Finite Group Databasebeta

← #30650471

Subgroups — #30650471

Order 16256.

482 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16861232C211yesmaximal, characteristic, complemented
2#22550537C212yesmaximal, complemented
3#22550261C211yesmaximal, characteristic, complemented
4#22550114C211yesmaximal, characteristic, complemented
5#22550112C211yesmaximal, characteristic, complemented
6#22550100C212yesmaximal, complemented
7#22550037C212yesmaximal, complemented
8#22550015C212yesmaximal, complemented
9Q16×Dic127C211yesmaximal, characteristic, complemented
10C8×C22⋊D127C211yesmaximal, characteristic, complemented
11C22×Q16×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13#16861232—24no—
14C2×#14523235C2212yes—
15C2×#14523235C2212yes—
16C2×D1016C2211yescharacteristic
17C2×C8×D127C2211yescharacteristic
18#16861198—22no—
19#16861198C2212yes—
20#16861185—22no—
21#16861185C2212yes—
22#16861183—22no—
23#16861183C2212yes—
24Q8×Dic127C2212yes—
25C4×C22⋊D127—22no—
26C4×C22⋊D127C2211yescharacteristic
27#16861143C2211yescharacteristic
28Dic127⋊C8C2211yescharacteristic
29#16861133C2212yes—
30#16861133C2212yes—
31#16861131C2212yes—
32C22⋊(C127⋊C8)C2211yescharacteristic
33(C2×D127)⋊C8C2211yescharacteristic
34C8×Dic127C2211yescharacteristic
35C22×Q8×C127C2212yes—
36C2×Q16×C127—24no—
37C2×Q16×C127C2212yes—
38C2×Q16×C127C2211yescharacteristic
39C2×Q16×C127C2211yescharacteristic
40C22×C8×C127C2211yescharacteristic
41#14523249—24no—
42#14523249—42no—
43C2×D508—22no—
44C2×D508C2312yes—
45C2×C22⋊D127—22no—
46C2×C22⋊D127D411yescharacteristic
47C2×C4×D127—22no—
48C2×C4×D127C2311yescharacteristic
49#14523235—24no—
50#14523235—42no—
51D1016—22no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C8×D127—22no—
55C4⋊Dic127—22no—
56C4⋊Dic127C2312yes—
57Dic127⋊C4—22no—
58Dic127⋊C4D411yescharacteristic
59C22⋊Dic127D411yescharacteristic
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—22no—
64(C2×D127)⋊C4D411yescharacteristic
65C4×Dic127—22no—
66C4×Dic127C2311yescharacteristic
67C2×Q8×C127—22no—
68C2×Q8×C127—24no—
69C2×Q8×C127—42no—
70C2×Q8×C127C2312yes—
71C2×Q8×C127C2312yes—
72C22×C4×C127—22no—
73C22×C4×C127C2311yescharacteristic
74Q16×C127—24no—
75Q16×C127—22no—
76Q16×C127D414yescomplemented
77C2×C8×C127—22no—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C22×D127—22no—
81C22×D127C2×D411yescharacteristic
82D508—22no—
83D508—22no—
84D508—44no—
85C22⋊D127—22no—
86C22⋊D127—44no—
87C2×Dic127—22no—
88C2×Dic127C2×D411yescharacteristic
89C2×Dic127C2×D411yescharacteristic
90C2×Dic127C2×D411yescharacteristic
91C4×D127—21no—
92C4×D127—21no—
93C4×D127—44no—
94C127⋊C8—21no—
95C127⋊C8—21no—
96C23×C127C2×D411yescharacteristic
97Q8×C127—22no—
98Q8×C127—22no—
99Q8×C127—22no—
100Q8×C127—44no—
101Q8×C127C2×D414yes—
102C2×C4×C127—22no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—22no—
106C2×C4×C127—44no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2411yescharacteristic
109C8×C127—21no—
110C8×C127C2×D412yes—
111C22×C127—21no—
112C22×C127—21no—
113C22×C127#13512yescomplemented
114C22×C127C22×D411yescharacteristic
115C2×D127—21no—
116C2×D127—21no—
117C2×D127—21no—
118C2×D127—42no—
119C2×D127—42no—
120C2×D127—42no—
121C4×C127—21no—
122C4×C127—24no—
123C4×C127—42no—
124C4×C127C22×D411yescharacteristic
125C4×C127H(5,𝔽2)11yescharacteristic
126Dic127—21no—
127Dic127—21no—
128Dic127—21no—
129Dic127—42no—
130C2×C127—22no—
131C2×C127D4211yescharacteristic
132C2×C127C2×#13511yescharacteristic
133C2×C127#57811yescharacteristic
134D127—41no—
135D127—82no—
136#3285—1271nomaximal, Sylow
137C127#328511yescharacteristic, complemented, Sylow
138C2×#135—1271no—
139C22×Q16D12711yescharacteristic, complemented
140#548—1272no—
141#495—1271no—
142#467—1271no—
143#466—1271no—
144#464—1272no—
145#452—1272no—
146#448—1272no—
147C4×Q16—1271no—
148C8×D4—1271no—
149C2×#41—1272no—
150C22×Q8C2×D12712yes—
151#135—2544no—
152C2×Q16—24no—
153C2×Q16C2×D12712yes—
154C2×Q16C2×D12711yescharacteristic
155C2×Q16C2×D12711yescharacteristic
156C2×SD16—1272no—
157C2×SD16—1272no—
158C2×D8—1271no—
159C22×C8—1271no—
160C22×C8C2×D12711yescharacteristic
161#124—1272no—
162#124—2542no—
163C22⋊Q8—1272no—
164C22⋊Q8—2542no—
165#121—1272no—
166#121—2542no—
167C4×Q8—1272no—
168C4×D4—1271no—
169C4×D4—2542no—
170#107—1271no—
171C4⋊C8—1271no—
172#103—1272no—
173#103—1272no—
174#102—1272no—
175C22⋊C8—1271no—
176C22⋊C8—1271no—
177C4×C8—1271no—
178#41—5082no—
179#41—2544no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8C22×D12712yes—
184C2×Q8C22×D12712yes—
185C2×D4—1271no—
186C2×D4—1272no—
187C2×D4—2542no—
188C2×D4—2542no—
189C22×C4—22no—
190C22×C4—1271no—
191C22×C4—2542no—
192C22×C4C22×D12711yescharacteristic
193Q16—24no—
194Q16—22no—
195Q16C22⋊D12714yescomplemented
196SD16—5082no—
197SD16—2544no—
198D8—2542no—
199C2×C8—22no—
200C2×C8—1271no—
201C2×C8—1271no—
202C2×C8—2542no—
203C2×C8C22×D12711yescharacteristic
204C2×C8C22×D12711yescharacteristic
205C4⋊C4—1271no—
206C4⋊C4—1272no—
207C4⋊C4—2542no—
208C4⋊C4—2542no—
209C22⋊C4—1271no—
210C22⋊C4—1271no—
211C22⋊C4—2542no—
212C22⋊C4—2542no—
213C22⋊C4—2542no—
214C22⋊C4—2542no—
215C42—1271no—
216C42—2542no—
217C23—1271no—
218C23—2542no—
219C23D4×D12711yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—44no—
224Q8C2×C22⋊D12714yes—
225D4—5084no—
226D4—5084no—
227D4—2542no—
228D4—2542no—
229D4—2542no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—1271no—
236C2×C4—1271no—
237C2×C4—1271no—
238C2×C4—5084no—
239C2×C4—2542no—
240C2×C4—2541no—
241C2×C4—2541no—
242C2×C4D4×D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—21no—
245C8—2541no—
246C8—2541no—
247C8C2×C22⋊D12712yes—
248C22—21no—
249C22—21no—
250C22—5082no—
251C22—5082no—
252C22—5082no—
253C22—2541no—
254C22—2541no—
255C22—2541no—
256C22#1686123212yescomplemented
257C22C2×D4×D12711yescharacteristic
258C4—21no—
259C4—24no—
260C4—42no—
261C4—5082no—
262C4—2541no—
263C4—2541no—
264C4—2541no—
265C4C22×C22⋊D12711yescharacteristic
266C4#1686125411yescharacteristic
267C2—22no—
268C2—10162no—
269C2—5081no—
270C2D4×C22⋊D12711yescharacteristic
271C2C2×#1686123211yescharacteristic
272C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×Q16×C127C2
socleC22×C127C22×D4
hypercenterC22×Q16D127
hypocenterC127#3285
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. #22550537C2semidirect
3. #22550261C2semidirect
4. #22550114C2semidirect
5. #22550112C2semidirect
6. #22550100C2semidirect
7. #22550037C2semidirect
8. #22550015C2semidirect
9. Q16×Dic127C2semidirect
10. C8×C22⋊D127C2semidirect
11. C22×Q16×C127C2semidirect
76. Q16×C127D4semidirect
113. C22×C127#135semidirect
137. C127#3285semidirect
139. C22×Q16D127semidirect
195. Q16C22⋊D127semidirect
256. C22#16861232semidirect

Maximal subgroups

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

subgroupindex
C2×#168612322
#225505372
#225502612
#225501142
#225501122
#225501002
#225500372
#225500152
Q16×Dic1272
C8×C22⋊D1272
C22×Q16×C1272
#3285127