Finite Group Databasebeta

← #30651379

Subgroups — #30651379

Order 16256.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550599C211yesmaximal, characteristic, complemented
2#22550233C212yesmaximal, complemented
3#22550228C211yesmaximal, characteristic, complemented
4#22550180C212yesmaximal, complemented
5#22550163C211yesmaximal, characteristic, complemented
6#22550091C212yesmaximal
7#22550081C211yesmaximal, characteristic, complemented
8Q8×C127⋊C8C211yesmaximal, characteristic, complemented
9C4×#14523236C212yesmaximal, complemented
10C4×#14523233C211yesmaximal, characteristic
11#554×C127C211yesmaximal, characteristic, complemented
12C2×#14523236C2212yes—
13C2×#14523233C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861204—22no—
16#16861197C2212yes—
17#16861185—21no—
18C4×Dic254C2212yes—
19C4×D508C2211yescharacteristic
20#16861167—21no—
21#16861142C2211yescharacteristic
22#16861140C2212yes—
23C4⋊(C127⋊C8)C2211yescharacteristic
24C4⋊(C127⋊C8)C2212yes—
25#16861132C2212yes—
26#16861132C2212yes—
27#16861132C2212yes—
28#16861130C2211yescharacteristic
29#16861129C2212yes—
30#16861129C2211yescharacteristic
31C4×C127⋊C8C2211yescharacteristic
32C4×C127⋊C8C2212yes—
33C4⋊Q8×C127C2211yescharacteristic
34#125×C127C2212yes—
35C22⋊Q8×C127—21no—
36C22⋊Q8×C127—22no—
37C4×Q8×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4×D4×C127C2212yes—
40C2×C4⋊C4×C127—21no—
41C2×Dic254C2312yes—
42C2×D508C2311yescharacteristic
43C2×C4×D127—21no—
44#14523236—24no—
45#14523233—22no—
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2311yescharacteristic
49C4⋊Dic127—21no—
50C4⋊Dic127—21no—
51C4⋊Dic127C2311yescharacteristic
52Dic127⋊C4—22no—
53(C2×D127)⋊C4—22no—
54(C2×D127)⋊C4—22no—
55(C2×D127)⋊C4—21no—
56C4×Dic127—22no—
57C2×Q8×C127—21no—
58C2×Q8×C127D411yescharacteristic
59C2×D4×C127C2311yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127—22no—
62C4⋊C4×C127—22no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127—22no—
65C4⋊C4×C127D412yes—
66C4⋊C4×C127D411yescharacteristic
67C4⋊C4×C127C2311yescharacteristic
68C4⋊C4×C127C2312yes—
69C22⋊C4×C127—21no—
70C22⋊C4×C127—22no—
71C42×C127C2312yes—
72C42×C127C2311yescharacteristic
73C22×D127—21no—
74Dic254—22no—
75Dic254—22no—
76D508—21no—
77D508—21no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C4×D127—42no—
81C127⋊C8—21no—
82C127⋊C8—22no—
83C127⋊C8—21no—
84C23×C127—21no—
85Q8×C127—22no—
86Q8×C127—81no—
87D4×C127—21no—
88D4×C127#4112yes—
89C2×C4×C127—21no—
90C2×C4×C127—22no—
91C2×C4×C127—42no—
92C2×C4×C127—82no—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2411yescharacteristic
98C22×C127—22no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127C22×D411yescharacteristic
102C2×D127—41no—
103C2×D127—41no—
104C2×D127—41no—
105C4×C127—22no—
106C4×C127—22no—
107C4×C127—21no—
108C4×C127—41no—
109C4×C127—42no—
110C4×C127#13512yes—
111C4×C127C2×#4111yescharacteristic
112C4×C1272−1+411yescharacteristic
113Dic127—42no—
114Dic127—41no—
115C2×C127—42no—
116C2×C127#54811yescharacteristic
117C2×C127C2×#13511yescharacteristic
118C2×C127#57711yescharacteristic
119D127—81no—
120#3360—1271nomaximal, Sylow
121C127#336011yescharacteristic, complemented, Sylow
122#556—1271no—
123#554D12711yescharacteristic, complemented
124#488—1272no—
125#487—1271no—
126#479—1272no—
127#476—1271no—
128#463—1272no—
129#461—1271no—
130C8×Q8—1271no—
131C4×SD16—1272no—
132C4×D8—1271no—
133C2×SD16—1272no—
134C2×D8—1271no—
135C4⋊Q8—1271no—
136C4⋊Q8C2×D12711yescharacteristic
137#126—2542no—
138#125C2×D12712yes—
139#124—1272no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—2541no—
143C4×Q8—1272no—
144C4×Q8C2×D12711yescharacteristic
145C4×D4—1271no—
146C4×D4C2×D12711yescharacteristic
147C4×D4C2×D12712yes—
148#117—2541no—
149C2×C4⋊C4—21no—
150#107—1271no—
151#106—1272no—
152C4⋊C8—1271no—
153C4⋊C8—1272no—
154#103—1272no—
155#103—1272no—
156#103—1272no—
157#102—1271no—
158#102—1271no—
159#102—1272no—
160C4×C8—1271no—
161C4×C8—1272no—
162C2×Q8—21no—
163C2×Q8—1272no—
164C2×Q8C22⋊D12711yescharacteristic
165C2×D4—1271no—
166C2×D4C22×D12711yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—2541no—
170SD16—2544no—
171D8—2542no—
172C2×C8—1271no—
173C2×C8—1272no—
174C2×C8—1271no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—1271no—
179C4⋊C4—2542no—
180C4⋊C4—2541no—
181C4⋊C4—2541no—
182C4⋊C4C22⋊D12711yescharacteristic
183C4⋊C4C22⋊D12712yes—
184C4⋊C4C22×D12712yes—
185C4⋊C4C22×D12711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—2542no—
189C22⋊C4—2541no—
190C22⋊C4—2542no—
191C42—2542no—
192C42C22×D12711yescharacteristic
193C42C22×D12712yes—
194C23—21no—
195C23—2541no—
196Q8—22no—
197Q8—81no—
198Q8—2542no—
199Q8—2542no—
200D4—21no—
201D4—2541no—
202D4—2541no—
203D4#1452324912yes—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—5082no—
209C2×C4—2541no—
210C2×C4—2542no—
211C2×C4C2×C22⋊D12712yes—
212C2×C4C2×C22⋊D12712yes—
213C2×C4C2×C22⋊D12711yescharacteristic
214C2×C4C2×C22⋊D12711yescharacteristic
215C2×C4C23×D12711yescharacteristic
216C8—2541no—
217C8—2542no—
218C8—2541no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—5081no—
223C22—5081no—
224C22—5081no—
225C22C22×C22⋊D12711yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—5082no—
232C4—5081no—
233C4#1686122912yes—
234C4C2×#1452324911yescharacteristic
235C4#1686125711yescharacteristic
236C2—42no—
237C2—10161no—
238C2#2255053711yescharacteristic
239C2C2×#1686122911yescharacteristic
240C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#554×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523233C22
hypercenter#554D127
hypocenterC127#3360
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #22550599C2semidirect
2. #22550233C2semidirect
3. #22550228C2semidirect
4. #22550180C2semidirect
5. #22550163C2semidirect
7. #22550081C2semidirect
8. Q8×C127⋊C8C2semidirect
9. C4×#14523236C2semidirect
11. #554×C127C2semidirect
121. C127#3360semidirect
123. #554D127semidirect

Maximal subgroups

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

subgroupindex
#225505992
#225502332
#225502282
#225501802
#225501632
#225500912
#225500812
Q8×C127⋊C82
C4×#145232362
C4×#145232332
#554×C1272
#3360127