Finite Group Databasebeta

← #30648417

Subgroups — #30648417

Order 16256.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861216C211yesmaximal, characteristic, complemented
3#22550454C211yesmaximal, characteristic
4#22550132C212yesmaximal, complemented
5#22550124C212yesmaximal, complemented
6#22550115C212yesmaximal, complemented
7#22550107C212yesmaximal, complemented
8#22549919C211yesmaximal, characteristic, complemented
9#22549870C211yesmaximal, characteristic, complemented
10C2×#16861133C211yesmaximal, characteristic, complemented
11#536×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861240—24no—
14C2×#14523238C2212yes—
15C2×#14523235C2212yes—
16#16861216—24no—
17C2×#14523232C2212yes—
18C2×#14523232C2211yescharacteristic
19C22×C127⋊C8C2212yes—
20C22×C127⋊C8C2211yescharacteristic
21#16861188C2212yes—
22#16861185—21no—
23#16861183C2212yes—
24#16861181—22no—
25#16861180—21no—
26C2×C4⋊Dic127C2211yescharacteristic
27#16861142C2212yes—
28#16861140C2212yes—
29#16861133C2212yes—
30#16861133C2212yes—
31#16861132C2211yescharacteristic
32#16861132C2211yescharacteristic
33#16861129C2211yescharacteristic
34#16861129C2211yescharacteristic
35C2×#41×C127C2211yescharacteristic
36C22×Q8×C127C2211yescharacteristic
37C4⋊Q8×C127—21no—
38#124×C127—21no—
39#123×C127—22no—
40C22⋊Q8×C127C2212yes—
41C22⋊Q8×C127C2212yes—
42#117×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254C2311yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C2×C22⋊D127—22no—
49C22×Dic127—21no—
50C2×C4×D127—21no—
51#14523238—24no—
52#14523235—24no—
53#14523232—24no—
54#14523232D414yescomplemented
55C2×C127⋊C8—24no—
56C2×C127⋊C8—22no—
57C2×C127⋊C8C2311yescharacteristic
58C2×C127⋊C8C2311yescharacteristic
59C2×C127⋊C8C2312yes—
60C4⋊Dic127—21no—
61C4⋊Dic127C2312yes—
62Dic127⋊C4—22no—
63C22⋊Dic127—22no—
64(C2×D127)⋊C4—22no—
65#41×C127—22no—
66#41×C127—22no—
67C2×Q8×C127—21no—
68C2×Q8×C127—22no—
69C2×Q8×C127—22no—
70C2×Q8×C127D411yescharacteristic
71C2×Q8×C127C2312yes—
72C2×D4×C127D412yes—
73C2×D4×C127D411yescharacteristic
74C22×C4×C127—21no—
75C22×C4×C127C2312yes—
76C22×C4×C127C2311yescharacteristic
77C4⋊C4×C127—22no—
78C4⋊C4×C127—22no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2311yescharacteristic
81C22⋊C4×C127—21no—
82C22⋊C4×C127—22no—
83C22⋊C4×C127—22no—
84C42×C127—21no—
85C22×D127—21no—
86Dic254—21no—
87Dic254—21no—
88D508—21no—
89D508—21no—
90C22⋊D127—42no—
91C22⋊D127—82no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—81no—
95C4×D127—42no—
96C127⋊C8—21no—
97C127⋊C8—21no—
98C127⋊C8C2×D414yes—
99C23×C127C2×D412yes—
100C23×C127C2×D411yescharacteristic
101Q8×C127—22no—
102Q8×C127—22no—
103Q8×C127—22no—
104Q8×C127—42no—
105D4×C127—24no—
106D4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—42no—
113C2×C4×C127C2×D412yes—
114C2×C4×C127C2×D412yes—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127#4112yes—
117C2×C4×C127C2411yescharacteristic
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127#12112yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—41no—
128C4×C127—22no—
129C4×C127—42no—
130C4×C127—41no—
131C4×C127—41no—
132C4×C127#12112yes—
133C4×C127C22×D411yescharacteristic
134C4×C127C2×#4111yescharacteristic
135Dic127—41no—
136Dic127—42no—
137C2×C127—22no—
138C2×C127—42no—
139C2×C127C2×#12111yescharacteristic
140C2×C127#57711yescharacteristic
141C2×C127#57811yescharacteristic
142D127—81no—
143#3063—1271nomaximal, Sylow
144C127#306311yescharacteristic, complemented, Sylow
145C2×#137—1271no—
146C2×#131—1271no—
147#537—1271no—
148#536D12711yescharacteristic, complemented
149#470—1272no—
150#468—1272no—
151#467—1272no—
152#465—1272no—
153#428—1271no—
154#416—1271no—
155C2×#103—1271no—
156C2×#41—1271no—
157C2×#41C2×D12711yescharacteristic
158C22×Q8C2×D12711yescharacteristic
159#137—2544no—
160C2×Q16—1272no—
161C2×SD16—1272no—
162#131—2544no—
163C2×M16—1271no—
164C2×M16—1272no—
165C22×C8—1271no—
166C22×C8—1272no—
167C4⋊Q8—21no—
168#124—21no—
169#123—22no—
170C22⋊Q8—1272no—
171C22⋊Q8—2541no—
172C22⋊Q8C2×D12712yes—
173C22⋊Q8C2×D12712yes—
174#121—1272no—
175#121—2541no—
176#121—2542no—
177#117C2×D12711yescharacteristic
178C2×C4⋊C4—1271no—
179#107—1272no—
180#106—1272no—
181#103—1272no—
182#103—1272no—
183#103—1271no—
184#103—1271no—
185#102—1271no—
186#102—1271no—
187#41—22no—
188#41—22no—
189#41—2542no—
190#41—2542no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1271no—
195C2×Q8C22⋊D12711yescharacteristic
196C2×Q8C22×D12712yes—
197C2×D4—1271no—
198C2×D4—2542no—
199C2×D4—2541no—
200C2×D4C22⋊D12712yes—
201C2×D4C22⋊D12711yescharacteristic
202C22×C4—21no—
203C22×C4—2541no—
204C22×C4—2541no—
205C22×C4C22×D12712yes—
206C22×C4C22×D12711yescharacteristic
207Q16—2544no—
208SD16—2544no—
209M16—1274no—
210M16—2544no—
211C2×C8—1272no—
212C2×C8—1271no—
213C2×C8—1271no—
214C2×C8—2544no—
215C2×C8—2542no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—1272no—
219C4⋊C4—2542no—
220C4⋊C4—2541no—
221C4⋊C4C22×D12711yescharacteristic
222C4⋊C4C22×D12711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—2542no—
227C22⋊C4—2542no—
228C42—21no—
229C23—2541no—
230C23C2×C22⋊D12712yes—
231C23C2×C22⋊D12711yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—22no—
235Q8—42no—
236Q8—2541no—
237Q8—2541no—
238D4—24no—
239D4—22no—
240D4—10162no—
241D4—5082no—
242D4—2541no—
243D4—2541no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—10161no—
251C2×C4—5082no—
252C2×C4—2542no—
253C2×C4—2541no—
254C2×C4D4×D12712yes—
255C2×C4C2×C22⋊D12712yes—
256C2×C4C2×C22⋊D12711yescharacteristic
257C2×C4#1452324712yes—
258C2×C4C23×D12711yescharacteristic
259C8—1274no—
260C8—2541no—
261C8—2541no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—5081no—
268C22—5081no—
269C22—5081no—
270C22#1686118112yes—
271C22C22×C22⋊D12711yescharacteristic
272C4—22no—
273C4—42no—
274C4—41no—
275C4—41no—
276C4—5081no—
277C4—5082no—
278C4#1686118112yes—
279C4C2×D4×D12711yescharacteristic
280C4C2×#1452324711yescharacteristic
281C2—22no—
282C2—42no—
283C2—10161no—
284C2C2×#1686118111yescharacteristic
285C2#2255068411yescharacteristic
286C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
prime radical#22550454C2
hypercenter#536D127
hypocenterC127#3063
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C2×#16861216C2semidirect
4. #22550132C2semidirect
5. #22550124C2semidirect
6. #22550115C2semidirect
7. #22550107C2semidirect
8. #22549919C2semidirect
9. #22549870C2semidirect
10. C2×#16861133C2semidirect
11. #536×C127C2semidirect
54. #14523232D4semidirect
144. C127#3063semidirect
148. #536D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612162
#225504542
#225501322
#225501242
#225501152
#225501072
#225499192
#225498702
C2×#168611332
#536×C1272
#3063127