Finite Group Databasebeta

← #30641802

Subgroups — #30641802

Order 16256.

632 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861195C211yesmaximal, characteristic
2C2×C4×Dic254C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D127C211yesmaximal, characteristic
4#22549780C211yesmaximal, characteristic, complemented
5#22549776C211yesmaximal, characteristic
6#22549770C211yesmaximal, characteristic, complemented
7#22549769C211yesmaximal, characteristic, complemented
8#22549767C211yesmaximal, characteristic, complemented
9#22549767C211yesmaximal, characteristic, complemented
10#22549750C211yesmaximal, characteristic, complemented
11#22549749C211yesmaximal, characteristic, complemented
12C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
13#22549727C211yesmaximal, characteristic, complemented
14#22549723C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19#16861195C418yescomplemented
20C4×Dic254—24no—
21C4×C22⋊D127—24no—
22C2×C4⋊Dic127C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×Dic127⋊C4C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C22⋊Dic127C2211yescharacteristic
30C2×C22⋊Dic127C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×(C2×D127)⋊C4C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
37C2×C4×Dic127C2211yescharacteristic
38C2×C4×Dic127C2211yescharacteristic
39C2×C4×Dic127C2211yescharacteristic
40C2×C4×Dic127C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C127⋊#95C2211yescharacteristic
47C23×C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C42×C127C2211yescharacteristic
51C2×C42×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53#95×C127C2211yescharacteristic
54C23×D127C2311yescharacteristic
55C2×Dic254—22no—
56C2×Dic254—22no—
57C2×Dic254C2×C414yescomplemented
58C2×C22⋊D127—22no—
59C2×C22⋊D127—22no—
60C2×C22⋊D127C2×C414yes—
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C2×C4×D127—22no—
69C2×C4×D127—22no—
70C2×C4×D127—41no—
71C4⋊Dic127—22no—
72Dic127⋊C4—22no—
73Dic127⋊C4—22no—
74Dic127⋊C4—22no—
75Dic127⋊C4—22no—
76Dic127⋊C4D414yes—
77C22⋊Dic127—22no—
78C22⋊Dic127—41no—
79C22⋊Dic127C2×C414yescomplemented
80(C2×D127)⋊C4—22no—
81(C2×D127)⋊C4—22no—
82(C2×D127)⋊C4—41no—
83(C2×D127)⋊C4C2×C414yescomplemented
84(C2×D127)⋊C4C2×C414yescomplemented
85C4×Dic127—22no—
86C4×Dic127—22no—
87C4×Dic127—22no—
88C4×Dic127—41no—
89C4×Dic127C2×C414yescomplemented
90C24×C127C2311yescharacteristic
91C22×C4×C127—22no—
92C22×C4×C127—22no—
93C22×C4×C127—41no—
94C22×C4×C127C2311yescharacteristic
95C22×C4×C127C2311yescharacteristic
96C22×C4×C127C2311yescharacteristic
97C22×C4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C22×C4×C127C2311yescharacteristic
100C22⋊C4×C127—22no—
101C22⋊C4×C127C2×C414yescomplemented
102C42×C127—22no—
103C42×C127—41no—
104C22×D127—21no—
105C22×D127—21no—
106C22×D127—21no—
107C22×D127—21no—
108C22×D127—21no—
109C22×D127—21no—
110C22×D127C22×C412yes—
111Dic254—28no—
112C22⋊D127—28no—
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×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127C22×C412yes—
129C2×Dic127C22×C412yes—
130C2×Dic127C22×C412yes—
131C2×Dic127C2×D412yes—
132C2×Dic127C2×D412yes—
133C2×Dic127#4112yes—
134C4×D127—44no—
135C23×C127—21no—
136C23×C127—21no—
137C23×C127—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127C22×C412yes—
142C23×C127C2411yescharacteristic
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—44no—
157C2×C4×C127C22×C412yes—
158C2×C4×C127C22×C412yes—
159C2×C4×C127C2×D412yes—
160C2×C4×C127#4112yes—
161C2×C4×C127#4112yes—
162C22×C127—22no—
163C22×C127—22no—
164C22×C127—22no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127C23×C411yescharacteristic
170C22×C127C22×D411yescharacteristic
171C22×C127C2×#4111yescharacteristic
172C22×C127C2×#4111yescharacteristic
173C22×C127C2×#4111yescharacteristic
174C22×C127H(5,𝔽2)11yescharacteristic
175C22×C1272−1+411yescharacteristic
176C2×D127—22no—
177C2×D127—22no—
178C2×D127—22no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C4×C127—22no—
184C4×C127—22no—
185C4×C127—22no—
186C4×C127—22no—
187C4×C127—22no—
188C4×C127—41no—
189Dic127—22no—
190Dic127—22no—
191Dic127—22no—
192Dic127—41no—
193Dic127—41no—
194Dic127—41no—
195Dic127C4×D414yes—
196C2×C127—42no—
197C2×C127C2×C4×D411yescharacteristic
198C2×C127C4×#4111yescharacteristic
199C2×C127#52011yescharacteristic
200C2×C127#54611yescharacteristic
201C2×C127#54811yescharacteristic
202C2×C127#55611yescharacteristic
203C2×C127#55911yescharacteristic
204D127—42no—
205#2367—1271nomaximal, Sylow
206C127#236711yescharacteristic, complemented, Sylow
207C2×#124—1271no—
208C2×C4×Q8—1271no—
209C2×C4×D4—1271no—
210#391—1271no—
211#390—1271no—
212#388—1271no—
213#388—1271no—
214#388—1271no—
215#388—1271no—
216#385—1271no—
217#385—1271no—
218C4×C4⋊C4—1271no—
219C4×C4⋊C4—1271no—
220C4×C22⋊C4—1271no—
221C4×C22⋊C4D12711yescharacteristic, complemented
222C22×Q8—1271no—
223C22×D4—1271no—
224C23×C4—1271no—
225C23×C4C2×D12711yescharacteristic
226#124—1278no—
227C4×Q8—2544no—
228C4×D4—2544no—
229C2×C4⋊C4—1271no—
230C2×C4⋊C4—1271no—
231C2×C4⋊C4—1271no—
232C2×C4⋊C4—1271no—
233C2×C4⋊C4—1271no—
234C2×C4⋊C4—1271no—
235C2×C22⋊C4—1271no—
236C2×C22⋊C4—1271no—
237C2×C22⋊C4—1271no—
238C2×C22⋊C4—1271no—
239C2×C22⋊C4—1271no—
240C2×C22⋊C4—1271no—
241C2×C22⋊C4—1271no—
242C2×C22⋊C4—1271no—
243C2×C22⋊C4C2×D12711yescharacteristic
244C2×C22⋊C4C2×D12711yescharacteristic
245C2×C42—1271no—
246C2×C42—1271no—
247C2×C42—1271no—
248C2×C42—1271no—
249C2×C42—1271no—
250C2×C42C2×D12711yescharacteristic
251C2×C42C2×D12711yescharacteristic
252#95—1271no—
253#95—1271no—
254#95—1271no—
255#95—1271no—
256#95—1271no—
257#95—1271no—
258#95C2×D12711yescharacteristic
259#95C2×D12711yescharacteristic
260C24—1271no—
261C24C22×D12711yescharacteristic
262C2×Q8—1274no—
263C2×Q8—2542no—
264C2×Q8—2542no—
265C2×D4—1274no—
266C2×D4—2542no—
267C2×D4—2542no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—1271no—
272C22×C4—1271no—
273C22×C4—1271no—
274C22×C4—1271no—
275C22×C4—1271no—
276C22×C4—1271no—
277C22×C4—1271no—
278C22×C4—5081no—
279C22×C4—2542no—
280C22×C4—2542no—
281C22×C4C22×D12711yescharacteristic
282C22×C4C22×D12711yescharacteristic
283C22×C4C22×D12711yescharacteristic
284C22×C4C22×D12711yescharacteristic
285C22×C4C22×D12711yescharacteristic
286C22×C4C22×D12711yescharacteristic
287C4⋊C4—1274no—
288C4⋊C4—2542no—
289C4⋊C4—2542no—
290C4⋊C4—2542no—
291C4⋊C4—2542no—
292C4⋊C4—2542no—
293C22⋊C4—22no—
294C22⋊C4—1274no—
295C22⋊C4—1274no—
296C22⋊C4—1274no—
297C22⋊C4—5081no—
298C22⋊C4—5081no—
299C22⋊C4—2542no—
300C22⋊C4—2542no—
301C22⋊C4—2542no—
302C22⋊C4C4×D12714yescomplemented
303C42—22no—
304C42—41no—
305C42—1274no—
306C42—5081no—
307C42—2542no—
308C42—2542no—
309C42—2542no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—1272no—
317C23—2541no—
318C23—2541no—
319C23—2541no—
320C23—2541no—
321C23—2541no—
322C23—2541no—
323C23C2×C4×D12712yes—
324C23C23×D12711yescharacteristic
325Q8—2548no—
326D4—2548no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—1272no—
342C2×C4—1272no—
343C2×C4—1272no—
344C2×C4—1272no—
345C2×C4—1272no—
346C2×C4—1272no—
347C2×C4—5084no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4—2541no—
361C2×C4—2541no—
362C2×C4—2541no—
363C2×C4C2×C4×D12712yes—
364C2×C4C2×C4×D12712yes—
365C2×C4D4×D12712yes—
366C2×C4#1452324812yes—
367C2×C4#1452324812yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—5081no—
376C22—5081no—
377C22—5081no—
378C22—5081no—
379C22—2542no—
380C22—2542no—
381C22—2542no—
382C22C22×C4×D12711yescharacteristic
383C22C2×D4×D12711yescharacteristic
384C22#41×D12711yescharacteristic
385C22C2×#1452324811yescharacteristic
386C22C2×#1452324811yescharacteristic
387C22#1686125411yescharacteristic
388C22#1686125611yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—1274no—
396C4—5081no—
397C4—5081no—
398C4—5081no—
399C4—2542no—
400C4—2542no—
401C4—2542no—
402C2—42no—
403C2—5082no—
404C2C4×D4×D12711yescharacteristic
405C2C4×#1452324811yescharacteristic
406C2#2255035511yescharacteristic
407C2#2255051611yescharacteristic
408C2#2255053511yescharacteristic
409C2#2255059411yescharacteristic
410C2#2255060911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC4×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC22×C22⋊D127C22
hypercenterC4×C22⋊C4D127
hypocenterC127#2367
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic254C2semidirect
4. #22549780C2semidirect
6. #22549770C2semidirect
7. #22549769C2semidirect
8. #22549767C2semidirect
9. #22549767C2semidirect
10. #22549750C2semidirect
11. #22549749C2semidirect
12. C4×Dic127⋊C4C2semidirect
13. #22549727C2semidirect
14. #22549723C2semidirect
15. C4×C22⋊C4×C127C2semidirect
19. #16861195C4semidirect
57. C2×Dic254C2×C4semidirect
79. C22⋊Dic127C2×C4semidirect
83. (C2×D127)⋊C4C2×C4semidirect
84. (C2×D127)⋊C4C2×C4semidirect
89. C4×Dic127C2×C4semidirect
101. C22⋊C4×C127C2×C4semidirect
206. C127#2367semidirect
221. C4×C22⋊C4D127semidirect
302. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611952
C2×C4×Dic2542
C2×C4×C22⋊D1272
#225497802
#225497762
#225497702
#225497692
#225497672
#225497502
#225497492
C4×Dic127⋊C42
#225497272
#225497232
C4×C22⋊C4×C1272
#2367127