no—
114C2×C4×C127—21no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127#4112yes—
119C2×C4×C127#4112yes—
120C2×C4×C127C2411yescharacteristic
121C8×C127—22no—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127#12412yes—
128C22×C127C22×D411yescharacteristic
129C2×D127—41no—
130C2×D127—41no—
131C2×D127—41no—
132C4×C127—41no—
133C4×C127—42no—
134C4×C127#12412yes—
135C4×C127C2×#4111yescharacteristic
136C4×C127C2×#4111yescharacteristic
137Dic127—21no—
138Dic127—21no—
139Dic127—41no—
140Dic127—42no—
141Dic127AGL(1,ℤ/8)12yes—
142Dic127#13712yes—
143C2×C127—22no—
144C2×C127—81no—
145C2×C127C2×#12411yescharacteristic
146C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C127C2×#13711yescharacteristic
148D127—81no—
149#3141—1271nomaximal, Sylow
150C127#314111yescharacteristic, complemented, Sylow
151#532—1271no—
152C2×C4⋊Q8—1271no—
153#489—1272no—
154#489—1272no—
155#487—1272no—
156#486—1272no—
157#417—1271no—
158#417—1271no—
159#417—1271no—
160#417D12711yescharacteristic, complemented
161C4×M16—1271no—
162C2×#41—1271no—
163C2×#41C2×D12711yescharacteristic
164C22×Q8—2541no—
165C2×M16—1271no—
166C2×M16C2×D12711yescharacteristic
167C4⋊Q8—1272no—
168C4⋊Q8—1272no—
169C4⋊Q8—1271no—
170C4⋊Q8—2541no—
171C4⋊Q8—2541no—
172#127—1271no—
173#124—1271no—
174#124—1271no—
175#121—2541no—
176#121—2541no—
177C4×D4—2541no—
178C4×D4—2541no—
179C2×C4⋊C4—1271no—
180C2×C4⋊C4—2541no—
181C2×C4⋊C4C2×D12711yescharacteristic
182C2×C42—1271no—
183#103—1272no—
184#103—1272no—
185#103—1272no—
186#103C2×D12712yes—
187#102—1272no—
188#102—1272no—
189#102—1272no—
190#102C2×D12712yes—
191#97—1272no—
192C4×C8—1272no—
193#41—22no—
194#41—22no—
195#41—2542no—
196#41—2542no—
197C2×Q8—1271no—
198C2×Q8—5084no—
199C2×Q8—2542no—
200C2×Q8—2542no—
201C2×Q8C22×D12711yescharacteristic
202C2×D4—21no—
203C2×D4—1271no—
204C2×D4—2541no—
205C2×D4—2541no—
206C2×D4—2541no—
207C2×D4C22×D12711yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—1271no—
211C22×C4—1271no—
212C22×C4—2541no—
213C22×C4—2541no—
214C22×C4C22×D12711yescharacteristic
215M16—22no—
216M16—2542no—
217C2×C8—1272no—
218C2×C8C22×D12712yes—
219C4⋊C4—21no—
220C4⋊C4—1272no—
221C4⋊C4—2542no—
222C4⋊C4—2542no—
223C4⋊C4—2541no—
224C4⋊C4—2541no—
225C4⋊C4—2541no—
226C4⋊C4C22×D12712yes—
227C22⋊C4—2541no—
228C22⋊C4—2541no—
229C22⋊C4—2541no—
230C22⋊C4—2541no—
231C42—1271no—
232C42—1271no—
233C42—1271no—
234C42—1271no—
235C23—21no—
236C23—2541no—
237C23D4×D12711yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—5084no—
241Q8—5084no—
242Q8—2541no—
243Q8—2541no—
244D4—21no—
245D4—21no—
246D4—42no—
247D4—5082no—
248D4—5082no—
249D4—5082no—
250D4—2541no—
251D4—2541no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—1271no—
257C2×C4—1271no—
258C2×C4—1271no—
259C2×C4—1271no—
260C2×C4—5082no—
261C2×C4—5082no—
262C2×C4—2542no—
263C2×C4—2542no—
264C2×C4—2541no—
265C2×C4—2542no—
266C2×C4D4×D12711yescharacteristic
267C2×C4#1452324712yes—
268C2×C4#1452324812yes—
269C2×C4C23×D12711yescharacteristic
270C8—22no—
271C8—2542no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—5081no—
278C22—5081no—
279C22—5081no—
280C22#1686119512yes—
281C22C2×D4×D12711yescharacteristic
282C4—41no—
283C4—42no—
284C4—1272no—
285C4—1272no—
286C4—5081no—
287C4—5082no—
288C4—2541no—
289C4—2541no—
290C4#1686119512yes—
291C4C2×#1452324711yescharacteristic
292C4C2×#1452324811yescharacteristic
293C2—22no—
294C2—81no—
295C2—10161no—
296C2C2×#1686119511yescharacteristic
297C2AGL(1,ℤ/8)×D12711yescharacteristic
298C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
prime radical#22550422C2
hypercenter#417D127
hypocenterC127#3141
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#16861208C2semidirect
3. #22550237C2semidirect
4. #22550234C2semidirect
5. #22550225C2semidirect
6. #22550221C2semidirect
7. #22549878C2semidirect
8. #22549877C2semidirect
9. #22549875C2semidirect
10. M16×Dic127C2semidirect
11. #417×C127C2semidirect
150. C127#3141semidirect
160. #417D127semidirect

Maximal subgroups

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

subgroupindex
#225504222
C2×#168612082
#225502372
#225502342
#225502252
#225502212
#225498782
#225498772
#225498752
M16×Dic1272
#417×C1272
#3141127