Finite Group Databasebeta

← #30650305

Subgroups — #30650305

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×#16861242C211yesmaximal, characteristic, complemented
2C22×Dic508C211yesmaximal, characteristic, complemented
3D4×Dic254C211yesmaximal, characteristic, complemented
4#22550535C211yesmaximal, characteristic
5#22550270C211yesmaximal, characteristic, complemented
6#22550134C211yesmaximal, characteristic, complemented
7#22550105C211yesmaximal, characteristic, complemented
8#22550099C211yesmaximal, characteristic, complemented
9#22550087C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550034C211yesmaximal, characteristic, complemented
12#22550034C211yesmaximal, characteristic, complemented
13#22550017C211yesmaximal, characteristic, complemented
14#22549984C211yesmaximal, characteristic, complemented
15#435×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C22×Dic254C2211yescharacteristic
19C22×Dic254C2211yescharacteristic
20#16861242—24no—
21C2×Dic508—24no—
22C2×Dic508C2211yescharacteristic
23C2×Dic508C2211yescharacteristic
24C2×Dic508C2211yescharacteristic
25C2×Dic508C2211yescharacteristic
26C2×Dic508C2211yescharacteristic
27C2×Dic508C2211yescharacteristic
28C2×#14523237C2211yescharacteristic
29C2×#14523237C2211yescharacteristic
30C2×#14523237C2211yescharacteristic
31#16861209C2211yescharacteristic
32#16861208—21no—
33#16861197C2211yescharacteristic
34#16861195—21no—
35#16861188C2211yescharacteristic
36#16861188C2211yescharacteristic
37#16861188C2211yescharacteristic
38#16861187—21no—
39#16861186—21no—
40#16861186—21no—
41#16861183C2211yescharacteristic
42#16861181—21no—
43C4×Dic254C2211yescharacteristic
44C4×Dic254C2211yescharacteristic
45D4×Dic127—21no—
46(C22⋊D127)⋊C4—21no—
47#16861141C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861134C2211yescharacteristic
50#16861134C2211yescharacteristic
51#16861134C2211yescharacteristic
52#16861134C2211yescharacteristic
53#16861131C2211yescharacteristic
54C2×M16×C127C2211yescharacteristic
55C22×C8×C127C2211yescharacteristic
56C4×D4×C127C2211yescharacteristic
57C4⋊C8×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59C22⋊C8×C127C2211yescharacteristic
60#97×C127C2211yescharacteristic
61#14523248—22no—
62#14523248—41no—
63C2×Dic254—22no—
64C2×Dic254—22no—
65C2×Dic254—22no—
66C2×Dic254—21no—
67C2×Dic254—21no—
68C2×Dic254—21no—
69C2×Dic254—41no—
70C2×Dic254—41no—
71C2×Dic254—41no—
72C2×Dic254C2311yescharacteristic
73C2×Dic254C2311yescharacteristic
74C2×Dic254C2311yescharacteristic
75C2×Dic254C2311yescharacteristic
76C2×Dic254C2311yescharacteristic
77C2×D508C2311yescharacteristic
78C2×C22⋊D127—21no—
79C2×C22⋊D127—21no—
80C22×Dic127—21no—
81C22×Dic127—21no—
82C22×Dic127—21no—
83C2×C4×D127—21no—
84Dic508—22no—
85Dic508—22no—
86Dic508—22no—
87Dic508—22no—
88Dic508—41no—
89Dic508D414yescomplemented
90#14523237—22no—
91#14523237—22no—
92#14523237—41no—
93C4⋊Dic127—21no—
94C4⋊Dic127—21no—
95C4⋊Dic127C2311yescharacteristic
96C4⋊Dic127C2311yescharacteristic
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101Dic127⋊C4—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127—21no—
104C22⋊Dic127—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108C4×Dic127—21no—
109C4×Dic127—21no—
110C2×D4×C127D411yescharacteristic
111C22×C4×C127C2311yescharacteristic
112C22×C4×C127C2311yescharacteristic
113M16×C127—22no—
114C2×C8×C127—22no—
115C2×C8×C127C2311yescharacteristic
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127D411yescharacteristic
120C22⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—42no—
133Dic254—42no—
134Dic254—42no—
135Dic254C2×D412yes—
136Dic254C2×D412yes—
137D508—21no—
138D508—21no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—21no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—21no—
148C2×Dic127—42no—
149C2×Dic127—42no—
150C2×Dic127—42no—
151C4×D127—42no—
152C23×C127C2×D411yescharacteristic
153C23×C127C2×D411yescharacteristic
154D4×C127—22no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127C2×D411yescharacteristic
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2×D411yescharacteristic
162C2×C4×C127C2×D411yescharacteristic
163C2×C4×C127C2411yescharacteristic
164C8×C127—21no—
165C8×C127—21no—
166C8×C127—21no—
167C8×C127C2×D412yes—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127#13712yescomplemented
174C22×C127C22×D411yescharacteristic
175C2×D127—41no—
176C2×D127—41no—
177C2×D127—41no—
178C4×C127—21no—
179C4×C127—21no—
180C4×C127—21no—
181C4×C127—21no—
182C4×C127C22×D411yescharacteristic
183C4×C127H(5,𝔽2)11yescharacteristic
184Dic127—22no—
185Dic127—22no—
186Dic127—41no—
187Dic127—41no—
188Dic127—41no—
189Dic127—41no—
190Dic127—41no—
191C2×C127—22no—
192C2×C127—41no—
193C2×C127D4211yescharacteristic
194C2×C127C2×#13711yescharacteristic
195C2×C127#57811yescharacteristic
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#441—1271no—
213#435D12711yescharacteristic, complemented
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—1271no—
225C2×Q16—2544no—
226C2×SD16—1271no—
227C2×SD16—1271no—
228C2×SD16—1271no—
229C2×M16C2×D12711yescharacteristic
230C22×C8C2×D12711yescharacteristic
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×Q8—1271no—
244C4×Q8—1271no—
245C4×D4—2541no—
246C4×D4—2541no—
247C4×D4C2×D12711yescharacteristic
248#106—1271no—
249C4⋊C8C2×D12711yescharacteristic
250#103—1271no—
251#103—1271no—
252#103—1271no—
253#103—1271no—
254#103—1271no—
255#102—1271no—
256C22⋊C8C2×D12711yescharacteristic
257C22⋊C8C2×D12711yescharacteristic
258#97C2×D12711yescharacteristic
259#41—5081no—
260#41—2542no—
261C2×Q8—1271no—
262C2×Q8—1271no—
263C2×Q8—1271no—
264C2×Q8—1271no—
265C2×Q8—1271no—
266C2×Q8—5081no—
267C2×Q8—5081no—
268C2×Q8—5081no—
269C2×Q8—2542no—
270C2×Q8—2542no—
271C2×Q8—2542no—
272C2×Q8—2541no—
273C2×Q8—2541no—
274C2×Q8—2541no—
275C2×D4—1271no—
276C2×D4—2541no—
277C2×D4—2541no—
278C2×D4D50811yescharacteristic
279C22×C4—2541no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4C22×D12711yescharacteristic
284C22×C4C22×D12711yescharacteristic
285Q16—1274no—
286Q16—5081no—
287Q16—2542no—
288Q16—2542no—
289Q16—2542no—
290Q16—2542no—
291SD16—5081no—
292SD16—2542no—
293SD16—2542no—
294M16—22no—
295C2×C8—22no—
296C2×C8C22×D12711yescharacteristic
297C2×C8C22×D12711yescharacteristic
298C2×C8C22×D12711yescharacteristic
299C2×C8C22×D12711yescharacteristic
300C4⋊C4—1271no—
301C4⋊C4—1271no—
302C4⋊C4—2541no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4—2541no—
308C4⋊C4—2541no—
309C4⋊C4D50811yescharacteristic
310C22⋊C4—2541no—
311C22⋊C4—2541no—
312C22⋊C4—2541no—
313C22⋊C4—2541no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4D50811yescharacteristic
317C22⋊C4D50811yescharacteristic
318C42—2541no—
319C42—2541no—
320C42C22×D12711yescharacteristic
321C23—2541no—
322C23C2×D50811yescharacteristic
323C23C2×D50811yescharacteristic
324Q8—1272no—
325Q8—1272no—
326Q8—5082no—
327Q8—5082no—
328Q8—5082no—
329Q8—2541no—
330Q8—2541no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8—2541no—
335Q8—2541no—
336Q8—2541no—
337D4—22no—
338D4—5082no—
339D4—5082no—
340D4—2541no—
341D4—2541no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—5082no—
347C2×C4—5082no—
348C2×C4—5082no—
349C2×C4—5082no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4C2×D50811yescharacteristic
358C2×C4C2×D50811yescharacteristic
359C2×C4C2×D50811yescharacteristic
360C2×C4C2×D50811yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8D4×D12712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—5081no—
372C22—5081no—
373C22—5081no—
374C22#1686124212yescomplemented
375C22C22×D50811yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—5081no—
381C4—5081no—
382C4—5081no—
383C4—5081no—
384C4—5081no—
385C4—2542no—
386C4—2542no—
387C4C2×D4×D12711yescharacteristic
388C4#1686125511yescharacteristic
389C2—22no—
390C2—41no—
391C2—10161no—
392C2D4×D50811yescharacteristic
393C2C2×#1686124211yescharacteristic
394C2#2255069311yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#435×C127C2
socleC22×C127C22×D4
prime radical#22550535C2
hypercenter#435D127
hypocenterC127#3270
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C22×Dic508C2semidirect
3. D4×Dic254C2semidirect
5. #22550270C2semidirect
6. #22550134C2semidirect
7. #22550105C2semidirect
8. #22550099C2semidirect
9. #22550087C2semidirect
10. #22550040C2semidirect
11. #22550034C2semidirect
12. #22550034C2semidirect
13. #22550017C2semidirect
14. #22549984C2semidirect
15. #435×C127C2semidirect
89. Dic508D4semidirect
173. C22×C127#137semidirect
198. C127#3270semidirect
213. #435D127semidirect
374. C22#16861242semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C22×Dic5082
D4×Dic2542
#225505352
#225502702
#225501342
#225501052
#225500992
#225500872
#225500402
#225500342
#225500172
#225499842
#435×C1272
#3270127