Finite Group Databasebeta

← #30638643

Subgroups — #30638643

Order 16256.

880 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 105 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×D4×D127C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic, complemented
3C2×#16861123C212yesmaximal, complemented
4#22549731C212yesmaximal, complemented
5C2×#99×C127C211yesmaximal, characteristic, complemented
6C24×D127C411yescharacteristic, complemented
7C24×D127C2211yescharacteristic
8C22×D508C411yescharacteristic, complemented
9C22×C22⋊D127C411yescharacteristic, complemented
10C22×C22⋊D127C2211yescharacteristic
11C2×D4×D127—24no—
12C2×D4×D127—24no—
13C2×D4×D127—22no—
14C2×D4×D127—22no—
15C22×C4×D127C411yescharacteristic, complemented
16C2×C22⋊Dic127C2212yescomplemented
17C2×(C2×D127)⋊C4—24no—
18#16861124—22no—
19#16861123—24no—
20C22×D4×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2212yescomplemented
22#99×C127—22no—
23C23×D127—22no—
24C23×D127—22no—
25C23×D127—22no—
26C23×D127—22no—
27C23×D127—21no—
28C23×D127—21no—
29C23×D127—44no—
30C23×D127—44no—
31C23×D127C2×C411yescharacteristic
32C23×D127C2×C411yescharacteristic
33C23×D127C2×C411yescharacteristic
34C23×D127D414yes—
35C2×D508—22no—
36C2×D508—22no—
37C2×D508—22no—
38C2×C22⋊D127—22no—
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C2×C22⋊D127—22no—
42C2×C22⋊D127—21no—
43C2×C22⋊D127—21no—
44C2×C22⋊D127D414yes—
45D4×D127—216no—
46D4×D127—216no—
47C22×Dic127—22no—
48C22×Dic127C2×C411yescharacteristic
49C2×C4×D127—22no—
50C2×C4×D127—22no—
51C2×C4×D127—22no—
52C22⋊Dic127—22no—
53C22⋊Dic127—22no—
54(C2×D127)⋊C4—48no—
55C24×C127C2×C411yescharacteristic
56C24×C127C2311yescharacteristic
57C2×D4×C127—22no—
58C2×D4×C127—22no—
59C2×D4×C127D412yes—
60C2×D4×C127D412yes—
61C22×C4×C127—22no—
62C22×C4×C127C2×C411yescharacteristic
63C22⋊C4×C127—22no—
64C22⋊C4×C127—22no—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—24no—
68C22×D127—28no—
69C22×D127—21no—
70C22×D127—21no—
71C22×D127—21no—
72C22×D127—21no—
73C22×D127—24no—
74C22×D127—24no—
75C22×D127—28no—
76C22×D127—42no—
77C22×D127—44no—
78C22×D127—48no—
79C22×D127—44no—
80C22×D127—44no—
81C22×D127—48no—
82C22×D127—48no—
83C22×D127—44no—
84C22×D127—44no—
85C22×D127—42no—
86C22×D127C22⋊C412yes—
87C22×D127C22⋊C412yes—
88C22×D127C22⋊C412yescomplemented
89D508—28no—
90C22⋊D127—28no—
91C22⋊D127—24no—
92C22⋊D127—24no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—44no—
96C2×Dic127—42no—
97C2×Dic127C22⋊C412yescomplemented
98C4×D127—28no—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—42no—
102C23×C127—42no—
103C23×C127C22⋊C412yescomplemented
104C23×C127C22×C411yescharacteristic
105C23×C127C2×D412yes—
106C23×C127C2×D412yes—
107C23×C127C2×D411yescharacteristic
108C23×C127C2×D411yescharacteristic
109D4×C127—24no—
110D4×C127—24no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—44no—
114C2×C4×C127—42no—
115C2×C4×C127C22⋊C412yescomplemented
116C22×C127—24no—
117C22×C127—22no—
118C22×C127—22no—
119C22×C127—42no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127C2×C22⋊C411yescharacteristic
126C22×C127C2×C22⋊C411yescharacteristic
127C22×C127C2×C22⋊C411yescharacteristic
128C22×C127#12012yes—
129C22×C127#12012yes—
130C2×D127—24no—
131C2×D127—24no—
132C2×D127—22no—
133C2×D127—22no—
134C2×D127—22no—
135C2×D127—22no—
136C2×D127—24no—
137C2×D127—48no—
138C2×D127—48no—
139C2×D127—416no—
140C2×D127—42no—
141C2×D127—42no—
142C2×D127—42no—
143C2×D127—42no—
144C2×D127—42no—
145C2×D127#9914yescomplemented
146C4×C127—22no—
147C4×C127—82no—
148Dic127—22no—
149Dic127—82no—
150C2×C127—22no—
151C2×C127—22no—
152C2×C127—21no—
153C2×C127—21no—
154C2×C127—42no—
155C2×C127#37911yescharacteristic
156C2×C127C2×#9912yes—
157D127—24no—
158D127—42no—
159D127—42no—
160#1781—1271nomaximal, Sylow
161C127#178111yescharacteristic, complemented, Sylow
162C23×D4—1271no—
163C2×#99—1271no—
164C2×#99—1272no—
165C2×#99D12711yescharacteristic, complemented
166#379—1272no—
167C25—1271no—
168C25—1271no—
169C22×D4—1271no—
170C22×D4—1271no—
171C22×D4—1271no—
172C22×D4—2544no—
173C22×D4—2544no—
174C22×D4—2542no—
175C22×D4—2542no—
176C22×D4C2×D12711yescharacteristic
177C23×C4—1271no—
178C2×C22⋊C4—1272no—
179C2×C22⋊C4—2544no—
180C2×C22⋊C4C2×D12712yescomplemented
181#99—22no—
182#99—2544no—
183#99—2542no—
184C24—1271no—
185C24—1271no—
186C24—1274no—
187C24—1271no—
188C24—5084no—
189C24—5084no—
190C24—2542no—
191C24—2542no—
192C24—2542no—
193C24—2542no—
194C24—2541no—
195C24—2541no—
196C24C4×D12711yescharacteristic
197C24C22×D12711yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—1274no—
201C2×D4—2542no—
202C2×D4—2542no—
203C2×D4—2542no—
204C2×D4—25416no—
205C2×D4—25416no—
206C2×D4—2542no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4—2541no—
210C2×D4—2541no—
211C2×D4—2542no—
212C2×D4C22⋊D12712yes—
213C2×D4D50812yes—
214C22×C4—22no—
215C22×C4—1271no—
216C22×C4—2542no—
217C22×C4—2542no—
218C22×C4—2542no—
219C22×C4—2542no—
220C22×C4C4×D12711yescharacteristic
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—5088no—
224C22⋊C4—2542no—
225C22⋊C4—2542no—
226C23—21no—
227C23—21no—
228C23—42no—
229C23—42no—
230C23—1272no—
231C23—1272no—
232C23—1272no—
233C23—5082no—
234C23—5084no—
235C23—5082no—
236C23—5084no—
237C23—5088no—
238C23—5084no—
239C23—5084no—
240C23—5088no—
241C23—5088no—
242C23—5084no—
243C23—2548no—
244C23—2544no—
245C23—2548no—
246C23—2541no—
247C23—2541no—
248C23—2541no—
249C23—2541no—
250C23—2544no—
251C23—2544no—
252C23—2541no—
253C23—2541no—
254C23(C2×D127)⋊C412yescomplemented
255C23C2×C4×D12711yescharacteristic
256C23D4×D12712yes—
257C23D4×D12712yes—
258C23C2×C22⋊D12711yescharacteristic
259C23C2×D50811yescharacteristic
260D4—24no—
261D4—24no—
262D4—2548no—
263D4—2548no—
264D4—2544no—
265D4—2544no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—1272no—
271C2×C4—5084no—
272C2×C4—5082no—
273C2×C4—2548no—
274C2×C4—2541no—
275C2×C4—2541no—
276C2×C4(C2×D127)⋊C412yescomplemented
277C22—24no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—1274no—
287C22—5082no—
288C22—5082no—
289C22—5082no—
290C22—5082no—
291C22—5088no—
292C22—5088no—
293C22—50816no—
294C22—5082no—
295C22—2544no—
296C22—2544no—
297C22—2544no—
298C22—2542no—
299C22—2542no—
300C22—2542no—
301C22—2542no—
302C22C22⋊C4×D12711yescharacteristic
303C22C22⋊C4×D12711yescharacteristic
304C22C2×(C2×D127)⋊C411yescharacteristic
305C22#1686117712yes—
306C22#1686117912yes—
307C4—22no—
308C4—82no—
309C4—10162no—
310C4—2542no—
311C2—22no—
312C2—22no—
313C2—21no—
314C2—21no—
315C2—42no—
316C2—5082no—
317C2—5082no—
318C2—2544no—
319C2#2254973111yescharacteristic
320C2#99×D12712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC2×#99×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×D127C2
cosocleC24×C127C23
hypercenterC2×#99D127
hypocenterC127#1781
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16861124C2semidirect
3. C2×#16861123C2semidirect
4. #22549731C2semidirect
5. C2×#99×C127C2semidirect
6. C24×D127C4semidirect
8. C22×D508C4semidirect
9. C22×C22⋊D127C4semidirect
15. C22×C4×D127C4semidirect
16. C2×C22⋊Dic127C22semidirect
21. C2×C22⋊C4×C127C22semidirect
88. C22×D127C22⋊C4semidirect
97. C2×Dic127C22⋊C4semidirect
103. C23×C127C22⋊C4semidirect
115. C2×C4×C127C22⋊C4semidirect
145. C2×D127#99semidirect
161. C127#1781semidirect
165. C2×#99D127semidirect
180. C2×C22⋊C4C2×D127semidirect
254. C23(C2×D127)⋊C4semidirect
276. C2×C4(C2×D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×#168611242
C2×#168611232
#225497312
C2×#99×C1272
#1781127