Finite Group Databasebeta

← #30641808

Subgroups — #30641808

Order 16256.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861186C211yesmaximal, characteristic
2#22549782C211yesmaximal, characteristic, complemented
3#22549780C211yesmaximal, characteristic, complemented
4#22549763C211yesmaximal, characteristic, complemented
5#22549763C211yesmaximal, characteristic, complemented
6#22549760C211yesmaximal, characteristic, complemented
7#22549759C211yesmaximal, characteristic, complemented
8#22549751C211yesmaximal, characteristic
9#22549749C211yesmaximal, characteristic
10C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic127C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic127C211yesmaximal, characteristic
13C22⋊C4×Dic127C211yesmaximal, characteristic
14C2×C127⋊#95C211yesmaximal, characteristic
15C4×C22⋊C4×C127C211yesmaximal, characteristic
16C22×Dic254C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18C23×Dic127C2211yescharacteristic
19#16861186C418yescomplemented
20C2×C4⋊Dic127C2211yescharacteristic
21C2×C4⋊Dic127C2211yescharacteristic
22C2×C4⋊Dic127C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C4×Dic127C2211yescharacteristic
30C2×C4×Dic127C2211yescharacteristic
31C2×C4×Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C127⋊#95C2211yescharacteristic
34C127⋊#95C2211yescharacteristic
35C127⋊#95C2211yescharacteristic
36C127⋊#95C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C23×C4×C127C2211yescharacteristic
46C2×C22⋊C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C42×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C2×Dic254—41no—
53C2×Dic254—41no—
54C2×Dic254C2×C414yescomplemented
55C22×Dic127—22no—
56C22×Dic127—22no—
57C22×Dic127—41no—
58C22×Dic127—41no—
59C22×Dic127—41no—
60C22×Dic127C2×C414yes—
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C4⋊Dic127—22no—
70C4⋊Dic127—41no—
71C4⋊Dic127C2×C414yescomplemented
72Dic127⋊C4—22no—
73Dic127⋊C4—41no—
74Dic127⋊C4C2×C414yescomplemented
75Dic127⋊C4C2×C414yescomplemented
76C22⋊Dic127—22no—
77C22⋊Dic127C2×C414yes—
78C4×Dic127—22no—
79C4×Dic127—41no—
80C4×Dic127—41no—
81C4×Dic127—41no—
82C24×C127C2311yescharacteristic
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—41no—
86C22×C4×C127C2311yescharacteristic
87C22×C4×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22⋊C4×C127C2×C414yes—
93C22⋊C4×C127Q814yescomplemented
94C42×C127—22no—
95C42×C127—41no—
96Dic254—44no—
97C2×Dic127—21no—
98C2×Dic127—21no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—21no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—21no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—44no—
117C2×Dic127—44no—
118C2×Dic127C22×C412yes—
119C2×Dic127C22×C412yes—
120C2×Dic127C22×C412yes—
121C2×Dic127C22×C412yes—
122C2×Dic127#4112yes—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127C22×C412yes—
128C23×C127C2×Q812yes—
129C23×C127#4112yes—
130C23×C127C2411yescharacteristic
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—44no—
145C2×C4×C127C22×C412yes—
146C2×C4×C127C22×C412yes—
147C2×C4×C127C2×Q812yes—
148C2×C4×C127C2×Q812yes—
149C2×C4×C127#4112yes—
150C22×C127—22no—
151C22×C127—22no—
152C22×C127—22no—
153C22×C127—22no—
154C22×C127—22no—
155C22×C127—22no—
156C22×C127C4×Q814yes—
157C22×C127C23×C411yescharacteristic
158C22×C127C22×Q811yescharacteristic
159C22×C127C2×#4111yescharacteristic
160C22×C127C2×#4111yescharacteristic
161C22×C127C2×#4111yescharacteristic
162C22×C127H(5,𝔽2)11yescharacteristic
163C22×C127H(5,𝔽2)11yescharacteristic
164C4×C127—22no—
165C4×C127—22no—
166C4×C127—22no—
167C4×C127—22no—
168C4×C127—22no—
169C4×C127—41no—
170Dic127—22no—
171Dic127—22no—
172Dic127—22no—
173Dic127—22no—
174Dic127—22no—
175Dic127—41no—
176Dic127—41no—
177Dic127—41no—
178C2×C127—24no—
179C2×C127C2×C4×Q811yescharacteristic
180C2×C127C4×#4111yescharacteristic
181C2×C127#51811yescharacteristic
182C2×C127#55111yescharacteristic
183C2×C127#55111yescharacteristic
184C2×C127#55411yescharacteristic
185C2×C127#55411yescharacteristic
186#2368—1271nomaximal, Sylow
187C127#236811yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—1271no—
189#391—1271no—
190#391—1271no—
191#387—1271no—
192#387—1271no—
193#387—1271no—
194#387—1271no—
195#385—1271no—
196#385—1271no—
197C4×C4⋊C4—1271no—
198C4×C4⋊C4—1271no—
199C4×C22⋊C4—1271no—
200C4×C22⋊C4—1271no—
201C4×C22⋊C4D12711yescharacteristic
202C2×#95—1271no—
203C22×Q8—1271no—
204C23×C4—1271no—
205C23×C4—1271no—
206C23×C4C2×D12711yescharacteristic
207C22⋊Q8—1278no—
208C2×C4⋊C4—1271no—
209C2×C4⋊C4—1271no—
210C2×C4⋊C4—1271no—
211C2×C4⋊C4—1271no—
212C2×C4⋊C4—1271no—
213C2×C4⋊C4—1271no—
214C2×C4⋊C4—1271no—
215C2×C22⋊C4—1271no—
216C2×C22⋊C4—1271no—
217C2×C22⋊C4C2×D12711yescharacteristic
218C2×C22⋊C4C2×D12711yescharacteristic
219C2×C42—1271no—
220C2×C42—1271no—
221C2×C42—1271no—
222C2×C42—1271no—
223C2×C42C2×D12711yescharacteristic
224C2×C42C2×D12711yescharacteristic
225#95—1271no—
226#95—1271no—
227#95—1271no—
228#95—1271no—
229#95—1271no—
230#95—1271no—
231#95—1271no—
232#95—1271no—
233#95—1271no—
234#95—1271no—
235#95—1271no—
236#95—1271no—
237#95C2×D12711yescharacteristic
238#95C2×D12711yescharacteristic
239C24C22×D12711yescharacteristic
240C2×Q8—1274no—
241C2×Q8—5081no—
242C2×Q8—5081no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—1274no—
247C22×C4—1271no—
248C22×C4—1271no—
249C22×C4—1271no—
250C22×C4—1271no—
251C22×C4—1271no—
252C22×C4—1271no—
253C22×C4—1271no—
254C22×C4—1271no—
255C22×C4—5081no—
256C22×C4—5081no—
257C22×C4—5081no—
258C22×C4—2542no—
259C22×C4—2542no—
260C22×C4C22×D12711yescharacteristic
261C22×C4C22×D12711yescharacteristic
262C22×C4C22×D12711yescharacteristic
263C22×C4C22×D12711yescharacteristic
264C22×C4C22×D12711yescharacteristic
265C22×C4C22×D12711yescharacteristic
266C4⋊C4—1274no—
267C4⋊C4—1274no—
268C4⋊C4—1274no—
269C4⋊C4—5081no—
270C4⋊C4—5081no—
271C4⋊C4—2542no—
272C4⋊C4—2542no—
273C22⋊C4—1274no—
274C22⋊C4—2542no—
275C22⋊C4C4×D12714yes—
276C22⋊C4Dic25414yescomplemented
277C42—22no—
278C42—41no—
279C42—5081no—
280C42—5081no—
281C42—5081no—
282C42—2542no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23C2×C4×D12712yes—
288C23C2×Dic25412yes—
289C23#1452324812yes—
290C23C23×D12711yescharacteristic
291Q8—5084no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—44no—
306C2×C4—1272no—
307C2×C4—1272no—
308C2×C4—1272no—
309C2×C4—1272no—
310C2×C4—1272no—
311C2×C4—5084no—
312C2×C4—5084no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4C2×C4×D12712yes—
333C2×C4C2×C4×D12712yes—
334C2×C4C2×Dic25412yes—
335C2×C4C2×Dic25412yes—
336C2×C4#1452324712yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22C4×Dic25414yes—
344C22C22×C4×D12711yescharacteristic
345C22C22×Dic25411yescharacteristic
346C22#41×D12711yescharacteristic
347C22C2×#1452324711yescharacteristic
348C22C2×#1452324811yescharacteristic
349C22#1686125411yescharacteristic
350C22#1686125511yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—5081no—
358C4—5081no—
359C4—5081no—
360C4—2542no—
361C4—2542no—
362C4—2542no—
363C4—2542no—
364C4—2542no—
365C2—24no—
366C2C2×C4×Dic25411yescharacteristic
367C2C4×#1452324711yescharacteristic
368C2#2255034811yescharacteristic
369C2#2255055011yescharacteristic
370C2#2255055211yescharacteristic
371C2#2255058311yescharacteristic
372C2#2255058511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC4×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenterC4×C22⋊C4D127
hypocenterC127#2368
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22549782C2semidirect
3. #22549780C2semidirect
4. #22549763C2semidirect
5. #22549763C2semidirect
6. #22549760C2semidirect
7. #22549759C2semidirect
10. C4×Dic127⋊C4C2semidirect
11. C4×C4⋊Dic127C2semidirect
19. #16861186C4semidirect
54. C2×Dic254C2×C4semidirect
71. C4⋊Dic127C2×C4semidirect
74. Dic127⋊C4C2×C4semidirect
75. Dic127⋊C4C2×C4semidirect
93. C22⋊C4×C127Q8semidirect
187. C127#2368semidirect
276. C22⋊C4Dic254semidirect

Maximal subgroups

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

subgroupindex
C2×#168611862
#225497822
#225497802
#225497632
#225497602
#225497592
#225497512
#225497492
C4×Dic127⋊C42
C4×C4⋊Dic1272
C22⋊C4×Dic1272
C2×C127⋊#952
C4×C22⋊C4×C1272
#2368127