Finite Group Databasebeta

← #30651386

Subgroups — #30651386

Order 16256.

362 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 123 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#22550586C211yesmaximal, characteristic, complemented
2#22550230C212yesmaximal, complemented
3#22550222C211yesmaximal, characteristic, complemented
4#22550176C212yesmaximal, complemented
5#22550171C211yesmaximal, characteristic, complemented
6#22550088C212yesmaximal, complemented
7#22550068C211yesmaximal, characteristic
8Q8×C127⋊C8C211yesmaximal, characteristic, complemented
9C4×#14523235C211yesmaximal, characteristic, complemented
10C4×#14523233C212yesmaximal, complemented
11#559×C127C211yesmaximal, characteristic, complemented
12C2×#14523235C2211yescharacteristic
13C2×#14523233C2212yes—
14#16861209C2211yescharacteristic
15#16861201C2212yes—
16#16861187—22no—
17#16861185—21no—
18C4×D508C2211yescharacteristic
19C4×D508C2212yes—
20C4⋊C4×D127—21no—
21#16861142C2212yes—
22#16861140C2211yescharacteristic
23C4⋊(C127⋊C8)C2211yescharacteristic
24C4⋊(C127⋊C8)C2212yes—
25#16861133C2212yes—
26#16861133C2211yescharacteristic
27#16861130C2212yes—
28#16861130C2212yes—
29#16861130C2212yes—
30#16861129C2211yescharacteristic
31C4×C127⋊C8C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33#127×C127C2211yescharacteristic
34#124×C127—21no—
35#124×C127C2212yes—
36#123×C127—22no—
37C4×Q8×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127—21no—
40C4×D4×C127C2212yes—
41C2×Dic254—21no—
42C2×D508C2311yescharacteristic
43C2×C4×D127—21no—
44C2×C4×D127—22no—
45#14523235—22no—
46#14523233—24no—
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2311yescharacteristic
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127—21no—
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2312yes—
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55(C2×D127)⋊C4—22no—
56(C2×D127)⋊C4—21no—
57C2×Q8×C127D411yescharacteristic
58C2×Q8×C127C2311yescharacteristic
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127C2312yes—
62C22×C4×C127—22no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127D412yes—
65C4⋊C4×C127D411yescharacteristic
66C4⋊C4×C127C2311yescharacteristic
67C22⋊C4×C127—22no—
68C22⋊C4×C127—22no—
69C22⋊C4×C127—22no—
70C42×C127—21no—
71C42×C127C2312yes—
72C42×C127C2311yescharacteristic
73C22×D127—21no—
74Dic254—81no—
75D508—21no—
76D508#4112yes—
77C2×Dic127—21no—
78C2×Dic127—22no—
79C4×D127—42no—
80C4×D127—82no—
81C127⋊C8—22no—
82C127⋊C8—21no—
83C127⋊C8—21no—
84C23×C127—22no—
85Q8×C127—22no—
86Q8×C127—21no—
87Q8×C127—21no—
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—42no—
91D4×C127—81no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—82no—
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D412yes—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2411yescharacteristic
100C22×C127—42no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127C22×D411yescharacteristic
104C2×D127—22no—
105C2×D127—41no—
106C2×D127—41no—
107C4×C127—22no—
108C4×C127—22no—
109C4×C127—21no—
110C4×C127—41no—
111C4×C127—41no—
112C4×C127#13512yes—
113C4×C127C2×#4111yescharacteristic
114C4×C1272−1+411yescharacteristic
115Dic127—41no—
116Dic127—42no—
117C2×C127—82no—
118C2×C127#54811yescharacteristic
119C2×C127C2×#13511yescharacteristic
120C2×C127#57611yescharacteristic
121D127—42no—
122#3361—1271nomaximal, Sylow
123C127#336111yescharacteristic, complemented, Sylow
124#559D12711yescharacteristic, complemented
125#554—1271no—
126#488—1272no—
127#486—1271no—
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⋊Q8—1271no—
138#127C2×D12711yescharacteristic
139#125—1272no—
140#124—21no—
141#124C2×D12712yes—
142#123—22no—
143C22⋊Q8—2541no—
144C22⋊Q8—2542no—
145C4×Q8C2×D12711yescharacteristic
146C4×Q8C2×D12711yescharacteristic
147C4×D4—21no—
148C4×D4—1272no—
149C4×D4—1271no—
150C4×D4C2×D12712yes—
151C2×C4⋊C4—2541no—
152#107—1272no—
153#106—1271no—
154C4⋊C8—1271no—
155C4⋊C8—1272no—
156#103—1271no—
157#103—1272no—
158#102—1272no—
159#102—1272no—
160#102—1272no—
161#102—1271no—
162C4×C8—1272no—
163C4×C8—1271no—
164C2×Q8—2541no—
165C2×Q8C22⋊D12711yescharacteristic
166C2×Q8C22×D12711yescharacteristic
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—1271no—
170C2×D4C22×D12712yes—
171C22×C4—22no—
172C22×C4—2542no—
173C22×C4—2541no—
174SD16—2542no—
175D8—2544no—
176C2×C8—1271no—
177C2×C8—1271no—
178C2×C8—1272no—
179C4⋊C4—21no—
180C4⋊C4—1271no—
181C4⋊C4—1272no—
182C4⋊C4—2542no—
183C4⋊C4—2542no—
184C4⋊C4—2541no—
185C4⋊C4C22⋊D12711yescharacteristic
186C4⋊C4C22⋊D12712yes—
187C4⋊C4C22×D12711yescharacteristic
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—2541no—
192C22⋊C4—2542no—
193C42—21no—
194C42C22×D12712yes—
195C42C22×D12711yescharacteristic
196C23—22no—
197C23—2541no—
198Q8—22no—
199Q8—21no—
200Q8—21no—
201Q8—10161no—
202D4—22no—
203D4—22no—
204D4—42no—
205D4—81no—
206D4—1272no—
207D4—2541no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—82no—
211C2×C4—10162no—
212C2×C4—5082no—
213C2×C4—2541no—
214C2×C4—2542no—
215C2×C4C2×C22⋊D12712yes—
216C2×C4C2×C22⋊D12712yes—
217C2×C4C2×C22⋊D12711yescharacteristic
218C2×C4C2×C22⋊D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—2542no—
221C8—2541no—
222C8—2541no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—5081no—
227C22—5081no—
228C22—2542no—
229C22C22×C22⋊D12711yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—5081no—
236C4—5082no—
237C4#1686122912yes—
238C4#41×D12711yescharacteristic
239C4#1686125611yescharacteristic
240C2—82no—
241C2—5082no—
242C2#2255053311yescharacteristic
243C2C2×#1686122911yescharacteristic
244C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#559×C127C2
socleC22×C127C22×D4
prime radical#22550068C2
hypercenter#559D127
hypocenterC127#3361
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #22550586C2semidirect
2. #22550230C2semidirect
3. #22550222C2semidirect
4. #22550176C2semidirect
5. #22550171C2semidirect
6. #22550088C2semidirect
8. Q8×C127⋊C8C2semidirect
9. C4×#14523235C2semidirect
10. C4×#14523233C2semidirect
11. #559×C127C2semidirect
123. C127#3361semidirect
124. #559D127semidirect

Maximal subgroups

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

subgroupindex
#225505862
#225502302
#225502222
#225501762
#225501712
#225500882
#225500682
Q8×C127⋊C82
C4×#145232352
C4×#145232332
#559×C1272
#3361127