Finite Group Databasebeta

← #30641544

Subgroups — #30641544

Order 16256.

540 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861195C211yesmaximal, characteristic
2#22549781C211yesmaximal, characteristic, complemented
3#22549780C211yesmaximal, characteristic, complemented
4#22549769C211yesmaximal, characteristic, complemented
5#22549766C211yesmaximal, characteristic, complemented
6#22549766C211yesmaximal, characteristic, complemented
7#22549764C211yesmaximal, characteristic, complemented
8#22549755C211yesmaximal, characteristic
9#22549752C211yesmaximal, characteristic
10#22549750C211yesmaximal, characteristic, complemented
11#22549743C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic127C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
14#22549723C211yesmaximal, characteristic, complemented
15#385×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19#16861195C418yescomplemented
20C2×C4⋊Dic127C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×C22⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×(C2×D127)⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2211yescharacteristic
27C2×(C2×D127)⋊C4C2211yescharacteristic
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×C4×Dic127C2211yescharacteristic
31C2×C4×Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
35C127⋊#95C2211yescharacteristic
36C127⋊#95C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C23×C4×C127C2211yescharacteristic
46C2×C4⋊C4×C127C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C23×D127C2311yescharacteristic
53C2×Dic254—41no—
54C2×Dic254—41no—
55C2×Dic254C2×C414yescomplemented
56C2×C22⋊D127—41no—
57C2×C22⋊D127—41no—
58C2×C22⋊D127C2×C414yes—
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C2×C4×D127—22no—
67C2×C4×D127—41no—
68C2×C4×D127—41no—
69C4⋊Dic127—22no—
70Dic127⋊C4—41no—
71C22⋊Dic127—22no—
72C22⋊Dic127—41no—
73C22⋊Dic127C2×C414yescomplemented
74(C2×D127)⋊C4—22no—
75(C2×D127)⋊C4—41no—
76(C2×D127)⋊C4—41no—
77(C2×D127)⋊C4C2×C414yescomplemented
78(C2×D127)⋊C4C2×C414yescomplemented
79C4×Dic127—41no—
80C4×Dic127—41no—
81C4×Dic127—41no—
82C4×Dic127—41no—
83C4×Dic127C2×C414yescomplemented
84C24×C127C2311yescharacteristic
85C22×C4×C127—22no—
86C22×C4×C127—41no—
87C22×C4×C127—41no—
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C4⋊C4×C127—22no—
95C4⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127C2×C414yescomplemented
98C22×D127—21no—
99C22×D127—21no—
100C22×D127—21no—
101C22×D127—21no—
102C22×D127—21no—
103C22×D127—21no—
104C22×D127C22×C412yes—
105Dic254—44no—
106C22⋊D127—44no—
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×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127C22×C412yes—
125C2×Dic127C22×C412yes—
126C2×Dic127C22×C412yes—
127C2×Dic127#4112yes—
128C4×D127—44no—
129C23×C127—21no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127—21no—
135C23×C127C22×C412yes—
136C23×C127C2411yescharacteristic
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—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—44no—
153C2×C4×C127C22×C412yes—
154C2×C4×C127C22×C412yes—
155C2×C4×C127#4112yes—
156C22×C127—22no—
157C22×C127—22no—
158C22×C127—22no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127C23×C411yescharacteristic
164C22×C127C2×#4111yescharacteristic
165C22×C127C2×#4111yescharacteristic
166C22×C127H(5,𝔽2)11yescharacteristic
167C22×C127H(5,𝔽2)11yescharacteristic
168C22×C127H(5,𝔽2)11yescharacteristic
169C22×C1272−1+411yescharacteristic
170C2×D127—22no—
171C2×D127—22no—
172C2×D127—22no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—41no—
177C4×C127—22no—
178C4×C127—22no—
179C4×C127—22no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183Dic127—22no—
184Dic127—22no—
185Dic127—22no—
186Dic127—22no—
187Dic127—41no—
188Dic127—41no—
189Dic127—41no—
190C2×C127—42no—
191C2×C127C4×#4111yescharacteristic
192C2×C127#51811yescharacteristic
193C2×C127#52011yescharacteristic
194C2×C127#53811yescharacteristic
195C2×C127#53811yescharacteristic
196C2×C127#54211yescharacteristic
197C2×C127#54211yescharacteristic
198D127—42no—
199#2337—1271nomaximal, Sylow
200C127#233711yescharacteristic, complemented, Sylow
201C2×#124—1271no—
202#391—1271no—
203#391—1271no—
204#388—1271no—
205#388—1271no—
206#388—1271no—
207#388—1271no—
208#386—1271no—
209#386—1271no—
210#385—1271no—
211#385D12711yescharacteristic, complemented
212#383—1271no—
213C4×C4⋊C4—1271no—
214C4×C22⋊C4—1271no—
215C4×C22⋊C4—1271no—
216C22×Q8—1271no—
217C22×D4—1271no—
218C23×C4—1271no—
219C23×C4C2×D12711yescharacteristic
220#124—1278no—
221C2×C4⋊C4—1271no—
222C2×C4⋊C4—1271no—
223C2×C4⋊C4C2×D12711yescharacteristic
224C2×C4⋊C4C2×D12711yescharacteristic
225C2×C22⋊C4—1271no—
226C2×C22⋊C4—1271no—
227C2×C22⋊C4—1271no—
228C2×C22⋊C4—1271no—
229C2×C22⋊C4—1271no—
230C2×C22⋊C4—1271no—
231C2×C22⋊C4—1271no—
232C2×C22⋊C4—1271no—
233C2×C22⋊C4C2×D12711yescharacteristic
234C2×C22⋊C4C2×D12711yescharacteristic
235C2×C42—1271no—
236C2×C42—1271no—
237C2×C42—1271no—
238C2×C42—1271no—
239C2×C42—1271no—
240#95—1271no—
241#95—1271no—
242#95—1271no—
243#95—1271no—
244#95—1271no—
245#95—1271no—
246#95—1271no—
247#95—1271no—
248#95—1271no—
249#95—1271no—
250#95C2×D12711yescharacteristic
251#95C2×D12711yescharacteristic
252C24—1271no—
253C24C22×D12711yescharacteristic
254C2×Q8—1274no—
255C2×Q8—5081no—
256C2×Q8—5081no—
257C2×D4—1274no—
258C2×D4—5081no—
259C2×D4—5081no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—1271no—
264C22×C4—1271no—
265C22×C4—1271no—
266C22×C4—1271no—
267C22×C4—1271no—
268C22×C4—1271no—
269C22×C4—1271no—
270C22×C4—5081no—
271C22×C4—5081no—
272C22×C4—2542no—
273C22×C4C22×D12711yescharacteristic
274C22×C4C22×D12711yescharacteristic
275C22×C4C22×D12711yescharacteristic
276C22×C4C22×D12711yescharacteristic
277C22×C4C22×D12711yescharacteristic
278C22×C4C22×D12711yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—5081no—
282C4⋊C4—2542no—
283C22⋊C4—41no—
284C22⋊C4—1274no—
285C22⋊C4—1274no—
286C22⋊C4—1274no—
287C22⋊C4—5081no—
288C22⋊C4—5081no—
289C22⋊C4—5081no—
290C22⋊C4—2542no—
291C22⋊C4—2542no—
292C22⋊C4C4×D12714yescomplemented
293C42—1274no—
294C42—5081no—
295C42—5081no—
296C42—5081no—
297C42—5081no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—1272no—
305C23—2541no—
306C23—2541no—
307C23—2541no—
308C23—2541no—
309C23—2541no—
310C23—2541no—
311C23C2×C4×D12712yes—
312C23C23×D12711yescharacteristic
313Q8—5084no—
314D4—5084no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—1272no—
332C2×C4—1272no—
333C2×C4—1272no—
334C2×C4—1272no—
335C2×C4—5084no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2541no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4C2×C4×D12712yes—
354C2×C4C2×C4×D12712yes—
355C2×C4#1452324712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—5081no—
364C22—5081no—
365C22—5081no—
366C22—5081no—
367C22—2542no—
368C22—2542no—
369C22—2542no—
370C22C22×C4×D12711yescharacteristic
371C22#41×D12711yescharacteristic
372C22C2×#1452324711yescharacteristic
373C22#1686125411yescharacteristic
374C22#1686125411yescharacteristic
375C22#1686125511yescharacteristic
376C22#1686125711yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—5081no—
384C4—5081no—
385C4—5081no—
386C4—2542no—
387C4—2542no—
388C4—2542no—
389C4—2542no—
390C2—42no—
391C2—5082no—
392C2C4×#1452324711yescharacteristic
393C2#2255034811yescharacteristic
394C2#2255035711yescharacteristic
395C2#2255046111yescharacteristic
396C2#2255046211yescharacteristic
397C2#2255049611yescharacteristic
398C2#2255049911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#385×C127C2
socle, cosocleC23×C127C24
prime radicalC22×C22⋊D127C22
hypercenter#385D127
hypocenterC127#2337
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #22549781C2semidirect
3. #22549780C2semidirect
4. #22549769C2semidirect
5. #22549766C2semidirect
6. #22549766C2semidirect
7. #22549764C2semidirect
10. #22549750C2semidirect
11. #22549743C2semidirect
12. C4⋊C4×Dic127C2semidirect
13. C4×C22⋊Dic127C2semidirect
14. #22549723C2semidirect
15. #385×C127C2semidirect
19. #16861195C4semidirect
55. C2×Dic254C2×C4semidirect
73. C22⋊Dic127C2×C4semidirect
77. (C2×D127)⋊C4C2×C4semidirect
78. (C2×D127)⋊C4C2×C4semidirect
83. C4×Dic127C2×C4semidirect
97. C22⋊C4×C127C2×C4semidirect
200. C127#2337semidirect
211. #385D127semidirect
292. 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
#225497812
#225497802
#225497692
#225497662
#225497642
#225497552
#225497522
#225497502
#225497432
C4⋊C4×Dic1272
C4×C22⋊Dic1272
#225497232
#385×C1272
#2337127