Finite Group Databasebeta

← #30651426

Subgroups — #30651426

Order 16256.

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

normal subgroups only

This group has 348 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#22550611C211yesmaximal, characteristic
2#22550233C212yesmaximal, complemented
3#22550224C211yesmaximal, characteristic
4#22550173C212yesmaximal, complemented
5#22550158C211yesmaximal, characteristic, complemented
6#22550098C212yesmaximal, complemented
7#22550076C211yesmaximal, characteristic
8Q8×C127⋊C8C211yesmaximal, characteristic, complemented
9C4×#14523238C211yesmaximal, characteristic, complemented
10C4×#14523235C212yesmaximal, complemented
11#557×C127C211yesmaximal, characteristic, complemented
12C2×#14523238C2211yescharacteristic
13C2×#14523235C2212yes—
14#16861206C2211yescharacteristic
15#16861198—21no—
16#16861197C2212yes—
17#16861194—22no—
18C4×Dic254C2211yescharacteristic
19C4×D508C2212yes—
20#16861171—21no—
21#16861142C2211yescharacteristic
22#16861140C2212yes—
23C4⋊(C127⋊C8)C2211yescharacteristic
24C4⋊(C127⋊C8)C2212yes—
25#16861133C2211yescharacteristic
26#16861132C2212yes—
27#16861132C2211yescharacteristic
28#16861129C2212yes—
29#16861129C2212yes—
30#16861129C2212yes—
31C4×C127⋊C8C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127—21no—
34C4⋊Q8×C127C2211yescharacteristic
35#125×C127—22no—
36#125×C127C2212yes—
37C4×Q8×C127—21no—
38C4×Q8×C127C2211yescharacteristic
39C4×Q8×C127C2212yes—
40C4×Q8×C127C2211yescharacteristic
41C2×Dic254C2311yescharacteristic
42C2×D508—21no—
43C2×D508—21no—
44C2×D508C2312yes—
45C2×C4×D127—22no—
46#14523238—22no—
47#14523235—24no—
48C2×C127⋊C8C2312yes—
49C2×C127⋊C8C2311yescharacteristic
50C2×C127⋊C8C2311yescharacteristic
51C4⋊Dic127C2311yescharacteristic
52Dic127⋊C4—21no—
53(C2×D127)⋊C4—22no—
54(C2×D127)⋊C4—22no—
55(C2×D127)⋊C4—22no—
56C4×Dic127—21no—
57C2×Q8×C127—21no—
58C2×Q8×C127D411yescharacteristic
59C2×Q8×C127C2311yescharacteristic
60C4⋊C4×C127—22no—
61C4⋊C4×C127—21no—
62C4⋊C4×C127—22no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127—22no—
65C4⋊C4×C127D412yes—
66C4⋊C4×C127D411yescharacteristic
67C4⋊C4×C127C2311yescharacteristic
68C4⋊C4×C127C2312yes—
69C42×C127—22no—
70C42×C127—21no—
71C42×C127C2312yes—
72C42×C127C2311yescharacteristic
73C22×D127—22no—
74Dic254—21no—
75Dic254—21no—
76D508—22no—
77D508—22no—
78D508—42no—
79D508—81no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C4×D127—82no—
83C127⋊C8—22no—
84C127⋊C8—21no—
85C127⋊C8—21no—
86Q8×C127—22no—
87Q8×C127—21no—
88Q8×C127—42no—
89Q8×C127#4112yes—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—22no—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2411yescharacteristic
98C22×C127C22×D411yescharacteristic
99C2×D127—42no—
100C2×D127—42no—
101C2×D127—42no—
102C4×C127—22no—
103C4×C127—22no—
104C4×C127—22no—
105C4×C127—21no—
106C4×C127—42no—
107C4×C127—41no—
108C4×C127#13512yes—
109C4×C127C2×#4111yescharacteristic
110C4×C1272−1+411yescharacteristic
111Dic127—41no—
112Dic127—41no—
113C2×C127#54811yescharacteristic
114C2×C127C2×#13511yescharacteristic
115C2×C127#57711yescharacteristic
116D127—82no—
117#3365—1271nomaximal, Sylow
118C127#336511yescharacteristic, complemented, Sylow
119#559—1271no—
120#557D12711yescharacteristic, complemented
121#488—1272no—
122#486—1271no—
123#478—1272no—
124#475—1271no—
125#464—1272no—
126#460—1271no—
127C8×Q8—1271no—
128C4×Q16—1271no—
129C4×SD16—1272no—
130C2×Q16—1271no—
131C2×SD16—1272no—
132C4⋊Q8—21no—
133C4⋊Q8C2×D12711yescharacteristic
134#127—1271no—
135#125—22no—
136#125C2×D12712yes—
137#124—1272no—
138#124—2541no—
139#123—2542no—
140C4×Q8—21no—
141C4×Q8—1271no—
142C4×Q8C2×D12712yes—
143C4×Q8C2×D12711yescharacteristic
144C4×Q8C2×D12711yescharacteristic
145C4×D4—1272no—
146C4×D4—2541no—
147#107—1271no—
148#106—1272no—
149C4⋊C8—1271no—
150C4⋊C8—1272no—
151#103—1271no—
152#103—1271no—
153#103—1272no—
154#102—1272no—
155#102—1272no—
156#102—1272no—
157C4×C8—1272no—
158C4×C8—1271no—
159C2×Q8—21no—
160C2×Q8—1271no—
161C2×Q8C22⋊D12711yescharacteristic
162C2×Q8C22×D12711yescharacteristic
163C2×D4—1272no—
164C2×D4—2541no—
165C2×D4—2541no—
166C22×C4—2542no—
167Q16—2542no—
168SD16—2544no—
169C2×C8—1271no—
170C2×C8—1271no—
171C2×C8—1272no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—1271no—
178C4⋊C4—2541no—
179C4⋊C4C22⋊D12711yescharacteristic
180C4⋊C4C22⋊D12712yes—
181C4⋊C4C22×D12712yes—
182C4⋊C4C22×D12711yescharacteristic
183C22⋊C4—2542no—
184C22⋊C4—2542no—
185C22⋊C4—2542no—
186C42—22no—
187C42—21no—
188C42—2541no—
189C42C22×D12712yes—
190C42C22×D12711yescharacteristic
191C23—2542no—
192Q8—22no—
193Q8—21no—
194Q8—42no—
195Q8—2541no—
196Q8—2541no—
197Q8#1452324912yes—
198D4—10161no—
199D4—5082no—
200D4—2542no—
201D4—2542no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—10162no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4C2×C22⋊D12712yes—
209C2×C4C2×C22⋊D12712yes—
210C2×C4C2×C22⋊D12711yescharacteristic
211C2×C4C2×C22⋊D12711yescharacteristic
212C2×C4C23×D12711yescharacteristic
213C8—2542no—
214C8—2541no—
215C8—2541no—
216C22—5082no—
217C22—5082no—
218C22—5082no—
219C22C22×C22⋊D12711yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—21no—
224C4—42no—
225C4—41no—
226C4—5081no—
227C4—5081no—
228C4#1686122912yes—
229C4C2×#1452324911yescharacteristic
230C4#1686125711yescharacteristic
231C2—10162no—
232C2#2255053711yescharacteristic
233C2C2×#1686122911yescharacteristic
234C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#557×C127C2
socleC22×C127C22×D4
prime radical#16861206C22
hypercenter#557D127
hypocenterC127#3365
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
2. #22550233C2semidirect
4. #22550173C2semidirect
5. #22550158C2semidirect
6. #22550098C2semidirect
8. Q8×C127⋊C8C2semidirect
9. C4×#14523238C2semidirect
10. C4×#14523235C2semidirect
11. #557×C127C2semidirect
118. C127#3365semidirect
120. #557D127semidirect

Maximal subgroups

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

subgroupindex
#225506112
#225502332
#225502242
#225501732
#225501582
#225500982
#225500762
Q8×C127⋊C82
C4×#145232382
C4×#145232352
#557×C1272
#3365127