Finite Group Databasebeta

← #30641400

Subgroups — #30641400

Order 16256.

632 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861191C211yesmaximal, characteristic
2C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic
3C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549774C211yesmaximal, characteristic, complemented
5#22549771C211yesmaximal, characteristic, complemented
6#22549770C211yesmaximal, characteristic, complemented
7#22549763C211yesmaximal, characteristic, complemented
8#22549759C211yesmaximal, characteristic, complemented
9#22549752C211yesmaximal, characteristic
10#22549751C211yesmaximal, characteristic, complemented
11#22549750C211yesmaximal, characteristic, complemented
12#22549748C211yesmaximal, characteristic, complemented
13#22549738C211yesmaximal, characteristic, complemented
14C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21#16861191C418yescomplemented
22(C22⋊D127)⋊C4—24no—
23C2×C4⋊Dic127C2211yescharacteristic
24C2×C4⋊Dic127C2211yescharacteristic
25C2×C4⋊Dic127C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×Dic127⋊C4C2211yescharacteristic
28C2×Dic127⋊C4C2211yescharacteristic
29C2×Dic127⋊C4C2211yescharacteristic
30C2×Dic127⋊C4C2211yescharacteristic
31C4⋊C4×D127—24no—
32C2×C22⋊Dic127C2211yescharacteristic
33C2×C22⋊Dic127C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×(C2×D127)⋊C4C2211yescharacteristic
37C2×C4×Dic127C2211yescharacteristic
38C2×C4×Dic127C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C127⋊#95C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C22⋊C4×C127C2211yescharacteristic
51C2×C42×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53#95×C127C2211yescharacteristic
54C23×D127C2311yescharacteristic
55C2×C22⋊D127—22no—
56C2×C22⋊D127—22no—
57C2×C22⋊D127C2×C414yes—
58C22×Dic127—22no—
59C22×Dic127—41no—
60C22×Dic127—41no—
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—22no—
71C2×C4×D127—41no—
72C2×C4×D127—41no—
73C2×C4×D127—41no—
74C2×C4×D127—41no—
75C2×C4×D127C2×C414yescomplemented
76C2×C4×D127D414yes—
77C4⋊Dic127—22no—
78C4⋊Dic127—41no—
79C4⋊Dic127C2×C414yescomplemented
80Dic127⋊C4—22no—
81Dic127⋊C4—22no—
82Dic127⋊C4—22no—
83Dic127⋊C4—41no—
84Dic127⋊C4C2×C414yescomplemented
85C22⋊Dic127—41no—
86C22⋊Dic127C2×C414yescomplemented
87(C2×D127)⋊C4—22no—
88(C2×D127)⋊C4—22no—
89(C2×D127)⋊C4C2×C414yescomplemented
90C4×Dic127—22no—
91C4×Dic127—41no—
92C24×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C22×C4×C127C2311yescharacteristic
95C22×C4×C127C2311yescharacteristic
96C22×C4×C127C2311yescharacteristic
97C22×C4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C4⋊C4×C127—22no—
100C22⋊C4×C127—22no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127C2×C414yescomplemented
103C42×C127—41no—
104C22×D127—21no—
105C22×D127—21no—
106C22×D127—21no—
107C22×D127—21no—
108C22×D127C22×C412yes—
109C22×D127C2×D412yes—
110C22×D127#4112yes—
111C22⋊D127—28no—
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×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—44no—
130C2×Dic127C22×C412yes—
131C2×Dic127C22×C412yes—
132C2×Dic127C22×C412yes—
133C2×Dic127C2×D412yes—
134C4×D127—28no—
135C4×D127—44no—
136C4×D127—44no—
137C23×C127—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127—21no—
142C23×C127—21no—
143C23×C127C22×C412yes—
144C23×C127C2411yescharacteristic
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—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127C22×C412yes—
161C2×C4×C127C22×C412yes—
162C2×C4×C127C2×D412yes—
163C22×C127—22no—
164C22×C127—22no—
165C22×C127—22no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127C23×C411yescharacteristic
171C22×C127C22×D411yescharacteristic
172C22×C127C2×#4111yescharacteristic
173C22×C127H(5,𝔽2)11yescharacteristic
174C22×C127H(5,𝔽2)11yescharacteristic
175C22×C1272−1+411yescharacteristic
176C22×C1272−1+411yescharacteristic
177C2×D127—22no—
178C2×D127—22no—
179C2×D127—22no—
180C2×D127—22no—
181C2×D127—22no—
182C2×D127—22no—
183C2×D127C4×D414yes—
184C4×C127—22no—
185C4×C127—22no—
186C4×C127—22no—
187C4×C127—41no—
188C4×C127—41no—
189C4×C127—41no—
190Dic127—22no—
191Dic127—22no—
192Dic127—22no—
193Dic127—22no—
194Dic127—41no—
195Dic127—41no—
196Dic127—41no—
197C2×C127—42no—
198C2×C127C2×C4×D411yescharacteristic
199C2×C127#52011yescharacteristic
200C2×C127#52011yescharacteristic
201C2×C127#53411yescharacteristic
202C2×C127#53611yescharacteristic
203C2×C127#53911yescharacteristic
204C2×C127#53911yescharacteristic
205D127—24no—
206#2322—1271nomaximal, Sylow
207C127#232211yescharacteristic, complemented, Sylow
208C2×#123—1271no—
209C2×C4×D4—1271no—
210C22×C4⋊C4—1271no—
211#389—1271no—
212#389—1271no—
213#388—1271no—
214#388D12711yescharacteristic, complemented
215#387—1271no—
216#387—1271no—
217#386—1271no—
218#385—1271no—
219#385—1271no—
220#385—1271no—
221#381—1271no—
222C4×C22⋊C4—1271no—
223C22×D4—1271no—
224C23×C4—1271no—
225C23×C4—1271no—
226C23×C4—1271no—
227C23×C4—1271no—
228#123—1278no—
229C4×D4—2544no—
230C2×C4⋊C4—1271no—
231C2×C4⋊C4—1271no—
232C2×C4⋊C4—1271no—
233C2×C4⋊C4—1271no—
234C2×C4⋊C4—1271no—
235C2×C4⋊C4—1271no—
236C2×C4⋊C4—1271no—
237C2×C4⋊C4—1271no—
238C2×C4⋊C4—2544no—
239C2×C4⋊C4C2×D12711yescharacteristic
240C2×C22⋊C4—1271no—
241C2×C22⋊C4—1271no—
242C2×C22⋊C4—1271no—
243C2×C22⋊C4—1271no—
244C2×C22⋊C4—1271no—
245C2×C22⋊C4C2×D12711yescharacteristic
246C2×C22⋊C4C2×D12711yescharacteristic
247C2×C22⋊C4C2×D12711yescharacteristic
248C2×C42—1271no—
249C2×C42—1271no—
250C2×C42C2×D12711yescharacteristic
251#95—1271no—
252#95—1271no—
253#95—1271no—
254#95—1271no—
255#95—1271no—
256#95—1271no—
257#95—1271no—
258#95—1271no—
259#95C2×D12711yescharacteristic
260#95C2×D12711yescharacteristic
261C24—1271no—
262C24C22×D12711yescharacteristic
263C2×D4—1274no—
264C2×D4—2542no—
265C2×D4—2542no—
266C22×C4—1271no—
267C22×C4—1271no—
268C22×C4—1271no—
269C22×C4—1271no—
270C22×C4—1274no—
271C22×C4—1274no—
272C22×C4—1271no—
273C22×C4—1271no—
274C22×C4—1271no—
275C22×C4—5081no—
276C22×C4—5081no—
277C22×C4—5081no—
278C22×C4—5081no—
279C22×C4—5081no—
280C22×C4—5081no—
281C22×C4—2542no—
282C22×C4—2542no—
283C22×C4—2542no—
284C22×C4—2542no—
285C22×C4C22×D12711yescharacteristic
286C22×C4C22×D12711yescharacteristic
287C22×C4C22×D12711yescharacteristic
288C22×C4C22×D12711yescharacteristic
289C22×C4C22×D12711yescharacteristic
290C22×C4C22×D12711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—1274no—
293C4⋊C4—1274no—
294C4⋊C4—5081no—
295C4⋊C4—5081no—
296C4⋊C4—2542no—
297C4⋊C4—2542no—
298C4⋊C4—2542no—
299C4⋊C4—2542no—
300C22⋊C4—22no—
301C22⋊C4—41no—
302C22⋊C4—1274no—
303C22⋊C4—1274no—
304C22⋊C4—5081no—
305C22⋊C4—2542no—
306C22⋊C4—2542no—
307C22⋊C4C4×D12714yescomplemented
308C42—41no—
309C42—5081no—
310C42—2542no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—1272no—
318C23—1272no—
319C23—1272no—
320C23—2541no—
321C23—2541no—
322C23—2541no—
323C23—2541no—
324C23C2×C4×D12712yes—
325C23C23×D12711yescharacteristic
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—21no—
341C2×C4—21no—
342C2×C4—1272no—
343C2×C4—1272no—
344C2×C4—1272no—
345C2×C4—1272no—
346C2×C4—5084no—
347C2×C4—5084no—
348C2×C4—5084no—
349C2×C4—2548no—
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×C4—2541no—
364C2×C4—2541no—
365C2×C4—2541no—
366C2×C4—2541no—
367C2×C4C2×C4×D12712yes—
368C2×C4C2×C4×D12712yes—
369C2×C4D4×D12712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—1274no—
378C22—2542no—
379C22—2542no—
380C22—2542no—
381C22—2542no—
382C22—2542no—
383C22—2542no—
384C22C22×C4×D12711yescharacteristic
385C22C2×D4×D12711yescharacteristic
386C22#41×D12711yescharacteristic
387C22#1686125411yescharacteristic
388C22#1686125511yescharacteristic
389C22#1686125611yescharacteristic
390C22#1686125611yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—5081no—
398C4—5081no—
399C4—5081no—
400C4—2542no—
401C4—2542no—
402C4—2542no—
403C4—2542no—
404C2—42no—
405C2—2544no—
406C2C4×D4×D12711yescharacteristic
407C2#2255035511yescharacteristic
408C2#2255035611yescharacteristic
409C2#2255043411yescharacteristic
410C2#2255044611yescharacteristic
411C2#2255046711yescharacteristic
412C2#2255046911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC22×C22⋊D127C22
hypercenter#388D127
hypocenterC127#2322
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D127C2semidirect
4. #22549774C2semidirect
5. #22549771C2semidirect
6. #22549770C2semidirect
7. #22549763C2semidirect
8. #22549759C2semidirect
10. #22549751C2semidirect
11. #22549750C2semidirect
12. #22549748C2semidirect
13. #22549738C2semidirect
14. C4×(C2×D127)⋊C4C2semidirect
15. #388×C127C2semidirect
21. #16861191C4semidirect
75. C2×C4×D127C2×C4semidirect
79. C4⋊Dic127C2×C4semidirect
84. Dic127⋊C4C2×C4semidirect
86. C22⋊Dic127C2×C4semidirect
89. (C2×D127)⋊C4C2×C4semidirect
102. C22⋊C4×C127C2×C4semidirect
207. C127#2322semidirect
214. #388D127semidirect
307. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611912
C2×(C22⋊D127)⋊C42
C2×C4⋊C4×D1272
#225497742
#225497712
#225497702
#225497632
#225497592
#225497522
#225497512
#225497502
#225497482
#225497382
C4×(C2×D127)⋊C42
#388×C1272
#2322127