Finite Group Databasebeta

← #30638663

Subgroups — #30638663

Order 16256.

592 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 125 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×Q8×D127C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic, complemented
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×#16861132C211yesmaximal, characteristic, complemented
5C2×(C2×D127)⋊C8C211yesmaximal, characteristic
6#22549733C211yesmaximal, characteristic
7C2×#103×C127C211yesmaximal, characteristic, complemented
8C22×Dic254—21no—
9C22×Dic254C2211yescharacteristic
10C2×Q8×D127—22no—
11C2×Q8×D127—22no—
12C2×Q8×D127—22no—
13C2×Q8×D127—22no—
14C2×Q8×D127—22no—
15C2×Q8×D127C414yescomplemented
16C22×C4×D127—21no—
17C22×C4×D127C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×(C2×D127)⋊C4—21no—
21#16861134—22no—
22#16861133—22no—
23#16861132—22no—
24(C2×D127)⋊C8—22no—
25C127⋊#95—21no—
26C22×Q8×C127C2211yescharacteristic
27C22×C8×C127C2211yescharacteristic
28C2×C4⋊C4×C127C2211yescharacteristic
29#103×C127—22no—
30C23×D127D411yescharacteristic
31C2×Dic254—24no—
32C2×Dic254—24no—
33C2×Dic254—24no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254C2×C412yes—
38C2×Dic254D412yes—
39C2×Dic254D412yes—
40Q8×D127—28no—
41Q8×D127—28no—
42Q8×D127—28no—
43Q8×D127—28no—
44C22×Dic127—21no—
45C22×Dic127—21no—
46C22×Dic127D411yescharacteristic
47C2×C4×D127—24no—
48C2×C4×D127—24no—
49C2×C4×D127—24no—
50C2×C4×D127—21no—
51C2×C4×D127—21no—
52C2×C4×D127—21no—
53C2×C4×D127—21no—
54C2×C4×D127C2×C412yes—
55C2×C4×D127D412yes—
56C2×C127⋊C8—21no—
57C2×C127⋊C8—21no—
58C2×C127⋊C8—21no—
59C4⋊Dic127—21no—
60C4⋊Dic127—21no—
61(C2×D127)⋊C4—42no—
62C2×Q8×C127—21no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127C2×C412yes—
66C2×Q8×C127D412yes—
67C2×Q8×C127D412yes—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71C2×C8×C127—21no—
72C2×C8×C127—21no—
73C2×C8×C127—21no—
74C4⋊C4×C127—21no—
75C4⋊C4×C127—21no—
76C22×D127—21no—
77C22×D127—21no—
78C22×D127—21no—
79C22×D127—21no—
80C22×D127C22⋊C412yes—
81C22×D127SD1612yes—
82C22×D127Q1612yes—
83Dic254—216no—
84Dic254—22no—
85Dic254—22no—
86Dic254—22no—
87Dic254C22⋊C414yescomplemented
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—22no—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C2×Dic127—41no—
94C2×Dic127—41no—
95C2×Dic127—41no—
96C2×Dic127C22⋊C412yes—
97C4×D127—216no—
98C4×D127—22no—
99C4×D127—22no—
100C4×D127—22no—
101C4×D127—22no—
102C127⋊C8—41no—
103C23×C127C2×D411yescharacteristic
104Q8×C127—22no—
105Q8×C127—22no—
106Q8×C127—22no—
107Q8×C127C22⋊C414yescomplemented
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127C22×C411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C8×C127—41no—
121C22×C127C2×C22⋊C411yescharacteristic
122C22×C127#12011yescharacteristic
123C22×C127#12011yescharacteristic
124C22×C127C2×SD1611yescharacteristic
125C22×C127C2×Q1611yescharacteristic
126C22×C127#13711yescharacteristic
127C22×C127#13711yescharacteristic
128C2×D127—22no—
129C2×D127—22no—
130C2×D127—22no—
131C2×D127—22no—
132C2×D127—22no—
133C2×D127—22no—
134C2×D127#10314yescomplemented
135C4×C127—24no—
136C4×C127—81no—
137C4×C127C2×C22⋊C411yescharacteristic
138C4×C127C2×C22⋊C411yescharacteristic
139C4×C127#12011yescharacteristic
140C4×C127#12011yescharacteristic
141Dic127—24no—
142Dic127—22no—
143Dic127—81no—
144C2×C127#37911yescharacteristic
145C2×C127C2×#10311yescharacteristic
146C2×C127#41911yescharacteristic
147C2×C127#44811yescharacteristic
148C2×C127#44811yescharacteristic
149C2×C127#45111yescharacteristic
150C2×C127#45111yescharacteristic
151D127—24no—
152#1787—1271nomaximal, Sylow
153C127#178711yescharacteristic, complemented, Sylow
154C23×Q8—1271no—
155C2×#103—1271no—
156C2×#103—1271no—
157C2×#103—1271no—
158C2×#103D12711yescharacteristic, complemented
159C2×C22⋊C8—1271no—
160#380—1271no—
161C22×Q8—1274no—
162C22×Q8—1271no—
163C22×Q8—2541no—
164C22×Q8—2542no—
165C22×Q8—2542no—
166C22×Q8—2542no—
167C22×Q8—2542no—
168C22×Q8—2542no—
169C22×Q8C2×D12711yescharacteristic
170C23×C4—1271no—
171C23×C4—2541no—
172C22×C8—1271no—
173C22×C8C2×D12711yescharacteristic
174C2×C4⋊C4—1271no—
175C2×C4⋊C4C2×D12711yescharacteristic
176C2×C22⋊C4—2541no—
177#103—22no—
178#103—2542no—
179#103—2542no—
180#103—2542no—
181C22⋊C8—2542no—
182#95—2541no—
183C24—1271no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—1272no—
188C2×Q8—1272no—
189C2×Q8—1272no—
190C2×Q8—2544no—
191C2×Q8—2544no—
192C2×Q8—2544no—
193C2×Q8—2548no—
194C2×Q8—2548no—
195C2×Q8—2548no—
196C2×Q8—2548no—
197C2×Q8—2541no—
198C2×Q8—2541no—
199C2×Q8—2541no—
200C2×Q8C4×D12712yes—
201C2×Q8C22⋊D12712yes—
202C2×Q8D50812yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—1271no—
206C22×C4—1272no—
207C22×C4—1272no—
208C22×C4—2544no—
209C22×C4—2544no—
210C22×C4—2544no—
211C22×C4—2541no—
212C22×C4—2541no—
213C22×C4—2541no—
214C22×C4—2541no—
215C22×C4—2541no—
216C22×C4—2541no—
217C22×C4C22×D12711yescharacteristic
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—2541no—
222C2×C8—2541no—
223C2×C8—2541no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—2541no—
227C4⋊C4—2541no—
228C22⋊C4—5082no—
229C23—1272no—
230C23—1272no—
231C23—1272no—
232C23—2541no—
233C23—2541no—
234C23—2541no—
235C23—2541no—
236C23D4×D12711yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—22no—
240Q8—1274no—
241Q8—25416no—
242Q8—2542no—
243Q8—2542no—
244Q8—2542no—
245Q8(C2×D127)⋊C414yescomplemented
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—1272no—
253C2×C4—5081no—
254C2×C4—5081no—
255C2×C4—5081no—
256C2×C4—25416no—
257C2×C4—2542no—
258C2×C4—2541no—
259C2×C4—2542no—
260C2×C4—2541no—
261C2×C4—2542no—
262C2×C4—2542no—
263C2×C4—2542no—
264C2×C4—2542no—
265C2×C4—2542no—
266C2×C4C2×C4×D12711yescharacteristic
267C2×C4D4×D12711yescharacteristic
268C2×C4D4×D12711yescharacteristic
269C2×C4D4×D12711yescharacteristic
270C2×C4C2×C22⋊D12711yescharacteristic
271C2×C4C2×D50811yescharacteristic
272C8—41no—
273C8—5081no—
274C22—1274no—
275C22—2542no—
276C22—2542no—
277C22—2542no—
278C22—2542no—
279C22—2542no—
280C22—2542no—
281C22C22⋊C4×D12711yescharacteristic
282C22#1686117711yescharacteristic
283C22#1686117911yescharacteristic
284C22SD16×D12711yescharacteristic
285C22Q16×D12711yescharacteristic
286C22#1686123811yescharacteristic
287C22#1686124111yescharacteristic
288C4—24no—
289C4—81no—
290C4—10161no—
291C4—2542no—
292C4—2544no—
293C4C22⋊C4×D12711yescharacteristic
294C4C2×(C2×D127)⋊C411yescharacteristic
295C4#1686117711yescharacteristic
296C4#1686117911yescharacteristic
297C2—2544no—
298C2#2254973111yescharacteristic
299C2#103×D12711yescharacteristic
300C2#2254988311yescharacteristic
301C2#2255001311yescharacteristic
302C2#2255001611yescharacteristic
303C2#2255003011yescharacteristic
304C2#2255003311yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
prime radicalC23×D127D4
cosocleC22×C4×C127C23
hypercenterC2×#103D127
hypocenterC127#1787
LCSG ▷ C2×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
2. C2×#16861134C2semidirect
3. C2×#16861133C2semidirect
4. C2×#16861132C2semidirect
7. C2×#103×C127C2semidirect
15. C2×Q8×D127C4semidirect
87. Dic254C22⋊C4semidirect
107. Q8×C127C22⋊C4semidirect
134. C2×D127#103semidirect
153. C127#1787semidirect
158. C2×#103D127semidirect
245. Q8(C2×D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1272
C2×#168611342
C2×#168611332
C2×#168611322
C2×(C2×D127)⋊C82
#225497332
C2×#103×C1272
#1787127