Finite Group Databasebeta

← #30642006

Subgroups — #30642006

Order 16256.

528 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic254C211yesmaximal, characteristic, complemented
2C2×#16861166C211yesmaximal, characteristic
3#22549817C211yesmaximal, characteristic, complemented
4#22549811C211yesmaximal, characteristic
5#22549810C211yesmaximal, characteristic, complemented
6#22549807C211yesmaximal, characteristic, complemented
7#22549800C211yesmaximal, characteristic
8#22549798C211yesmaximal, characteristic
9#22549796C211yesmaximal, characteristic, complemented
10#22549789C211yesmaximal, characteristic
11C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
12C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
13C4×(C2×D127)⋊C4C211yesmaximal, characteristic
14C4×(C2×D127)⋊C4C211yesmaximal, characteristic
15C4×C4⋊C4×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18C4×Dic254—24no—
19#16861166—24no—
20C2×C4⋊Dic127C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×Dic127⋊C4C2211yescharacteristic
28C2×Dic127⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×(C2×D127)⋊C4C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×C4×Dic127C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
37C2×C4×Dic127C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C2×C4⋊C4×C127C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C42×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic
50C2×C42×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53C23×D127C2311yescharacteristic
54C2×Dic254—22no—
55C2×Dic254—22no—
56C2×Dic254—22no—
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C2×C4×D127—22no—
65C2×C4×D127—22no—
66C2×C4×D127Q814yescomplemented
67C4⋊Dic127—22no—
68Dic127⋊C4—22no—
69Dic127⋊C4—22no—
70Dic127⋊C4—22no—
71Dic127⋊C4—22no—
72Dic127⋊C4—22no—
73Dic127⋊C4—22no—
74Dic127⋊C4—22no—
75Dic127⋊C4—22no—
76(C2×D127)⋊C4—22no—
77(C2×D127)⋊C4—22no—
78(C2×D127)⋊C4—22no—
79(C2×D127)⋊C4—22no—
80(C2×D127)⋊C4—22no—
81(C2×D127)⋊C4—22no—
82C4×Dic127—22no—
83C4×Dic127—22no—
84C4×Dic127D414yes—
85C22×C4×C127C2311yescharacteristic
86C22×C4×C127C2311yescharacteristic
87C22×C4×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C4⋊C4×C127—22no—
93C4⋊C4×C127—22no—
94C42×C127—22no—
95C42×C127—22no—
96C42×C127—22no—
97C22×D127—21no—
98C22×D127—21no—
99C22×D127—21no—
100C22×D127—21no—
101C22×D127—21no—
102C22×D127—21no—
103C22×D127C2×Q812yes—
104Dic254—28no—
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—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127C2×D412yes—
122C2×Dic127C2×D412yes—
123C2×Dic127C2×Q812yes—
124C2×Dic127#4112yes—
125C2×Dic127#4112yes—
126C4×D127—28no—
127C23×C127C2411yescharacteristic
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
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×C127C2×D412yes—
145C2×C4×C127C2×Q812yes—
146C2×C4×C127#4112yes—
147C2×C4×C127#4112yes—
148C2×C4×C127#4112yes—
149C22×C127C22×D411yescharacteristic
150C22×C127C22×Q811yescharacteristic
151C22×C127C2×#4111yescharacteristic
152C22×C127C2×#4111yescharacteristic
153C22×C127C2×#4111yescharacteristic
154C22×C127C2×#4111yescharacteristic
155C22×C127C2×#4111yescharacteristic
156C2×D127—22no—
157C2×D127—22no—
158C2×D127—22no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C4×C127—22no—
164C4×C127—22no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127C22⋊Q814yes—
170Dic127—22no—
171Dic127—22no—
172Dic127—22no—
173Dic127—22no—
174Dic127—22no—
175Dic127—41no—
176Dic127—41no—
177C2×C127C2×C22⋊Q811yescharacteristic
178C2×C127#52511yescharacteristic
179C2×C127#52911yescharacteristic
180C2×C127#52911yescharacteristic
181C2×C127#53211yescharacteristic
182C2×C127#53311yescharacteristic
183C2×C127#53311yescharacteristic
184D127—42no—
185#2396—1271nomaximal, Sylow
186C127#239611yescharacteristic, complemented, Sylow
187C2×C4×Q8—1271no—
188C2×#117—1271no—
189#400—1271no—
190#399—1271no—
191#398—1271no—
192#398—1271no—
193#396—1271no—
194#396—1271no—
195#395—1271no—
196#393—1271no—
197C4×C4⋊C4—1271no—
198C4×C4⋊C4—1271no—
199C4×C4⋊C4D12711yescharacteristic, complemented
200C4×C22⋊C4—1271no—
201C4×C22⋊C4—1271no—
202C22×Q8—1271no—
203C23×C4—1271no—
204C4×Q8—2544no—
205#117—2544no—
206C2×C4⋊C4—1271no—
207C2×C4⋊C4—1271no—
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×C4⋊C4C2×D12711yescharacteristic
216C2×C4⋊C4C2×D12711yescharacteristic
217C2×C22⋊C4—1271no—
218C2×C22⋊C4—1271no—
219C2×C22⋊C4—1271no—
220C2×C22⋊C4—1271no—
221C2×C22⋊C4—1271no—
222C2×C22⋊C4—1271no—
223C2×C42—1271no—
224C2×C42—1271no—
225C2×C42—1271no—
226C2×C42C2×D12711yescharacteristic
227C2×C42C2×D12711yescharacteristic
228C2×C42C2×D12711yescharacteristic
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
239C24—1271no—
240C2×Q8—2542no—
241C2×Q8—2542no—
242C2×Q8—2542no—
243C22×C4—1271no—
244C22×C4—1271no—
245C22×C4—1271no—
246C22×C4—1271no—
247C22×C4—1271no—
248C22×C4—1274no—
249C22×C4—1271no—
250C22×C4—1271no—
251C22×C4—2542no—
252C22×C4—2542no—
253C22×C4C22×D12711yescharacteristic
254C22×C4C22×D12711yescharacteristic
255C22×C4C22×D12711yescharacteristic
256C22×C4C22×D12711yescharacteristic
257C22×C4C22×D12711yescharacteristic
258C22×C4C22×D12711yescharacteristic
259C22×C4C22×D12711yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—2542no—
263C4⋊C4—2542no—
264C4⋊C4—2542no—
265C4⋊C4—2542no—
266C4⋊C4—2542no—
267C4⋊C4—2542no—
268C4⋊C4—2542no—
269C4⋊C4—2542no—
270C4⋊C4—2542no—
271C22⋊C4—2542no—
272C22⋊C4—2542no—
273C22⋊C4—2542no—
274C22⋊C4—2542no—
275C22⋊C4—2542no—
276C22⋊C4—2542no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—1274no—
281C42—2542no—
282C42—2542no—
283C23—1272no—
284C23—2541no—
285C23—2541no—
286C23—2541no—
287C23—2541no—
288C23—2541no—
289C23—2541no—
290C23C23×D12711yescharacteristic
291Q8—2548no—
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—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—1272no—
309C2×C4—1272no—
310C2×C4—1272no—
311C2×C4—1272no—
312C2×C4—1272no—
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—2548no—
330C2×C4D4×D12712yes—
331C2×C4Q8×D12712yes—
332C2×C4#1452324812yes—
333C2×C4#1452324812yes—
334C2×C4#1452324812yes—
335C22—5081no—
336C22—5081no—
337C22—5081no—
338C22—5081no—
339C22—2542no—
340C22—2542no—
341C22—2542no—
342C22C2×D4×D12711yescharacteristic
343C22C2×Q8×D12711yescharacteristic
344C22#41×D12711yescharacteristic
345C22#41×D12711yescharacteristic
346C22C2×#1452324811yescharacteristic
347C22C2×#1452324811yescharacteristic
348C22C2×#1452324811yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—5081no—
356C4—5081no—
357C4—2542no—
358C4—2542no—
359C4—2542no—
360C4—2542no—
361C4—2542no—
362C4#1686118714yes—
363C2—5082no—
364C2C2×#1686118711yescharacteristic
365C2#2255038511yescharacteristic
366C2#2255040411yescharacteristic
367C2#2255041011yescharacteristic
368C2#2255042311yescharacteristic
369C2#2255043011yescharacteristic
370C2#2255043011yescharacteristic
subgroupG/H
centerC22×C4C22×D127
commutator, socle, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC4×C4⋊C4×C127C2
prime radicalC23×D127C23
hypercenterC4×C4⋊C4D127
hypocenterC127#2396
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic254C2semidirect
3. #22549817C2semidirect
5. #22549810C2semidirect
6. #22549807C2semidirect
9. #22549796C2semidirect
11. C4×Dic127⋊C4C2semidirect
12. C4×Dic127⋊C4C2semidirect
15. C4×C4⋊C4×C127C2semidirect
66. C2×C4×D127Q8semidirect
186. C127#2396semidirect
199. C4×C4⋊C4D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2542
C2×#168611662
#225498172
#225498112
#225498102
#225498072
#225498002
#225497982
#225497962
#225497892
C4×Dic127⋊C42
C4×(C2×D127)⋊C42
C4×C4⋊C4×C1272
#2396127