Finite Group Databasebeta

← #30650302

Subgroups — #30650302

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C211yesmaximal, characteristic, complemented
2C2×Q16×D127C211yesmaximal, characteristic, complemented
3Q8×D508C211yesmaximal, characteristic, complemented
4#22550538C211yesmaximal, characteristic
5#22550270C211yesmaximal, characteristic, complemented
6#22550135C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550098C211yesmaximal, characteristic, complemented
9#22550086C211yesmaximal, characteristic, complemented
10#22550041C211yesmaximal, characteristic, complemented
11#22550033C211yesmaximal, characteristic, complemented
12#22550033C211yesmaximal, characteristic, complemented
13#22550016C211yesmaximal, characteristic, complemented
14#22549949C211yesmaximal, characteristic, complemented
15#441×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19C2×Q8×D127C2211yescharacteristic
20#16861241—24no—
21C2×Dic508C2211yescharacteristic
22C2×Dic508C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25C2×#14523238C2211yescharacteristic
26Q16×D127—24no—
27C2×#14523237C2211yescharacteristic
28C2×#14523237C2211yescharacteristic
29C2×#14523235C2211yescharacteristic
30C2×#14523231C2211yescharacteristic
31C2×C8×D127C2211yescharacteristic
32#16861209—21no—
33#16861209C2211yescharacteristic
34#16861197—21no—
35#16861197C2211yescharacteristic
36#16861189—21no—
37#16861189—21no—
38#16861189—21no—
39#16861189C2211yescharacteristic
40#16861189C2211yescharacteristic
41#16861189C2211yescharacteristic
42#16861184—21no—
43#16861184C2211yescharacteristic
44C4×D508—21no—
45C4×D508—21no—
46C4×D508C2211yescharacteristic
47C4⋊(C127⋊C8)C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861132C2211yescharacteristic
50#16861132C2211yescharacteristic
51#16861129C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53(C2×D127)⋊C8C2211yescharacteristic
54C2×Q16×C127C2211yescharacteristic
55C4×Q8×C127C2211yescharacteristic
56C4×Q8×C127C2211yescharacteristic
57#106×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#97×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—41no—
63C2×Dic254—21no—
64C2×Dic254—21no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69Q8×D127—22no—
70Q8×D127—22no—
71Q8×D127—22no—
72Q8×D127—41no—
73Q8×D127—41no—
74Q8×D127—41no—
75C2×D508—21no—
76C2×D508—21no—
77C2×D508D411yescharacteristic
78C2×D508C2311yescharacteristic
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127C2311yescharacteristic
84C2×C4×D127C2311yescharacteristic
85Dic508—22no—
86Dic508—41no—
87#14523238—22no—
88#14523238—22no—
89#14523238—22no—
90#14523237—22no—
91#14523237—41no—
92#14523235—22no—
93#14523231—22no—
94C2×C127⋊C8C2311yescharacteristic
95C2×C127⋊C8C2311yescharacteristic
96C8×D127—22no—
97C4⋊Dic127—21no—
98C4⋊Dic127—21no—
99C4⋊Dic127—21no—
100C4⋊Dic127—21no—
101C4⋊Dic127—21no—
102C4⋊Dic127D411yescharacteristic
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109(C2×D127)⋊C4D411yescharacteristic
110(C2×D127)⋊C4D411yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×Q8×C127C2311yescharacteristic
113Q16×C127D414yescomplemented
114C2×C8×C127C2311yescharacteristic
115C2×C8×C127C2311yescharacteristic
116C4⋊C4×C127—21no—
117C4⋊C4×C127—21no—
118C4⋊C4×C127C2311yescharacteristic
119C4⋊C4×C127C2311yescharacteristic
120C42×C127—21no—
121C42×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127C2×D411yescharacteristic
125C22×D127C2×D411yescharacteristic
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—42no—
133Dic254—42no—
134Dic254—42no—
135D508—22no—
136D508—21no—
137D508—21no—
138D508—42no—
139D508—42no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C4×D127—21no—
146C4×D127—21no—
147C4×D127—21no—
148C4×D127—21no—
149C4×D127—42no—
150C4×D127—42no—
151C4×D127—42no—
152C4×D127—42no—
153C127⋊C8—21no—
154C127⋊C8—21no—
155Q8×C127—21no—
156Q8×C127—21no—
157Q8×C127C2×D412yes—
158Q8×C127C2×D412yes—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—21no—
167C8×C127C2×D412yes—
168C22×C127C22×D411yescharacteristic
169C2×D127—21no—
170C2×D127—21no—
171C2×D127—21no—
172C2×D127—21no—
173C2×D127—21no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—41no—
177C2×D127#13712yescomplemented
178C4×C127—22no—
179C4×C127—22no—
180C4×C127—21no—
181C4×C127—21no—
182C4×C127—41no—
183C4×C127—41no—
184C4×C127C22×D411yescharacteristic
185C4×C127H(5,𝔽2)11yescharacteristic
186Dic127—21no—
187Dic127—21no—
188Dic127—41no—
189Dic127—41no—
190Dic127—41no—
191C2×C127D4211yescharacteristic
192C2×C127C2×#13711yescharacteristic
193C2×C127#57811yescharacteristic
194D127—22no—
195D127—41no—
196D127—81no—
197#3270—1271nomaximal, Sylow
198C127#327011yescharacteristic, complemented, Sylow
199C2×#137—1271no—
200C22×Q16—1271no—
201D4×Q8—1271no—
202#548—1271no—
203#497—1271no—
204#470—1271no—
205#465—1271no—
206#464—1271no—
207#462—1271no—
208#452—1271no—
209#451—1271no—
210#451—1271no—
211#448—1271no—
212#441D12711yescharacteristic, complemented
213#435—1271no—
214C2×#41—1271no—
215C22×Q8—1271no—
216C22×Q8—1271no—
217C22×Q8—1271no—
218#137—2544no—
219C2×Q16—1271no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×Q16—1271no—
223C2×Q16—1271no—
224C2×Q16—2544no—
225C2×Q16C2×D12711yescharacteristic
226C2×SD16—1271no—
227C2×SD16—1271no—
228C2×SD16—1271no—
229C2×M16—1271no—
230C22×C8—1271no—
231C4⋊Q8—1271no—
232C4⋊Q8—2541no—
233#124—1271no—
234#124—2541no—
235C22⋊Q8—1271no—
236C22⋊Q8—1271no—
237C22⋊Q8—1271no—
238C22⋊Q8—2541no—
239C22⋊Q8—2541no—
240C22⋊Q8—2541no—
241#121—1271no—
242#121—2541no—
243C4×Q8C2×D12711yescharacteristic
244C4×Q8C2×D12711yescharacteristic
245C4×D4—1271no—
246C4×D4—2541no—
247C4×D4—2541no—
248#106C2×D12711yescharacteristic
249C4⋊C8—1271no—
250#103—1271no—
251#103—1271no—
252#103—1271no—
253#103C2×D12711yescharacteristic
254#103C2×D12711yescharacteristic
255#102—1271no—
256C22⋊C8—1271no—
257C22⋊C8—1271no—
258#97C2×D12711yescharacteristic
259#41—5081no—
260#41—2542no—
261C2×Q8—1271no—
262C2×Q8—1271no—
263C2×Q8—1271no—
264C2×Q8—5081no—
265C2×Q8—5081no—
266C2×Q8—5081no—
267C2×Q8—2541no—
268C2×Q8—2541no—
269C2×Q8—2541no—
270C2×Q8—2542no—
271C2×Q8—2542no—
272C2×Q8—2542no—
273C2×Q8C22×D12711yescharacteristic
274C2×Q8C22×D12711yescharacteristic
275C2×D4—1271no—
276C2×D4—1271no—
277C2×D4—2541no—
278C2×D4—2541no—
279C22×C4—1271no—
280C22×C4—1271no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284C22×C4—2541no—
285Q16—5081no—
286Q16—2542no—
287Q16—2542no—
288Q16—2542no—
289Q16—2542no—
290Q16D50814yescomplemented
291SD16—5081no—
292SD16—2542no—
293SD16—2542no—
294M16—2542no—
295C2×C8—1271no—
296C2×C8—1271no—
297C2×C8—2542no—
298C2×C8C22×D12711yescharacteristic
299C2×C8C22×D12711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—1271no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4—2541no—
308C4⋊C4C22×D12711yescharacteristic
309C4⋊C4C22×D12711yescharacteristic
310C22⋊C4—1271no—
311C22⋊C4—1271no—
312C22⋊C4—2541no—
313C22⋊C4—2541no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4—2541no—
317C22⋊C4—2541no—
318C42—21no—
319C42—21no—
320C42C22×D12711yescharacteristic
321C23—1271no—
322C23—1271no—
323C23—2541no—
324Q8—21no—
325Q8—21no—
326Q8—5082no—
327Q8—5082no—
328Q8—5082no—
329Q8—2541no—
330Q8—2541no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8—2541no—
335Q8C2×D50812yes—
336Q8C2×D50812yes—
337D4—5082no—
338D4—5082no—
339D4—2542no—
340D4—2541no—
341D4—2541no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—1271no—
347C2×C4—1271no—
348C2×C4—5082no—
349C2×C4—5082no—
350C2×C4—5082no—
351C2×C4—5082no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4D4×D12711yescharacteristic
360C2×C4D4×D12711yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—21no—
363C8—2541no—
364C8—2541no—
365C8C2×D50812yes—
366C22—1272no—
367C22—5081no—
368C22—5081no—
369C22—5081no—
370C22—2541no—
371C22—2541no—
372C22—2541no—
373C22—2541no—
374C22—2541no—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—5081no—
383C4—5081no—
384C4—5081no—
385C4—2541no—
386C4—2541no—
387C4C22×D50811yescharacteristic
388C4#1686125511yescharacteristic
389C2—10161no—
390C2—5081no—
391C2—2542no—
392C2D4×D50811yescharacteristic
393C2#137×D12711yescharacteristic
394C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#441×C127C2
socleC22×C127C22×D4
prime radical#22550538C2
hypercenter#441D127
hypocenterC127#3270
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
2. C2×Q16×D127C2semidirect
3. Q8×D508C2semidirect
5. #22550270C2semidirect
6. #22550135C2semidirect
7. #22550106C2semidirect
8. #22550098C2semidirect
9. #22550086C2semidirect
10. #22550041C2semidirect
11. #22550033C2semidirect
12. #22550033C2semidirect
13. #22550016C2semidirect
14. #22549949C2semidirect
15. #441×C127C2semidirect
113. Q16×C127D4semidirect
177. C2×D127#137semidirect
198. C127#3270semidirect
212. #441D127semidirect
290. Q16D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×Q16×D1272
Q8×D5082
#225505382
#225502702
#225501352
#225501062
#225500982
#225500862
#225500412
#225500332
#225500162
#225499492
#441×C1272
#3270127