Finite Group Databasebeta

← #30639732

Subgroups — #30639732

Order 16256.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#14523235C211yesmaximal, characteristic, complemented
2C2×#16861185C211yesmaximal, characteristic, complemented
3C2×#16861184C211yesmaximal, characteristic
4C2×#16861131C211yesmaximal, characteristic, complemented
5C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#103×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C22×D508C2211yescharacteristic
10C22×C4×D127C2211yescharacteristic
11C2×#14523235—22no—
12C2×#14523235—22no—
13C2×#14523235—22no—
14C22×C127⋊C8C2211yescharacteristic
15#16861185—22no—
16#16861185—22no—
17#16861184—22no—
18#16861184—22no—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4—21no—
21C2×(C2×D127)⋊C4—21no—
22C2×(C2×D127)⋊C4—21no—
23#16861131—22no—
24(C2×D127)⋊C8—22no—
25C22×Q8×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28#103×C127—22no—
29C23×D127—21no—
30C23×D127D411yescharacteristic
31C2×D508—24no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—42no—
38C2×D508—42no—
39C2×D508D412yes—
40C22×Dic127—21no—
41C22×Dic127D411yescharacteristic
42C2×C4×D127—21no—
43C2×C4×D127—21no—
44C2×C4×D127—21no—
45C2×C4×D127—21no—
46C2×C4×D127—21no—
47C2×C4×D127—21no—
48#14523235—28no—
49C2×C127⋊C8—21no—
50C2×C127⋊C8—21no—
51C2×C127⋊C8D412yescomplemented
52C4⋊Dic127—21no—
53C4⋊Dic127D412yescomplemented
54Dic127⋊C4—24no—
55(C2×D127)⋊C4—24no—
56(C2×D127)⋊C4—24no—
57C2×Q8×C127—21no—
58C2×Q8×C127—21no—
59C2×Q8×C127—21no—
60C2×Q8×C127—21no—
61C2×Q8×C127—21no—
62C2×Q8×C127D412yescomplemented
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65C22×C4×C127C2311yescharacteristic
66C2×C8×C127—21no—
67C2×C8×C127—21no—
68C2×C8×C127—21no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127D412yes—
71C22×D127—21no—
72C22×D127—21no—
73C22×D127—21no—
74C22×D127—21no—
75C22×D127—21no—
76C22×D127—21no—
77C22×D127—22no—
78C22×D127—41no—
79C22×D127—41no—
80C22×D127—41no—
81C22×D127—41no—
82C22×D127—41no—
83C22×D127—41no—
84C22×D127SD1612yescomplemented
85D508—22no—
86D508—22no—
87D508—48no—
88D508—41no—
89D508—41no—
90C2×Dic127—21no—
91C2×Dic127—22no—
92C2×Dic127—21no—
93C2×Dic127—21no—
94C2×Dic127—41no—
95C2×Dic127—41no—
96C4×D127—41no—
97C4×D127—41no—
98C4×D127—41no—
99C4×D127—41no—
100C127⋊C8—22no—
101C23×C127C2×D411yescharacteristic
102Q8×C127—22no—
103Q8×C127—22no—
104Q8×C127—41no—
105Q8×C127—41no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127#4111yescharacteristic
118C8×C127—41no—
119C22×C127#12011yescharacteristic
120C22×C127#12011yescharacteristic
121C22×C127#12111yescharacteristic
122C22×C127C2×SD1611yescharacteristic
123C22×C127#13511yescharacteristic
124C22×C127AGL(1,ℤ/8)11yescharacteristic
125C22×C127#13711yescharacteristic
126C2×D127—22no—
127C2×D127—22no—
128C2×D127—22no—
129C2×D127—41no—
130C2×D127—41no—
131C2×D127—41no—
132C2×D127—41no—
133C2×D127—42no—
134C2×D127—42no—
135C2×D127—42no—
136C2×D127—81no—
137C2×D127—81no—
138C2×D127—81no—
139C2×D127—81no—
140C4×C127—42no—
141C4×C127—42no—
142C4×C127#12011yescharacteristic
143C4×C127#12111yescharacteristic
144C4×C127#12111yescharacteristic
145C4×C127#12711yescharacteristic
146Dic127—41no—
147Dic127—42no—
148C2×C127#39211yescharacteristic
149C2×C127#44811yescharacteristic
150C2×C127#44911yescharacteristic
151C2×C127#45011yescharacteristic
152C2×C127#45211yescharacteristic
153C2×C127#46511yescharacteristic
154C2×C127#46811yescharacteristic
155D127—42no—
156D127—82no—
157#2000—1271nomaximal, Sylow
158C127#200011yescharacteristic, complemented, Sylow
159C22×SD16—1271no—
160C2×C22⋊Q8—1271no—
161C2×#121—1271no—
162C2×#103D12711yescharacteristic, complemented
163C2×#102—1271no—
164C2×C22⋊C8—1271no—
165#340—1271no—
166C22×Q8C2×D12711yescharacteristic
167C22×D4—1271no—
168C22×D4—2541no—
169C23×C4—1271no—
170C2×SD16—2542no—
171C2×SD16—2542no—
172C2×SD16—2542no—
173C22×C8—1271no—
174C22×C8C2×D12711yescharacteristic
175C22⋊Q8—2542no—
176C22⋊Q8—2542no—
177#121—2542no—
178#121—2542no—
179C2×C4⋊C4—1271no—
180C2×C4⋊C4—2541no—
181C2×C4⋊C4C2×D12711yescharacteristic
182C2×C22⋊C4—2541no—
183C2×C22⋊C4—2541no—
184#103—22no—
185#102—2542no—
186C22⋊C8—2542no—
187C24—1271no—
188C24—2541no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8D50812yescomplemented
195C2×D4—1272no—
196C2×D4—5082no—
197C2×D4—5082no—
198C2×D4—2544no—
199C2×D4—2541no—
200C2×D4—2541no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2541no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1271no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4—2541no—
210C22×C4—2541no—
211C22×C4—2541no—
212C22×C4—2541no—
213C22×C4—2541no—
214C22×C4C22×D12711yescharacteristic
215SD16—2548no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—1272no—
220C2×C8—2541no—
221C2×C8—2541no—
222C4⋊C4—21no—
223C4⋊C4—1272no—
224C4⋊C4—2544no—
225C4⋊C4—2541no—
226C4⋊C4C22⋊D12712yes—
227C22⋊C4—2544no—
228C22⋊C4—2544no—
229C23—1272no—
230C23—5081no—
231C23—5081no—
232C23—5081no—
233C23—5081no—
234C23—5081no—
235C23—5081no—
236C23—2542no—
237C23—2541no—
238C23—2541no—
239C23—2541no—
240C23—2541no—
241C23—2541no—
242C23—2541no—
243C23D4×D12711yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
248D4—5088no—
249D4—5081no—
250D4—5081no—
251D4—2542no—
252D4—2542no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—5081no—
260C2×C4—5081no—
261C2×C4—5081no—
262C2×C4—5081no—
263C2×C4—5081no—
264C2×C4—5081no—
265C2×C4—2541no—
266C2×C4—2542no—
267C2×C4—2541no—
268C2×C4—2541no—
269C2×C4D4×D12711yescharacteristic
270C2×C4D4×D12711yescharacteristic
271C2×C4D4×D12711yescharacteristic
272C2×C4C2×C22⋊D12711yescharacteristic
273C2×C4C2×D50811yescharacteristic
274C2×C4#1452324811yescharacteristic
275C8—41no—
276C8—2542no—
277C22—10161no—
278C22—10161no—
279C22—10161no—
280C22—10161no—
281C22—5082no—
282C22—5082no—
283C22—5082no—
284C22—5081no—
285C22—5081no—
286C22—5081no—
287C22—5081no—
288C22—2542no—
289C22—2542no—
290C22—2542no—
291C22#1686117711yescharacteristic
292C22#1686117911yescharacteristic
293C22#1686118211yescharacteristic
294C22SD16×D12711yescharacteristic
295C22#1686123211yescharacteristic
296C22#1686123611yescharacteristic
297C22#1686124111yescharacteristic
298C4—42no—
299C4—42no—
300C4—5082no—
301C4—5081no—
302C4#1686117911yescharacteristic
303C4#1686118211yescharacteristic
304C4#1686118311yescharacteristic
305C4#1686120511yescharacteristic
306C2—10162no—
307C2—5082no—
308C2#2254978511yescharacteristic
309C2#2255001611yescharacteristic
310C2#2255002411yescharacteristic
311C2#2255002511yescharacteristic
312C2#2255003711yescharacteristic
313C2#2255010811yescharacteristic
314C2#2255012511yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861184C2
hypercenterC2×#103D127
hypocenterC127#2000
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#14523235C2semidirect
2. C2×#16861185C2semidirect
4. C2×#16861131C2semidirect
5. C2×(C2×D127)⋊C8C2semidirect
6. #22549638C2semidirect
7. C2×#103×C127C2semidirect
51. C2×C127⋊C8D4semidirect
53. C4⋊Dic127D4semidirect
62. C2×Q8×C127D4semidirect
84. C22×D127SD16semidirect
158. C127#2000semidirect
162. C2×#103D127semidirect
194. C2×Q8D508semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
C2×#168611852
C2×#168611842
C2×#168611312
C2×(C2×D127)⋊C82
#225496382
C2×#103×C1272
#2000127