Finite Group Databasebeta

← #30651350

Subgroups — #30651350

Order 16256.

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

normal subgroups only

This group has 354 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#22550583C211yesmaximal, characteristic, complemented
2#22550279C211yesmaximal, characteristic, complemented
3#22550217C212yesmaximal, complemented
4#22550187C212yesmaximal, complemented
5#22550168C211yesmaximal, characteristic, complemented
6#22550166C211yesmaximal, characteristic, complemented
7#22550120C212yesmaximal, complemented
8C4×#14523236C211yesmaximal, characteristic, complemented
9#22549954C211yesmaximal, characteristic, complemented
10#22549919C212yesmaximal, complemented
11#555×C127C211yesmaximal, characteristic
12C2×#14523236C2211yescharacteristic
13C2×#14523232C2212yes—
14#16861209C2211yescharacteristic
15#16861200—21no—
16#16861188C2212yes—
17#16861186—22no—
18C4×Dic254C2211yescharacteristic
19D4×Dic127—21no—
20C2×C4⋊Dic127C2212yes—
21#16861142C2212yes—
22#16861142C2212yes—
23#16861142C2212yes—
24#16861140C2212yes—
25#16861140C2211yescharacteristic
26C4⋊(C127⋊C8)C2211yescharacteristic
27#16861132C2212yes—
28#16861132C2211yescharacteristic
29#16861130C2212yes—
30#16861130C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127C2211yescharacteristic
34#124×C127—22no—
35#121×C127—21no—
36#121×C127C2212yes—
37C4×D4×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39#117×C127—21no—
40#117×C127C2212yes—
41C2×Dic254—21no—
42C2×Dic254C2311yescharacteristic
43C22×Dic127—22no—
44#14523236—22no—
45#14523232—24no—
46C2×C127⋊C8C2312yes—
47C2×C127⋊C8C2311yescharacteristic
48C2×C127⋊C8C2311yescharacteristic
49C4⋊Dic127—22no—
50C4⋊Dic127—21no—
51C4⋊Dic127C2312yes—
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—21no—
54Dic127⋊C4—22no—
55C22⋊Dic127—22no—
56C4×Dic127—21no—
57C2×Q8×C127—21no—
58C2×D4×C127—22no—
59C2×D4×C127D411yescharacteristic
60C2×D4×C127C2311yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127C2312yes—
63C4⋊C4×C127—21no—
64C4⋊C4×C127D411yescharacteristic
65C4⋊C4×C127C2311yescharacteristic
66C4⋊C4×C127C2312yes—
67C22⋊C4×C127—22no—
68C22⋊C4×C127—22no—
69C22⋊C4×C127—21no—
70C22⋊C4×C127D412yes—
71C42×C127—22no—
72C42×C127C2311yescharacteristic
73Dic254—21no—
74Dic254—21no—
75Dic254—81no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—22no—
79C2×Dic127—82no—
80C127⋊C8—22no—
81C127⋊C8—21no—
82C127⋊C8#4112yes—
83C23×C127—21no—
84C23×C127C2×D412yes—
85Q8×C127—81no—
86D4×C127—22no—
87D4×C127—21no—
88D4×C127—21no—
89D4×C127—82no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—42no—
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2411yescharacteristic
99C22×C127—22no—
100C22×C127—22no—
101C22×C127—22no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127C22×D411yescharacteristic
106C4×C127—22no—
107C4×C127—21no—
108C4×C127—41no—
109C4×C127—42no—
110C4×C127#13712yes—
111C4×C127C2×#4111yescharacteristic
112C4×C1272−1+411yescharacteristic
113Dic127—41no—
114Dic127—41no—
115Dic127—42no—
116C2×C127—42no—
117C2×C127—81no—
118C2×C127#54711yescharacteristic
119C2×C127C2×#13711yescharacteristic
120C2×C127#57611yescharacteristic
121#3357—1271nomaximal, Sylow
122C127#335711yescharacteristic, complemented, Sylow
123#555D12711yescharacteristic
124#554—1271no—
125#500—1271no—
126#485—1272no—
127#480—1272no—
128#477—1271no—
129#476—1271no—
130#468—1272no—
131C4×SD16—1271no—
132#436—1271no—
133#428—1272no—
134C2×SD16—1271no—
135C2×M16—1272no—
136C4⋊Q8—1271no—
137C4⋊Q8C2×D12711yescharacteristic
138#125—2541no—
139#124—22no—
140C22⋊Q8—1272no—
141C22⋊Q8—2542no—
142#121—21no—
143#121C2×D12712yes—
144C4×Q8—1271no—
145C4×D4—2541no—
146C4×D4C2×D12711yescharacteristic
147C4×D4C2×D12711yescharacteristic
148#117—21no—
149#117C2×D12712yes—
150C2×C4⋊C4—1272no—
151#107—1272no—
152#107—1272no—
153#107—1272no—
154#106—1271no—
155#106—1272no—
156C4⋊C8—1271no—
157#103—1271no—
158#103—1272no—
159#102—1271no—
160#102—1272no—
161C22⋊C8—1272no—
162C4×C8—1271no—
163C2×Q8—21no—
164C2×Q8—1271no—
165C2×Q8—2541no—
166C2×D4—22no—
167C2×D4C22⋊D12711yescharacteristic
168C2×D4C22×D12711yescharacteristic
169C22×C4—21no—
170C22×C4—2542no—
171C22×C4C22×D12712yes—
172SD16—2542no—
173M16—2544no—
174C2×C8—1271no—
175C2×C8—1271no—
176C2×C8—1272no—
177C4⋊C4—21no—
178C4⋊C4—1271no—
179C4⋊C4—1272no—
180C4⋊C4—2541no—
181C4⋊C4—2542no—
182C4⋊C4—2542no—
183C4⋊C4—2541no—
184C4⋊C4C22⋊D12711yescharacteristic
185C4⋊C4C22×D12711yescharacteristic
186C4⋊C4C22×D12712yes—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—21no—
190C22⋊C4—2542no—
191C22⋊C4C22⋊D12712yes—
192C42—22no—
193C42—2541no—
194C42C22×D12711yescharacteristic
195C23—21no—
196C23C2×C22⋊D12712yes—
197Q8—81no—
198Q8—10161no—
199Q8—2541no—
200Q8—2541no—
201D4—22no—
202D4—21no—
203D4—21no—
204D4—82no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—10162no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4—2542no—
214C2×C4C2×C22⋊D12712yes—
215C2×C4C2×C22⋊D12711yescharacteristic
216C2×C4C2×C22⋊D12711yescharacteristic
217C2×C4C23×D12711yescharacteristic
218C8—1272no—
219C8—2542no—
220C8—2541no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22C22×C22⋊D12711yescharacteristic
228C4—21no—
229C4—22no—
230C4—41no—
231C4—42no—
232C4—5081no—
233C4—5082no—
234C4—5081no—
235C4#1686123912yes—
236C4#41×D12711yescharacteristic
237C4#1686125611yescharacteristic
238C2—42no—
239C2—81no—
240C2#2255052511yescharacteristic
241C2C2×#1686123911yescharacteristic
242C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#555×C127C2
socleC22×C127C22×D4
hypercenter#555D127
hypocenterC127#3357
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #22550583C2semidirect
2. #22550279C2semidirect
3. #22550217C2semidirect
4. #22550187C2semidirect
5. #22550168C2semidirect
6. #22550166C2semidirect
7. #22550120C2semidirect
8. C4×#14523236C2semidirect
9. #22549954C2semidirect
10. #22549919C2semidirect
122. C127#3357semidirect

Maximal subgroups

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

subgroupindex
#225505832
#225502792
#225502172
#225501872
#225501682
#225501662
#225501202
C4×#145232362
#225499542
#225499192
#555×C1272
#3357127