Finite Group Databasebeta

← #30651395

Subgroups — #30651395

Order 16256.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550612C211yesmaximal, characteristic, complemented
2#22550587C211yesmaximal, characteristic, complemented
3#22550229C212yesmaximal, complemented
4#22550178C212yesmaximal, complemented
5#22550168C211yesmaximal, characteristic, complemented
6#22550090C212yesmaximal, complemented
7#22550068C211yesmaximal, characteristic
8C8×Dic254C211yesmaximal, characteristic, complemented
9C4×#14523237C211yesmaximal, characteristic, complemented
10#22549956C212yesmaximal, complemented
11#486×C127C211yesmaximal, characteristic, complemented
12C2×#14523237C2211yescharacteristic
13C2×#14523233C2212yes—
14#16861200C2212yes—
15#16861196—21no—
16#16861196C2212yes—
17#16861190—22no—
18#16861189—21no—
19#16861185—22no—
20C4×Dic254C2211yescharacteristic
21C4×Dic254C2211yescharacteristic
22C4×D508C2211yescharacteristic
23D4×Dic127—21no—
24D4×Dic127C2212yes—
25#16861171C2212yes—
26C4⋊C4×D127—21no—
27#16861142C2212yes—
28#16861141C2211yescharacteristic
29C4⋊(C127⋊C8)C2211yescharacteristic
30Dic127⋊C8C2212yes—
31#16861134C2211yescharacteristic
32#16861132C2212yes—
33#16861131C2211yescharacteristic
34#16861130C2212yes—
35C8×Dic127C2212yes—
36C4⋊Q8×C127C2211yescharacteristic
37#127×C127C2211yescharacteristic
38#102×C127C2212yes—
39#102×C127C2212yes—
40C4×C8×C127C2211yescharacteristic
41C2×Dic254D411yescharacteristic
42C2×Dic254C2311yescharacteristic
43C2×D508C2311yescharacteristic
44C22×Dic127—22no—
45C2×C4×D127—22no—
46C2×C4×D127—21no—
47#14523237—22no—
48#14523233—24no—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127—22no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53C4⋊Dic127C2311yescharacteristic
54Dic127⋊C4—22no—
55Dic127⋊C4—21no—
56Dic127⋊C4D412yes—
57C22⋊Dic127—22no—
58C22⋊Dic127—22no—
59C22⋊Dic127—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—21no—
62C4×Dic127—21no—
63C4×Dic127C2312yes—
64C2×Q8×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127C2312yes—
68C2×C8×C127C2311yescharacteristic
69C2×C8×C127C2311yescharacteristic
70C4⋊C4×C127—21no—
71C4⋊C4×C127C2312yes—
72C42×C127C2311yescharacteristic
73C22×D127—21no—
74Dic254—22no—
75Dic254—21no—
76Dic254—21no—
77D508—21no—
78D508#4112yes—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—82no—
83C2×Dic127C2×D412yes—
84C2×Dic127C2×D412yes—
85C4×D127—42no—
86C4×D127—82no—
87C127⋊C8—22no—
88C23×C127—22no—
89Q8×C127—81no—
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—42no—
93D4×C127—81no—
94C2×C4×C127—22no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2411yescharacteristic
98C8×C127—21no—
99C8×C127—21no—
100C22×C127—42no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127C22×D411yescharacteristic
104C2×D127—22no—
105C2×D127—41no—
106C2×D127—41no—
107C4×C127—21no—
108C4×C127—42no—
109C4×C127#13512yes—
110C4×C127C2×#4111yescharacteristic
111C4×C1272−1+411yescharacteristic
112Dic127—22no—
113Dic127—22no—
114Dic127—41no—
115Dic127—41no—
116Dic127—41no—
117C2×C127—82no—
118C2×C127#54811yescharacteristic
119C2×C127C2×#13511yescharacteristic
120C2×C127#57611yescharacteristic
121D127—42no—
122#3361—1271nomaximal, Sylow
123C127#336111yescharacteristic, complemented, Sylow
124#559—1271no—
125#554—1271no—
126#488—1272no—
127#486D12711yescharacteristic, complemented
128#478—1272no—
129#477—1271no—
130#463—1272no—
131#459—1271no—
132C8×Q8—1271no—
133C4×SD16—1271no—
134C4×D8—1272no—
135C2×SD16—1271no—
136C2×D8—1272no—
137C4⋊Q8C2×D12711yescharacteristic
138#127C2×D12711yescharacteristic
139#125—1272no—
140#124—1272no—
141#124—2541no—
142#123—2542no—
143C22⋊Q8—2541no—
144C22⋊Q8—2542no—
145C4×Q8—1271no—
146C4×Q8—1271no—
147C4×D4—1272no—
148C4×D4—1272no—
149C4×D4—1271no—
150C4×D4—2541no—
151C2×C4⋊C4—2541no—
152#107—1272no—
153#106—1271no—
154C4⋊C8—1272no—
155C4⋊C8—1271no—
156#103—1271no—
157#103—1272no—
158#102—1272no—
159#102—1271no—
160#102C2×D12712yes—
161#102C2×D12712yes—
162C4×C8—1272no—
163C4×C8C2×D12711yescharacteristic
164C2×Q8—21no—
165C2×Q8—1271no—
166C2×Q8—1271no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—1271no—
170C2×D4C22×D12712yes—
171C22×C4—2542no—
172C22×C4—2542no—
173C22×C4—2541no—
174SD16—2542no—
175D8—2544no—
176C2×C8—1272no—
177C2×C8C22×D12711yescharacteristic
178C2×C8C22×D12711yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—1272no—
181C4⋊C4—1271no—
182C4⋊C4—1271no—
183C4⋊C4—1271no—
184C4⋊C4—2542no—
185C4⋊C4—2542no—
186C4⋊C4—2541no—
187C4⋊C4C22×D12712yes—
188C22⋊C4—2542no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C22⋊C4—2542no—
192C22⋊C4—2541no—
193C42—1272no—
194C42—2541no—
195C42C22×D12711yescharacteristic
196C23—22no—
197C23—2541no—
198Q8—81no—
199Q8—2541no—
200Q8—2541no—
201Q8—2542no—
202D4—22no—
203D4—22no—
204D4—42no—
205D4—81no—
206D4—1272no—
207D4—2541no—
208C2×C4—22no—
209C2×C4—1272no—
210C2×C4—1272no—
211C2×C4—10162no—
212C2×C4—10162no—
213C2×C4—5082no—
214C2×C4—2541no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4D4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—21no—
221C8—21no—
222C8—2542no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—5081no—
227C22—5081no—
228C22—2542no—
229C22C2×D4×D12711yescharacteristic
230C4—21no—
231C4—42no—
232C4—5081no—
233C4—5081no—
234C4—5081no—
235C4—2542no—
236C4—2542no—
237C4#1686123012yes—
238C4#41×D12711yescharacteristic
239C4#1686125611yescharacteristic
240C2—82no—
241C2—5082no—
242C2#2255053411yescharacteristic
243C2C2×#1686123011yescharacteristic
244C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#486×C127C2
socleC22×C127C22×D4
prime radical#22550068C2
hypercenter#486D127
hypocenterC127#3361
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #22550612C2semidirect
2. #22550587C2semidirect
3. #22550229C2semidirect
4. #22550178C2semidirect
5. #22550168C2semidirect
6. #22550090C2semidirect
8. C8×Dic254C2semidirect
9. C4×#14523237C2semidirect
10. #22549956C2semidirect
11. #486×C127C2semidirect
123. C127#3361semidirect
127. #486D127semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225505872
#225502292
#225501782
#225501682
#225500902
#225500682
C8×Dic2542
C4×#145232372
#225499562
#486×C1272
#3361127