Finite Group Databasebeta

← #30650236

Subgroups — #30650236

Order 16256.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861239C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3D4×Dic254C211yesmaximal, characteristic, complemented
4#22550271C211yesmaximal, characteristic, complemented
5#22550120C211yesmaximal, characteristic, complemented
6#22550115C211yesmaximal, characteristic, complemented
7#22550095C211yesmaximal, characteristic, complemented
8#22550081C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550031C211yesmaximal, characteristic, complemented
11#22550027C211yesmaximal, characteristic, complemented
12#22550014C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15#546×C127C211yesmaximal, characteristic
16C22×Dic254C2211yescharacteristic
17C22×Dic254C2211yescharacteristic, complemented
18#16861239—24no—
19C2×#14523238C2211yescharacteristic, complemented
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523236—24no—
23C2×#14523236C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523236C2211yescharacteristic
27C2×#14523236C2211yescharacteristic
28C2×#14523236C2211yescharacteristic
29C2×#14523232C2211yescharacteristic, complemented
30C22×C127⋊C8C2211yescharacteristic
31#16861209C2211yescharacteristic, complemented
32#16861208—21no—
33#16861188C2211yescharacteristic
34#16861188C2211yescharacteristic, complemented
35#16861186—21no—
36#16861186—21no—
37C4×Dic254C2211yescharacteristic, complemented
38D4×Dic127—21no—
39#16861142C2211yescharacteristic, complemented
40C4⋊(C127⋊C8)C2211yescharacteristic, complemented
41#16861133C2211yescharacteristic, complemented
42#16861132C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47C22⋊(C127⋊C8)C2211yescharacteristic
48C22⋊(C127⋊C8)C2211yescharacteristic, complemented
49#16861120C2211yescharacteristic, complemented
50C2×#41×C127C2211yescharacteristic
51C22×D4×C127C2211yescharacteristic
52#124×C127C2211yescharacteristic
53#123×C127—21no—
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127—21no—
56#121×C127C2211yescharacteristic
57#120×C127—21no—
58C4×D4×C127C2211yescharacteristic
59C4×D4×C127C2211yescharacteristic
60C2×C22⋊C4×C127—21no—
61C2×Dic254—22no—
62C2×Dic254—22no—
63C2×Dic254—21no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254—41no—
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70C22×Dic127—21no—
71C22×Dic127—21no—
72#14523238—22no—
73#14523238—22no—
74#14523238—41no—
75#14523236—22no—
76#14523236—22no—
77#14523236—22no—
78#14523236—22no—
79#14523236—41no—
80#14523236D414yescomplemented
81#14523232—22no—
82C2×C127⋊C8—22no—
83C2×C127⋊C8C2311yescharacteristic
84C2×C127⋊C8C2311yescharacteristic
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C4⋊Dic127—21no—
88C4⋊Dic127—21no—
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92Dic127⋊C4—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95C4×Dic127—21no—
96C24×C127—21no—
97#41×C127—22no—
98#41×C127—41no—
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—22no—
101C2×D4×C127—21no—
102C2×D4×C127—21no—
103C2×D4×C127—21no—
104C2×D4×C127—41no—
105C2×D4×C127D411yescharacteristic
106C2×D4×C127C2311yescharacteristic
107C2×D4×C127C2311yescharacteristic
108C22×C4×C127—21no—
109C22×C4×C127—21no—
110C22×C4×C127C2311yescharacteristic
111C22×C4×C127C2311yescharacteristic
112C4⋊C4×C127—21no—
113C4⋊C4×C127D411yescharacteristic
114C4⋊C4×C127C2311yescharacteristic
115C22⋊C4×C127—21no—
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127—21no—
120C22⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
122C42×C127C2311yescharacteristic
123Dic254—21no—
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129Dic254—42no—
130Dic254C2×D412yes—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—42no—
136C2×Dic127—42no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C127⋊C8—21no—
140C127⋊C8C2×D412yes—
141C23×C127—21no—
142C23×C127—21no—
143C23×C127—42no—
144C23×C127—42no—
145C23×C127—81no—
146C23×C127C2×D411yescharacteristic
147C23×C127C2×D411yescharacteristic
148Q8×C127—21no—
149Q8×C127—21no—
150D4×C127—22no—
151D4×C127—21no—
152D4×C127—21no—
153D4×C127—21no—
154D4×C127—42no—
155D4×C127—42no—
156D4×C127—81no—
157D4×C127C2×D412yes—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—42no—
165C2×C4×C127—81no—
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2×D411yescharacteristic
170C2×C4×C127C2411yescharacteristic
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—22no—
175C22×C127—21no—
176C22×C127—21no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—84no—
183C22×C127#13712yescomplemented
184C22×C127C22×D411yescharacteristic
185C4×C127—21no—
186C4×C127—21no—
187C4×C127—21no—
188C4×C127—21no—
189C4×C127—41no—
190C4×C127—41no—
191C4×C127C22×D411yescharacteristic
192C4×C127H(5,𝔽2)11yescharacteristic
193Dic127—22no—
194Dic127—41no—
195Dic127—41no—
196Dic127—41no—
197C2×C127—22no—
198C2×C127—41no—
199C2×C127—42no—
200C2×C127—81no—
201C2×C127D4211yescharacteristic
202C2×C127C2×#13711yescharacteristic
203C2×C127#57711yescharacteristic
204#3266—1271nomaximal, Sylow
205C127#326611yescharacteristic, complemented, Sylow
206C2×#137—1271no—
207C22×SD16—1271no—
208D4×Q8—1271no—
209#546D12711yescharacteristic
210#497—1271no—
211#468—1271no—
212#467—1271no—
213#464—1271no—
214#461—1271no—
215#452—1271no—
216#451—1271no—
217#450—1271no—
218#448—1271no—
219#440—1271no—
220#435—1271no—
221C2×#41C2×D12711yescharacteristic
222C22×Q8—1271no—
223C22×Q8—1271no—
224C22×D4C2×D12711yescharacteristic
225#137—2544no—
226C2×Q16—1271no—
227C2×Q16—1271no—
228C2×Q16—1271no—
229C2×SD16—1271no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—2544no—
236C2×M16—1271no—
237C22×C8—1271no—
238C4⋊Q8—1271no—
239C4⋊Q8—2541no—
240#124C2×D12711yescharacteristic
241#123—21no—
242C22⋊Q8—1271no—
243C22⋊Q8—1271no—
244C22⋊Q8—2541no—
245C22⋊Q8—2541no—
246C22⋊Q8C2×D12711yescharacteristic
247#121—21no—
248#121C2×D12711yescharacteristic
249#120—21no—
250C4×Q8—1271no—
251C4×D4—2541no—
252C4×D4C2×D12711yescharacteristic
253C4×D4C2×D12711yescharacteristic
254C2×C22⋊C4—21no—
255#107—1271no—
256C4⋊C8—1271no—
257#103—1271no—
258#103—1271no—
259#103—1271no—
260#103—1271no—
261#102—1271no—
262#102—1271no—
263C22⋊C8—1271no—
264C22⋊C8—1271no—
265#97—1271no—
266C24—21no—
267#41—22no—
268#41—41no—
269C2×Q8—1271no—
270C2×Q8—1271no—
271C2×Q8—1271no—
272C2×Q8—5081no—
273C2×Q8—5081no—
274C2×Q8—2542no—
275C2×Q8—2542no—
276C2×Q8—2541no—
277C2×Q8—2541no—
278C2×Q8C22×D12711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D12711yescharacteristic
285C2×D4C22×D12711yescharacteristic
286C2×D4C22×D12711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2541no—
290C22×C4—2541no—
291C22×C4C22×D12711yescharacteristic
292C22×C4C22×D12711yescharacteristic
293Q16—5081no—
294Q16—2542no—
295Q16—2542no—
296SD16—1274no—
297SD16—5081no—
298SD16—2542no—
299SD16—2542no—
300SD16—2542no—
301SD16—2542no—
302M16—2542no—
303C2×C8—1271no—
304C2×C8—1271no—
305C2×C8—1271no—
306C2×C8—1271no—
307C2×C8—2542no—
308C4⋊C4—21no—
309C4⋊C4—1271no—
310C4⋊C4—2541no—
311C4⋊C4—2541no—
312C4⋊C4—2541no—
313C4⋊C4—2541no—
314C4⋊C4—2541no—
315C4⋊C4C22⋊D12711yescharacteristic
316C4⋊C4C22×D12711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4C22⋊D12711yescharacteristic
325C22⋊C4C22⋊D12711yescharacteristic
326C42—2541no—
327C42C22×D12711yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D12711yescharacteristic
334C23C2×C22⋊D12711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—1272no—
338Q8—5082no—
339Q8—5082no—
340Q8—2541no—
341Q8—2541no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D12712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—5082no—
362C2×C4—5082no—
363C2×C4—2541no—
364C2×C4—2541no—
365C2×C4—2541no—
366C2×C4—2541no—
367C2×C4C2×C22⋊D12711yescharacteristic
368C2×C4C2×C22⋊D12711yescharacteristic
369C2×C4C2×C22⋊D12711yescharacteristic
370C2×C4C2×C22⋊D12711yescharacteristic
371C2×C4C23×D12711yescharacteristic
372C8—1272no—
373C8—2541no—
374C8—2541no—
375C8—2541no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#1686123912yescomplemented
389C22C22×C22⋊D12711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—5081no—
397C4—5081no—
398C4—5081no—
399C4—2542no—
400C4C2×D4×D12711yescharacteristic
401C4#1686125411yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D12711yescharacteristic
407C2C2×#1686123911yescharacteristic
408C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#546×C127C2
socleC22×C127C22×D4
hypercenter#546D127
hypocenterC127#3266
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C22×#14523236C2semidirect
3. D4×Dic254C2semidirect
4. #22550271C2semidirect
5. #22550120C2semidirect
6. #22550115C2semidirect
7. #22550095C2semidirect
8. #22550081C2semidirect
9. #22550038C2semidirect
10. #22550031C2semidirect
11. #22550027C2semidirect
12. #22550014C2semidirect
13. #22549976C2semidirect
14. #22549948C2semidirect
17. C22×Dic254C22semidirect
19. C2×#14523238C22semidirect
20. C2×#14523238C22semidirect
21. C2×#14523238C22semidirect
29. C2×#14523232C22semidirect
31. #16861209C22semidirect
34. #16861188C22semidirect
37. C4×Dic254C22semidirect
39. #16861142C22semidirect
40. C4⋊(C127⋊C8)C22semidirect
41. #16861133C22semidirect
42. #16861132C22semidirect
43. #16861132C22semidirect
44. #16861132C22semidirect
48. C22⋊(C127⋊C8)C22semidirect
49. #16861120C22semidirect
80. #14523236D4semidirect
183. C22×C127#137semidirect
205. C127#3266semidirect
388. C22#16861239semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232362
D4×Dic2542
#225502712
#225501202
#225501152
#225500952
#225500812
#225500382
#225500312
#225500272
#225500142
#225499762
#225499482
#546×C1272
#3266127