Finite Group Databasebeta

← #30650446

Subgroups — #30650446

Order 16256.

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

normal subgroups only

This group has 484 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×#16861228C211yesmaximal, characteristic, complemented
2#22550525C212yesmaximal, complemented
3#22550254C211yesmaximal, characteristic, complemented
4#22550117C211yesmaximal, characteristic, complemented
5#22550114C211yesmaximal, characteristic, complemented
6#22550079C212yesmaximal, complemented
7#22550038C212yesmaximal, complemented
8#22550035C212yesmaximal, complemented
9D8×Dic127C211yesmaximal, characteristic, complemented
10C8×C22⋊D127C211yesmaximal, characteristic, complemented
11C2×#135×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yes—
13#16861228—24no—
14C2×Dic508C2211yescharacteristic
15C2×#14523238C2212yes—
16C2×#14523236C2212yes—
17C2×C8×D127C2211yescharacteristic
18#16861207C2212yes—
19#16861192—22no—
20#16861190—22no—
21#16861188C2212yes—
22#16861185C2212yes—
23#16861181—22no—
24D4×Dic127C2212yes—
25C4×C22⋊D127C2211yescharacteristic
26C2×Dic127⋊C4—22no—
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
41#14523247—24no—
42#14523247—42no—
43C2×Dic254C2312yes—
44C2×C22⋊D127—22no—
45C2×C22⋊D127D411yescharacteristic
46C22×Dic127—22no—
47C22×Dic127—22no—
48C2×C4×D127C2311yescharacteristic
49Dic508—22no—
50#14523238—42no—
51#14523236—24no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2311yescharacteristic
54C8×D127—22no—
55C4⋊Dic127C2312yes—
56Dic127⋊C4—22no—
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59Dic127⋊C4—22no—
60Dic127⋊C4D411yescharacteristic
61C22⋊Dic127—22no—
62C22⋊Dic127—22no—
63C22⋊Dic127D411yescharacteristic
64(C2×D127)⋊C4—22no—
65(C2×D127)⋊C4D411yescharacteristic
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×C127—22no—
75SD16×C127—24no—
76D8×C127D414yescomplemented
77C2×C8×C127—22no—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C22×D127C2×D411yescharacteristic
81Dic254—22no—
82Dic254—22no—
83C22⋊D127—22no—
84C22⋊D127—44no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—44no—
88C2×Dic127—44no—
89C2×Dic127C2×D411yescharacteristic
90C2×Dic127C2×D411yescharacteristic
91C2×Dic127C2×D411yescharacteristic
92C4×D127—21no—
93C4×D127—21no—
94C127⋊C8—21no—
95C127⋊C8—21no—
96C23×C127—22no—
97C23×C127C2×D411yescharacteristic
98Q8×C127—22no—
99Q8×C127—22no—
100D4×C127—22no—
101D4×C127—44no—
102D4×C127C2×D414yes—
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—24no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—42no—
116C22×C127—42no—
117C22×C127C22×D411yescharacteristic
118C2×D127—21no—
119C2×D127—21no—
120C2×D127—21no—
121C4×C127—21no—
122C4×C127—42no—
123C4×C127C22×D411yescharacteristic
124C4×C127H(5,𝔽2)11yescharacteristic
125Dic127—21no—
126Dic127—21no—
127Dic127—42no—
128Dic127—42no—
129Dic127#13512yescomplemented
130C2×C127—41no—
131C2×C127—44no—
132C2×C127D4211yescharacteristic
133C2×C127C2×#13511yescharacteristic
134C2×C127#57811yescharacteristic
135D127—41no—
136#3283—1271nomaximal, Sylow
137C127#328311yescharacteristic, complemented, Sylow
138C2×#135—1271no—
139C2×#135D12711yescharacteristic, complemented
140#547—1272no—
141#494—1271no—
142#467—1271no—
143#467—1271no—
144#461—1272no—
145#452—1272no—
146#452—1272no—
147C4×D8—1271no—
148C8×D4—1271no—
149C2×#41—1272no—
150C2×#41C2×D12712yes—
151#135—24no—
152#135—2544no—
153C2×Q16—1272no—
154C2×Q16—1271no—
155C2×Q16C2×D12711yescharacteristic
156C2×SD16—1272no—
157C2×SD16C2×D12712yes—
158C2×D8C2×D12711yescharacteristic
159C22×C8—1271no—
160C22×C8C2×D12711yescharacteristic
161C4⋊Q8—1272no—
162#123—2542no—
163#123—2542no—
164C22⋊Q8—1272no—
165C22⋊Q8—1272no—
166#121—2542no—
167C4×D4—1271no—
168C4×D4—1272no—
169C2×C4⋊C4—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—
180#41—5082no—
181#41—2544no—
182C2×Q8—1272no—
183C2×Q8C22×D12712yes—
184C2×D4—22no—
185C2×D4—1271no—
186C2×D4—2542no—
187C2×D4C22×D12712yes—
188C22×C4—22no—
189C22×C4—1271no—
190C22×C4—2542no—
191C22×C4—2542no—
192C22×C4C22×D12711yescharacteristic
193Q16—22no—
194Q16—5082no—
195Q16—2542no—
196SD16—24no—
197SD16—2544no—
198D8C22⋊D12714yescomplemented
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—
209C4⋊C4—2542no—
210C4⋊C4—2542no—
211C22⋊C4—1271no—
212C22⋊C4—1271no—
213C22⋊C4—2542no—
214C22⋊C4—2542no—
215C22⋊C4—2542no—
216C42—1271no—
217C23—22no—
218C23—1271no—
219C23D4×D12711yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—2542no—
223Q8—2542no—
224D4—22no—
225D4—44no—
226D4—5084no—
227D4—2542no—
228D4C2×C22⋊D12714yes—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—1271no—
234C2×C4—1271no—
235C2×C4—1271no—
236C2×C4—5084no—
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—24no—
252C22—42no—
253C22—42no—
254C22—2541no—
255C22—2541no—
256C22—2541no—
257C22C2×D4×D12711yescharacteristic
258C4—21no—
259C4—42no—
260C4—1272no—
261C4—5082no—
262C4—5082no—
263C4—2541no—
264C4—2541no—
265C4C22×C22⋊D12711yescharacteristic
266C4#1686125411yescharacteristic
267C2—41no—
268C2—44no—
269C2—5081no—
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#3283
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×#16861228C2semidirect
2. #22550525C2semidirect
3. #22550254C2semidirect
4. #22550117C2semidirect
5. #22550114C2semidirect
6. #22550079C2semidirect
7. #22550038C2semidirect
8. #22550035C2semidirect
9. D8×Dic127C2semidirect
10. C8×C22⋊D127C2semidirect
11. C2×#135×C127C2semidirect
76. D8×C127D4semidirect
129. Dic127#135semidirect
137. C127#3283semidirect
139. C2×#135D127semidirect
198. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612282
#225505252
#225502542
#225501172
#225501142
#225500792
#225500382
#225500352
D8×Dic1272
C8×C22⋊D1272
C2×#135×C1272
#3283127