Finite Group Databasebeta

← #30648009

Subgroups — #30648009

Order 16256.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861255C211yesmaximal, characteristic
2C2×#16861242C211yesmaximal, characteristic, complemented
3C22×#14523237C211yesmaximal, characteristic, complemented
4#22550447C211yesmaximal, characteristic, complemented
5#22550040C214yesmaximal, complemented
6#22550029C214yesmaximal, complemented
7#22549882C211yesmaximal, characteristic, complemented
8#22549873C211yesmaximal, characteristic, complemented
9#408×C127C211yesmaximal, characteristic, complemented
10#16861255—24no—
11#16861255—24no—
12C2×#14523249—21no—
13C2×#14523248C2212yes—
14C2×#14523248C2211yescharacteristic
15C22×Dic254C2211yescharacteristic, complemented
16C22×D508C2212yes—
17C22×D508C2211yescharacteristic
18C2×D4×D127—22no—
19C2×D4×D127—21no—
20#16861242—24no—
21C2×Dic508C2212yescomplemented
22C2×#14523237—24no—
23C2×#14523237C2212yescomplemented
24C2×#14523237C2214yescomplemented
25#16861207—21no—
26#16861196—21no—
27#16861190—22no—
28#16861188C2214yescomplemented
29#16861168C2211yescharacteristic, complemented
30#16861134C2212yescomplemented
31#16861131C2212yescomplemented
32#16861131C2214yescomplemented
33C2×#41×C127C2211yescharacteristic
34C2×M16×C127C2211yescharacteristic, complemented
35C22×C8×C127C2211yescharacteristic, complemented
36C22⋊C8×C127C2214yescomplemented
37C23×D127—22no—
38C23×D127—21no—
39#14523249—28no—
40#14523248—24no—
41#14523248—24no—
42#14523248—22no—
43#14523248D414yes—
44C2×Dic254—21no—
45C2×Dic254—22no—
46C2×Dic254—22no—
47C2×Dic254C2312yes—
48C2×Dic254C2311yescharacteristic
49C2×D508—24no—
50C2×D508—24no—
51C2×D508—22no—
52C2×D508—21no—
53C2×D508—22no—
54C2×D508D414yes—
55C2×D508C2311yescharacteristic
56C2×D508C2312yes—
57C2×C22⋊D127—22no—
58C2×C22⋊D127—21no—
59D4×D127—216no—
60D4×D127—28no—
61C22×Dic127—21no—
62C2×C4×D127—22no—
63C2×C4×D127—21no—
64Dic508—24no—
65#14523237—28no—
66#14523237—24no—
67C4⋊Dic127C2312yes—
68Dic127⋊C4—24no—
69C22⋊Dic127—24no—
70C22⋊Dic127—21no—
71C4×Dic127—21no—
72#41×C127—22no—
73#41×C127—22no—
74C2×Q8×C127D411yescharacteristic
75C2×D4×C127D412yes—
76C2×D4×C127D411yescharacteristic
77C22×C4×C127C2312yes—
78C22×C4×C127C2311yescharacteristic
79M16×C127—22no—
80C2×C8×C127—22no—
81C2×C8×C127C2312yes—
82C2×C8×C127C2312yes—
83C22×D127—22no—
84C22×D127—21no—
85C22×D127—28no—
86C22×D127—24no—
87C22×D127—82no—
88C22×D127—84no—
89Dic254—21no—
90Dic254—22no—
91Dic254—22no—
92Dic254—42no—
93Dic254C2×D412yes—
94D508—28no—
95D508—24no—
96D508—22no—
97D508—21no—
98D508C2×D412yes—
99D508C2×D414yes—
100C22⋊D127—28no—
101C22⋊D127—24no—
102C2×Dic127—21no—
103C2×Dic127—22no—
104C2×Dic127—22no—
105C2×Dic127—42no—
106C4×D127—28no—
107C4×D127—24no—
108C23×C127C2×D412yes—
109C23×C127C2×D411yescharacteristic
110Q8×C127—22no—
111D4×C127—24no—
112D4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C8×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—24no—
125C22×C127#12012yes—
126C22×C127C22×D411yescharacteristic
127C2×D127—24no—
128C2×D127—22no—
129C2×D127—44no—
130C2×D127—42no—
131C2×D127—84no—
132C2×D127—88no—
133C4×C127—22no—
134C4×C127#12012yes—
135C4×C127C22×D411yescharacteristic
136C4×C127C22×D411yescharacteristic
137Dic127—22no—
138Dic127—42no—
139Dic127—42no—
140C2×C127—22no—
141C2×C127—42no—
142C2×C127C2×#12011yescharacteristic
143C2×C127#57712yes—
144D127—44no—
145D127—42no—
146#3009—1271nomaximal, Sylow
147C127#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—1271no—
149C2×#137—1271no—
150C22×SD16—1271no—
151#536—1271no—
152#452—1274no—
153#450—1274no—
154#418—1271no—
155#416—1271no—
156#408D12711yescharacteristic, complemented
157H(5,𝔽2)—2544no—
158H(5,𝔽2)—2544no—
159C2×#41—1271no—
160C2×#41—1272no—
161C2×#41—2541no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8—1271no—
164C22×D4—1271no—
165C22×D4—1272no—
166C22×D4—2542no—
167C22×D4—2541no—
168#137—2544no—
169C2×Q16—1272no—
170C2×SD16—1274no—
171C2×SD16—1272no—
172C2×SD16—2544no—
173C2×M16C2×D12711yescharacteristic, complemented
174C22×C8C2×D12711yescharacteristic, complemented
175C4⋊Q8—2541no—
176#124—2541no—
177#123—2542no—
178C22⋊Q8—1274no—
179#117—1271no—
180#103—1272no—
181#102—1272no—
182#102—1274no—
183C22⋊C8C2×D12714yescomplemented
184C24—2542no—
185C24—2541no—
186#41—22no—
187#41—22no—
188#41—1274no—
189#41—2544no—
190#41—2544no—
191#41—2548no—
192#41—2542no—
193C2×Q8—1271no—
194C2×Q8—1272no—
195C2×Q8—2542no—
196C2×Q8—2542no—
197C2×Q8—2541no—
198C2×Q8D50811yescharacteristic
199C2×D4—1274no—
200C2×D4—1272no—
201C2×D4—1271no—
202C2×D4—2544no—
203C2×D4—2544no—
204C2×D4—25416no—
205C2×D4—2548no—
206C2×D4—2542no—
207C2×D4—2541no—
208C2×D4—2542no—
209C2×D4—2542no—
210C2×D4—2541no—
211C2×D4D50812yes—
212C2×D4D50811yescharacteristic
213C22×C4—2541no—
214C22×C4—2542no—
215C22×C4—2541no—
216C22×C4C22×D12712yes—
217C22×C4C22×D12711yescharacteristic
218Q16—2544no—
219SD16—2548no—
220SD16—2544no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D12712yes—
224C2×C8C22×D12712yes—
225C4⋊C4—1272no—
226C4⋊C4—2544no—
227C22⋊C4—2541no—
228C22⋊C4—2544no—
229C42—2541no—
230C23—10162no—
231C23—10164no—
232C23—2548no—
233C23—2544no—
234C23—2542no—
235C23—2541no—
236C23C2×D50812yes—
237C23C2×D50811yescharacteristic
238Q8—22no—
239Q8—1272no—
240Q8—5082no—
241Q8—2542no—
242Q8—2542no—
243Q8—2541no—
244D4—24no—
245D4—22no—
246D4—1274no—
247D4—1272no—
248D4—2548no—
249D4—2548no—
250D4—2544no—
251D4—2541no—
252D4—2542no—
253D4—2544no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—5082no—
257C2×C4—2548no—
258C2×C4—2542no—
259C2×C4—2542no—
260C2×C4—2541no—
261C2×C4—2544no—
262C2×C4D4×D12712yes—
263C2×C4D4×D12712yes—
264C2×C4C2×D50812yes—
265C2×C4C2×D50811yescharacteristic
266C2×C4C23×D12711yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—10164no—
273C22—10168no—
274C22—5084no—
275C22—5082no—
276C22—2544no—
277C22—2542no—
278C22#1686117912yes—
279C22C22×D50811yescharacteristic
280C4—22no—
281C4—5082no—
282C4—5082no—
283C4—2542no—
284C4#1686117912yes—
285C4C2×D4×D12711yescharacteristic
286C4C2×D4×D12711yescharacteristic
287C2—22no—
288C2—42no—
289C2—5084no—
290C2—5082no—
291C2C2×#1686117911yescharacteristic
292C2#2255068812yes—
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#408×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861255C2
hypercenter#408D127
hypocenterC127#3009
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#16861242C2semidirect
3. C22×#14523237C2semidirect
4. #22550447C2semidirect
5. #22550040C2semidirect
6. #22550029C2semidirect
7. #22549882C2semidirect
8. #22549873C2semidirect
9. #408×C127C2semidirect
15. C22×Dic254C22semidirect
21. C2×Dic508C22semidirect
23. C2×#14523237C22semidirect
24. C2×#14523237C22semidirect
28. #16861188C22semidirect
29. #16861168C22semidirect
30. #16861134C22semidirect
31. #16861131C22semidirect
32. #16861131C22semidirect
34. C2×M16×C127C22semidirect
35. C22×C8×C127C22semidirect
36. C22⋊C8×C127C22semidirect
147. C127#3009semidirect
156. #408D127semidirect
173. C2×M16C2×D127semidirect
174. C22×C8C2×D127semidirect
183. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612422
C22×#145232372
#225504472
#225500402
#225500292
#225498822
#225498732
#408×C1272
#3009127