Finite Group Databasebeta

← #30651394

Subgroups — #30651394

Order 16256.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550583C211yesmaximal, characteristic, complemented
3#22550229C212yesmaximal, complemented
4#22550222C211yesmaximal, characteristic, complemented
5#22550174C212yesmaximal, complemented
6#22550172C211yesmaximal, characteristic, complemented
7#22550089C212yesmaximal, complemented
8#22550000C211yesmaximal, characteristic, complemented
9C4×#14523236C211yesmaximal, characteristic, complemented
10D8×Dic127C212yesmaximal, complemented
11#459×C127C211yesmaximal, characteristic
12C2×#14523236C2211yescharacteristic
13#16861209C2211yescharacteristic
14#16861200C2212yes—
15#16861196—21no—
16#16861196C2212yes—
17#16861190—22no—
18#16861188—21no—
19#16861186—22no—
20C4×Dic254C2211yescharacteristic
21C4×Dic254C2211yescharacteristic
22D4×Dic127—21no—
23D4×Dic127C2212yes—
24D4×Dic127C2212yes—
25C2×C4⋊Dic127—21no—
26#16861143C2212yes—
27#16861140C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861134C2212yes—
30#16861132C2211yescharacteristic
31#16861130C2212yes—
32#16861130C2211yescharacteristic
33#16861130C2212yes—
34C4×C127⋊C8C2211yescharacteristic
35C8×Dic127C2212yes—
36C2×D8×C127C2212yes—
37#127×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41C2×Dic254—21no—
42C2×Dic254D411yescharacteristic
43C2×Dic254C2311yescharacteristic
44C22×Dic127—22no—
45C22×Dic127—22no—
46#14523236—22no—
47C2×C127⋊C8C2311yescharacteristic
48C2×C127⋊C8C2311yescharacteristic
49C4⋊Dic127—22no—
50C4⋊Dic127—21no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2312yes—
53C4⋊Dic127C2311yescharacteristic
54Dic127⋊C4—22no—
55Dic127⋊C4—21no—
56Dic127⋊C4D412yes—
57C22⋊Dic127—22no—
58C22⋊Dic127—22no—
59C22⋊Dic127—22no—
60C22⋊Dic127—22no—
61C4×Dic127—21no—
62C4×Dic127C2312yes—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127C2312yes—
66C2×D4×C127C2311yescharacteristic
67C22×C4×C127—21no—
68D8×C127—24no—
69C2×C8×C127C2312yes—
70C4⋊C4×C127C2311yescharacteristic
71C22⋊C4×C127—21no—
72C42×C127C2311yescharacteristic
73Dic254—22no—
74Dic254—21no—
75Dic254—21no—
76Dic254—81no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—82no—
81C2×Dic127—82no—
82C2×Dic127C2×D412yes—
83C2×Dic127C2×D412yes—
84C127⋊C8—21no—
85C127⋊C8—21no—
86C23×C127—21no—
87C23×C127—22no—
88D4×C127—22no—
89D4×C127—21no—
90D4×C127—22no—
91D4×C127—42no—
92D4×C127—81no—
93D4×C127#4112yes—
94C2×C4×C127—21no—
95C2×C4×C127—42no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2411yescharacteristic
99C8×C127—22no—
100C22×C127—22no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—42no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127C22×D411yescharacteristic
107C4×C127—21no—
108C4×C127—41no—
109C4×C127#13512yes—
110C4×C127C2×#4111yescharacteristic
111C4×C1272−1+411yescharacteristic
112Dic127—22no—
113Dic127—22no—
114Dic127—41no—
115Dic127—41no—
116Dic127—42no—
117C2×C127—42no—
118C2×C127—82no—
119C2×C127#54811yescharacteristic
120C2×C127C2×#13511yescharacteristic
121C2×C127#57611yescharacteristic
122#3361—1271nomaximal, Sylow
123C127#336111yescharacteristic, complemented, Sylow
124#559—1271no—
125#554—1271no—
126#488—1272no—
127#486—1271no—
128#478—1272no—
129#477—1271no—
130#463—1272no—
131#459D12711yescharacteristic
132C8×Q8—1271no—
133C4×SD16—1271no—
134C4×D8—1272no—
135C2×SD16—1271no—
136C2×D8C2×D12712yes—
137C4⋊Q8—1271no—
138#127C2×D12711yescharacteristic
139#125—1272no—
140#124—1272no—
141#124—2541no—
142#123—2542no—
143C22⋊Q8—2542no—
144C22⋊Q8—2541no—
145C4×Q8—1271no—
146C4×Q8—1271no—
147C4×D4—1272no—
148C4×D4—1272no—
149C4×D4—2541no—
150C4×D4C2×D12711yescharacteristic
151C2×C4⋊C4—2541no—
152#107—1272no—
153#106—1271no—
154C4⋊C8—1272no—
155C4⋊C8C2×D12711yescharacteristic
156#103—1272no—
157#103—1271no—
158#102—1271no—
159#102—1272no—
160#102—1272no—
161#102C2×D12712yes—
162C4×C8—1272no—
163C4×C8—1271no—
164C2×Q8—1271no—
165C2×Q8—1271no—
166C2×Q8—2541no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4C22×D12712yes—
170C2×D4C22×D12711yescharacteristic
171C22×C4—21no—
172C22×C4—2542no—
173C22×C4—2542no—
174SD16—2542no—
175D8—24no—
176C2×C8—1271no—
177C2×C8—1271no—
178C2×C8C22×D12712yes—
179C4⋊C4—1272no—
180C4⋊C4—1271no—
181C4⋊C4—1272no—
182C4⋊C4—1271no—
183C4⋊C4—2542no—
184C4⋊C4—2542no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C4⋊C4C22×D12711yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C22⋊C4—2542no—
192C22⋊C4—2542no—
193C42—1272no—
194C42—2541no—
195C42C22×D12711yescharacteristic
196C23—21no—
197C23—22no—
198Q8—10161no—
199Q8—2541no—
200Q8—2541no—
201Q8—2542no—
202D4—22no—
203D4—21no—
204D4—22no—
205D4—42no—
206D4—81no—
207D4#1452324712yes—
208C2×C4—21no—
209C2×C4—42no—
210C2×C4—1272no—
211C2×C4—1272no—
212C2×C4—10162no—
213C2×C4—10162no—
214C2×C4—2541no—
215C2×C4—2541no—
216C2×C4—2542no—
217C2×C4D4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—22no—
221C8—2541no—
222C8—2541no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22C2×D4×D12711yescharacteristic
230C4—21no—
231C4—41no—
232C4—5081no—
233C4—5081no—
234C4—5082no—
235C4—2542no—
236C4—2542no—
237C4#1686122812yes—
238C4C2×#1452324711yescharacteristic
239C4#1686125611yescharacteristic
240C2—42no—
241C2—82no—
242C2#2255053211yescharacteristic
243C2C2×#1686122811yescharacteristic
244C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3361
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. #22550612C2semidirect
2. #22550583C2semidirect
3. #22550229C2semidirect
4. #22550222C2semidirect
5. #22550174C2semidirect
6. #22550172C2semidirect
7. #22550089C2semidirect
8. #22550000C2semidirect
9. C4×#14523236C2semidirect
10. D8×Dic127C2semidirect
123. C127#3361semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225505832
#225502292
#225502222
#225501742
#225501722
#225500892
#225500002
C4×#145232362
D8×Dic1272
#459×C1272
#3361127