Finite Group Databasebeta

← #30650465

Subgroups — #30650465

Order 16256.

482 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 131 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×Q16×D127C211yesmaximal, characteristic, complemented
2#22550533C212yesmaximal, complemented
3#22550259C211yesmaximal, characteristic, complemented
4#22550117C211yesmaximal, characteristic, complemented
5#22550109C211yesmaximal, characteristic, complemented
6#22550096C212yesmaximal, complemented
7#22550038C212yesmaximal, complemented
8#22550013C212yesmaximal, complemented
9Q16×Dic127C211yesmaximal, characteristic, complemented
10C8×C22⋊D127C211yesmaximal, characteristic, complemented
11C2×#135×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2212yes—
13C2×Dic508C2211yescharacteristic
14C2×#14523238C2212yes—
15Q16×D127—24no—
16C2×#14523236C2212yes—
17C2×C8×D127C2211yescharacteristic
18#16861196—22no—
19#16861196C2212yes—
20#16861188—22no—
21#16861188C2212yes—
22#16861180—22no—
23#16861180C2212yes—
24Q8×Dic127C2212yes—
25C4×C22⋊D127—22no—
26C4×C22⋊D127C2211yescharacteristic
27#16861143C2211yescharacteristic
28Dic127⋊C8C2211yescharacteristic
29#16861134C2212yes—
30#16861133C2212yes—
31#16861130C2212yes—
32C22⋊(C127⋊C8)C2211yescharacteristic
33(C2×D127)⋊C8C2211yescharacteristic
34C8×Dic127C2211yescharacteristic
35C2×#41×C127C2212yes—
36#135×C127—24no—
37C2×Q16×C127C2211yescharacteristic
38C2×SD16×C127C2212yes—
39C2×D8×C127C2211yescharacteristic
40C22×C8×C127C2211yescharacteristic
41C2×Dic254—22no—
42C2×Dic254C2312yes—
43Q8×D127—24no—
44Q8×D127—42no—
45C2×C22⋊D127—22no—
46C2×C22⋊D127D411yescharacteristic
47C2×C4×D127—22no—
48C2×C4×D127C2311yescharacteristic
49Dic508—22no—
50#14523238—24no—
51#14523236—42no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C8×D127—22no—
55C4⋊Dic127—22no—
56C4⋊Dic127C2312yes—
57Dic127⋊C4—22no—
58Dic127⋊C4D411yescharacteristic
59C22⋊Dic127—22no—
60C22⋊Dic127—22no—
61C22⋊Dic127—22no—
62C22⋊Dic127—22no—
63C22⋊Dic127D411yescharacteristic
64(C2×D127)⋊C4D411yescharacteristic
65C4×Dic127—22no—
66C4×Dic127C2311yescharacteristic
67#41×C127—24no—
68#41×C127—42no—
69C2×Q8×C127C2312yes—
70C2×D4×C127—22no—
71C2×D4×C127C2312yes—
72C22×C4×C127—22no—
73C22×C4×C127C2311yescharacteristic
74Q16×C127D414yescomplemented
75SD16×C127—24no—
76D8×C127—22no—
77C2×C8×C127—22no—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C22×D127C2×D411yescharacteristic
81Dic254—22no—
82Dic254—22no—
83Dic254—44no—
84C22⋊D127—22no—
85C22⋊D127—44no—
86C2×Dic127—22no—
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×C127—22no—
97C23×C127C2×D411yescharacteristic
98Q8×C127—22no—
99Q8×C127C2×D414yes—
100D4×C127—22no—
101D4×C127—22no—
102D4×C127—44no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
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—21no—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127—42no—
117C22×C127C22×D411yescharacteristic
118C2×D127—21no—
119C2×D127—21no—
120C2×D127#13512yescomplemented
121C4×C127—21no—
122C4×C127—24no—
123C4×C127C22×D411yescharacteristic
124C4×C127H(5,𝔽2)11yescharacteristic
125Dic127—21no—
126Dic127—21no—
127Dic127—21no—
128Dic127—42no—
129Dic127—42no—
130C2×C127—41no—
131C2×C127—82no—
132C2×C127D4211yescharacteristic
133C2×C127C2×#13511yescharacteristic
134C2×C127#57811yescharacteristic
135D127—22no—
136#3285—1271nomaximal, Sylow
137C127#328511yescharacteristic, complemented, Sylow
138C2×#135D12711yescharacteristic, complemented
139C22×Q16—1271no—
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×#41C2×D12712yes—
150C22×Q8—1272no—
151#135—24no—
152C2×Q16—1272no—
153C2×Q16—1271no—
154C2×Q16—2544no—
155C2×Q16C2×D12711yescharacteristic
156C2×SD16—1272no—
157C2×SD16C2×D12712yes—
158C2×D8C2×D12711yescharacteristic
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—24no—
179#41—42no—
180C2×Q8—1272no—
181C2×Q8—5082no—
182C2×Q8—2542no—
183C2×Q8—2544no—
184C2×Q8C22×D12712yes—
185C2×D4—22no—
186C2×D4—1271no—
187C2×D4—2542no—
188C2×D4C22×D12712yes—
189C22×C4—22no—
190C22×C4—1271no—
191C22×C4—2542no—
192C22×C4C22×D12711yescharacteristic
193Q16—2544no—
194Q16—2542no—
195Q16C22⋊D12714yescomplemented
196SD16—24no—
197SD16—5082no—
198D8—22no—
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—22no—
218C23—1271no—
219C23D4×D12711yescharacteristic
220Q8—22no—
221Q8—5084no—
222Q8—2542no—
223Q8—2542no—
224Q8C2×C22⋊D12714yes—
225D4—22no—
226D4—22no—
227D4—44no—
228D4—5084no—
229D4—2542no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—44no—
234C2×C4—1271no—
235C2×C4—1271no—
236C2×C4—1271no—
237C2×C4—5084no—
238C2×C4—2542no—
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—21no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—1272no—
255C22—2541no—
256C22—2541no—
257C22C2×D4×D12711yescharacteristic
258C4—21no—
259C4—24no—
260C4—5082no—
261C4—5082no—
262C4—2541no—
263C4—2541no—
264C4—2541no—
265C4C22×C22⋊D12711yescharacteristic
266C4#1686125411yescharacteristic
267C2—41no—
268C2—82no—
269C2—2542no—
270C2D4×C22⋊D12711yescharacteristic
271C2#135×D12711yescharacteristic
272C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#135×C127C2
socleC22×C127C22×D4
hypercenterC2×#135D127
hypocenterC127#3285
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D127C2semidirect
2. #22550533C2semidirect
3. #22550259C2semidirect
4. #22550117C2semidirect
5. #22550109C2semidirect
6. #22550096C2semidirect
7. #22550038C2semidirect
8. #22550013C2semidirect
9. Q16×Dic127C2semidirect
10. C8×C22⋊D127C2semidirect
11. C2×#135×C127C2semidirect
74. Q16×C127D4semidirect
120. C2×D127#135semidirect
137. C127#3285semidirect
138. C2×#135D127semidirect
195. Q16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1272
#225505332
#225502592
#225501172
#225501092
#225500962
#225500382
#225500132
Q16×Dic1272
C8×C22⋊D1272
C2×#135×C1272
#3285127