Finite Group Databasebeta

← #30639735

Subgroups — #30639735

Order 16256.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14523237C211yesmaximal, characteristic, complemented
2C2×#16861188C211yesmaximal, characteristic, complemented
3C2×#16861183C211yesmaximal, characteristic
4C2×#16861134C211yesmaximal, characteristic, complemented
5C2×#16861131C211yesmaximal, characteristic, complemented
6#22549639C211yesmaximal, characteristic, complemented
7C2×C22⋊C8×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C2×#14523237—22no—
12C2×#14523237—22no—
13C2×#14523237—22no—
14#16861188—22no—
15#16861188—22no—
16#16861183—22no—
17#16861183—22no—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4—21no—
21C2×C22⋊Dic127—21no—
22C2×(C2×D127)⋊C4—21no—
23#16861134—22no—
24#16861131—22no—
25C23×C4×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27C22×C8×C127C2211yescharacteristic
28C22⋊C8×C127—22no—
29C23×D127—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254D412yescomplemented
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508D412yes—
42C2×C22⋊D127—24no—
43C2×C22⋊D127—42no—
44C2×C22⋊D127—42no—
45C22×Dic127—21no—
46C22×Dic127—21no—
47C22×Dic127—21no—
48#14523237—28no—
49C4⋊Dic127—21no—
50C4⋊Dic127—21no—
51C4⋊Dic127D412yes—
52C4⋊Dic127D412yescomplemented
53Dic127⋊C4—24no—
54C22⋊Dic127—24no—
55(C2×D127)⋊C4—24no—
56C24×C127D411yescharacteristic
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127D411yescharacteristic
64C22×C4×C127C2311yescharacteristic
65C2×C8×C127—21no—
66C2×C8×C127—21no—
67C2×C8×C127—21no—
68C2×C8×C127—21no—
69C2×C8×C127—21no—
70C2×C8×C127D412yescomplemented
71C22×D127—22no—
72C22×D127—41no—
73C22×D127—41no—
74C22×D127—41no—
75C22×D127—41no—
76C22×D127—41no—
77C22×D127—41no—
78Dic254—22no—
79Dic254—22no—
80Dic254—41no—
81Dic254—41no—
82D508—22no—
83D508—22no—
84D508—41no—
85D508—41no—
86C22⋊D127—48no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—41no—
91C2×Dic127—41no—
92C2×Dic127—41no—
93C2×Dic127—41no—
94C2×Dic127—41no—
95C2×Dic127—41no—
96C23×C127—21no—
97C23×C127—21no—
98C23×C127—21no—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127SD1612yescomplemented
103C23×C127C2×D411yescharacteristic
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127#4111yescharacteristic
117C8×C127—22no—
118C8×C127—41no—
119C22×C127—22no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127#12011yescharacteristic
127C22×C127#12011yescharacteristic
128C22×C127#12111yescharacteristic
129C22×C127C2×SD1611yescharacteristic
130C22×C127#13511yescharacteristic
131C22×C127AGL(1,ℤ/8)11yescharacteristic
132C22×C127#13711yescharacteristic
133C2×D127—42no—
134C2×D127—42no—
135C2×D127—42no—
136C2×D127—81no—
137C2×D127—81no—
138C2×D127—81no—
139C2×D127—81no—
140C4×C127—41no—
141C4×C127#12011yescharacteristic
142C4×C127#12111yescharacteristic
143C4×C127#12111yescharacteristic
144C4×C127#12711yescharacteristic
145Dic127—42no—
146Dic127—42no—
147Dic127—42no—
148C2×C127—42no—
149C2×C127#39211yescharacteristic
150C2×C127#44811yescharacteristic
151C2×C127#44911yescharacteristic
152C2×C127#45011yescharacteristic
153C2×C127#45211yescharacteristic
154C2×C127#46511yescharacteristic
155C2×C127#46811yescharacteristic
156D127—82no—
157#2000—1271nomaximal, Sylow
158C127#200011yescharacteristic, complemented, Sylow
159C22×SD16—1271no—
160C2×C22⋊Q8—1271no—
161C2×#121—1271no—
162C2×#103—1271no—
163C2×#102—1271no—
164C2×C22⋊C8D12711yescharacteristic, complemented
165#340—1271no—
166C22×Q8—1271no—
167C22×D4—1271no—
168C22×D4—2541no—
169C23×C4C2×D12711yescharacteristic
170C2×SD16—2542no—
171C2×SD16—2542no—
172C2×SD16—2542no—
173C22×C8C2×D12711yescharacteristic
174C22×C8C2×D12711yescharacteristic
175C22⋊Q8—2542no—
176C22⋊Q8—2542no—
177#121—2542no—
178#121—2542no—
179C2×C4⋊C4—1271no—
180C2×C4⋊C4—1271no—
181C2×C4⋊C4—2541no—
182C2×C22⋊C4—2541no—
183C2×C22⋊C4—2541no—
184#103—2542no—
185#102—2542no—
186C22⋊C8—22no—
187C24—2541no—
188C24D50811yescharacteristic
189C2×Q8—1272no—
190C2×Q8—2541no—
191C2×Q8—2541no—
192C2×Q8—2541no—
193C2×Q8—2541no—
194C2×Q8—2541no—
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—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—2541no—
211C22×C4—2541no—
212C22×C4—2541no—
213C22×C4D50811yescharacteristic
214C22×C4C22×D12711yescharacteristic
215SD16—2548no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8C22⋊D12712yescomplemented
222C4⋊C4—1272no—
223C4⋊C4—1272no—
224C4⋊C4—2544no—
225C4⋊C4—2541no—
226C4⋊C4—2541no—
227C22⋊C4—2544no—
228C22⋊C4—2544no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—5081no—
236C23—5081no—
237C23—5081no—
238C23—5081no—
239C23—5081no—
240C23—5081no—
241C23—2542no—
242C23#1452323712yescomplemented
243C23C2×D50811yescharacteristic
244Q8—5081no—
245Q8—5081no—
246Q8—2542no—
247Q8—2542no—
248D4—5088no—
249D4—5081no—
250D4—5081no—
251D4—2542no—
252D4—2542no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—5081no—
261C2×C4—5081no—
262C2×C4—5081no—
263C2×C4—5081no—
264C2×C4—5081no—
265C2×C4—5081no—
266C2×C4—2542no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4D4×D12711yescharacteristic
270C2×C4D4×D12711yescharacteristic
271C2×C4D4×D12711yescharacteristic
272C2×C4D4×D12711yescharacteristic
273C2×C4C2×C22⋊D12711yescharacteristic
274C2×C4#1452324811yescharacteristic
275C8—22no—
276C8—41no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—10161no—
285C22—10161no—
286C22—10161no—
287C22—10161no—
288C22—5082no—
289C22—5082no—
290C22—5082no—
291C22#1686117911yescharacteristic
292C22#1686117911yescharacteristic
293C22#1686118311yescharacteristic
294C22C2×#1452323711yescharacteristic
295C22#1686123111yescharacteristic
296C22#1686123711yescharacteristic
297C22#1686124211yescharacteristic
298C4—41no—
299C4—5082no—
300C4—5082no—
301C4—5082no—
302C4#1686117711yescharacteristic
303C4#1686118211yescharacteristic
304C4#1686118211yescharacteristic
305C4#1686120511yescharacteristic
306C2—42no—
307C2—10162no—
308C2#2254978511yescharacteristic
309C2#2255001711yescharacteristic
310C2#2255002311yescharacteristic
311C2#2255002911yescharacteristic
312C2#2255004011yescharacteristic
313C2#2255010511yescharacteristic
314C2#2255012211yescharacteristic
subgroupG/H
centerC23C2×D508
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×C22⋊C8×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
hypercenterC2×C22⋊C8D127
hypocenterC127#2000
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×#14523237C2semidirect
2. C2×#16861188C2semidirect
4. C2×#16861134C2semidirect
5. C2×#16861131C2semidirect
6. #22549639C2semidirect
7. C2×C22⋊C8×C127C2semidirect
35. C2×Dic254D4semidirect
52. C4⋊Dic127D4semidirect
70. C2×C8×C127D4semidirect
102. C23×C127SD16semidirect
158. C127#2000semidirect
164. C2×C22⋊C8D127semidirect
221. C2×C8C22⋊D127semidirect
242. C23#14523237semidirect

Maximal subgroups

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

subgroupindex
C22×#145232372
C2×#168611882
C2×#168611832
C2×#168611342
C2×#168611312
#225496392
C2×C22⋊C8×C1272
#2000127