Finite Group Databasebeta

← #30650293

Subgroups — #30650293

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×#16861238C211yesmaximal, characteristic, complemented
2C22×#14523238C211yesmaximal, characteristic, complemented
3Q8×C22⋊D127C211yesmaximal, characteristic, complemented
4#22550532C211yesmaximal, characteristic
5#22550267C211yesmaximal, characteristic, complemented
6#22550133C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550096C211yesmaximal, characteristic, complemented
9#22550085C211yesmaximal, characteristic, complemented
10#22550035C211yesmaximal, characteristic, complemented
11#22550034C211yesmaximal, characteristic, complemented
12#22550033C211yesmaximal, characteristic, complemented
13#22549980C211yesmaximal, characteristic, complemented
14#22549944C211yesmaximal, characteristic, complemented
15#448×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861238—24no—
20C2×Dic508C2211yescharacteristic
21C2×Dic508C2211yescharacteristic
22C2×#14523238—24no—
23C2×#14523238C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25C2×#14523238C2211yescharacteristic
26C2×#14523238C2211yescharacteristic
27C2×#14523236C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29C22×C127⋊C8C2211yescharacteristic
30#16861207—21no—
31#16861207C2211yescharacteristic
32#16861196C2211yescharacteristic
33#16861195—21no—
34#16861189C2211yescharacteristic
35#16861188—21no—
36#16861188C2211yescharacteristic
37#16861186—21no—
38#16861185—21no—
39#16861185C2211yescharacteristic
40#16861181—21no—
41Q8×Dic127C2211yescharacteristic
42#16861174C2211yescharacteristic
43D4×Dic127—21no—
44C4×C22⋊D127—21no—
45C4×C22⋊D127C2211yescharacteristic
46#16861140C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861134C2211yescharacteristic
50#16861132C2211yescharacteristic
51#16861130C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53#16861119C2211yescharacteristic
54C22×Q8×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56C2×SD16×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61#14523247—22no—
62#14523247—41no—
63C2×Dic254—21no—
64C2×Dic254—22no—
65C2×Dic254—21no—
66C2×Dic254—41no—
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70Q8×D127—22no—
71Q8×D127—41no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127D411yescharacteristic
74C22×Dic127—21no—
75C22×Dic127—21no—
76C2×C4×D127—21no—
77C2×C4×D127C2311yescharacteristic
78Dic508—22no—
79Dic508—41no—
80#14523238—22no—
81#14523238—22no—
82#14523238—22no—
83#14523238D414yescomplemented
84#14523236—22no—
85#14523231—22no—
86C2×C127⋊C8—22no—
87C2×C127⋊C8C2311yescharacteristic
88C2×C127⋊C8C2311yescharacteristic
89C4⋊Dic127—21no—
90C4⋊Dic127—21no—
91C4⋊Dic127C2311yescharacteristic
92Dic127⋊C4—21no—
93Dic127⋊C4—21no—
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4D411yescharacteristic
98C22⋊Dic127—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127D411yescharacteristic
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4D411yescharacteristic
105C4×Dic127—21no—
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108C2×Q8×C127—21no—
109C2×Q8×C127—22no—
110C2×Q8×C127—41no—
111C2×Q8×C127C2311yescharacteristic
112C2×Q8×C127C2311yescharacteristic
113C2×D4×C127—21no—
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127—21no—
116C22×C4×C127C2311yescharacteristic
117SD16×C127—22no—
118SD16×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127C2×D411yescharacteristic
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—42no—
130Dic254—42no—
131Dic254C2×D412yes—
132C22⋊D127—22no—
133C22⋊D127—42no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—42no—
139C2×Dic127—42no—
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C2×Dic127C2×D411yescharacteristic
143C4×D127—21no—
144C4×D127—21no—
145C4×D127—42no—
146C127⋊C8—21no—
147C127⋊C8C2×D412yes—
148C23×C127—21no—
149C23×C127C2×D411yescharacteristic
150Q8×C127—21no—
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127—42no—
154Q8×C127C2×D412yes—
155D4×C127—21no—
156D4×C127—21no—
157D4×C127—42no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—42no—
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—21no—
167C8×C127—21no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127#13712yescomplemented
174C22×C127C22×D411yescharacteristic
175C2×D127—21no—
176C2×D127—21no—
177C2×D127—21no—
178C4×C127—21no—
179C4×C127—22no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127C22×D411yescharacteristic
183C4×C127H(5,𝔽2)11yescharacteristic
184Dic127—21no—
185Dic127—22no—
186Dic127—21no—
187Dic127—21no—
188Dic127—41no—
189Dic127—41no—
190Dic127—41no—
191C2×C127—22no—
192C2×C127—81no—
193C2×C127D4211yescharacteristic
194C2×C127C2×#13711yescharacteristic
195C2×C127#57811yescharacteristic
196D127—41no—
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#448D12711yescharacteristic, complemented
212#441—1271no—
213#435—1271no—
214C2×#41—1271no—
215C22×Q8—1271no—
216C22×Q8—1271no—
217C22×Q8C2×D12711yescharacteristic
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×SD16C2×D12711yescharacteristic
228C2×SD16C2×D12711yescharacteristic
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—2541no—
242#121C2×D12711yescharacteristic
243C4×Q8—1271no—
244C4×Q8—1271no—
245C4×D4—1271no—
246C4×D4—2541no—
247C4×D4—2541no—
248#106—1271no—
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⋊C8C2×D12711yescharacteristic
258#97—1271no—
259#41—5081no—
260#41—2542no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—1271no—
265C2×Q8—1271no—
266C2×Q8—1271no—
267C2×Q8—5081no—
268C2×Q8—5081no—
269C2×Q8—2541no—
270C2×Q8—2542no—
271C2×Q8—2542no—
272C2×Q8—2541no—
273C2×Q8C22×D12711yescharacteristic
274C2×Q8C22×D12711yescharacteristic
275C2×D4—21no—
276C2×D4—1271no—
277C2×D4—2541no—
278C2×D4C22×D12711yescharacteristic
279C22×C4—21no—
280C22×C4—1271no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284C22×C4C22×D12711yescharacteristic
285Q16—1274no—
286Q16—5081no—
287Q16—2542no—
288Q16—2542no—
289Q16—2542no—
290Q16—2542no—
291SD16—22no—
292SD16—41no—
293SD16—2542no—
294M16—2542no—
295C2×C8—1271no—
296C2×C8—1271no—
297C2×C8—2542no—
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⋊C4C22×D12711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—1271no—
312C22⋊C4—1271no—
313C22⋊C4—2541no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4—2541no—
317C22⋊C4—2541no—
318C42—1271no—
319C42—2541no—
320C42—2541no—
321C23—21no—
322C23—1271no—
323C23D4×D12711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—1272no—
329Q8—5082no—
330Q8—5082no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8—2541no—
335Q8—2541no—
336Q8D4×D12712yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—5082no—
341D4—2542no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—1271no—
349C2×C4—1271no—
350C2×C4—1271no—
351C2×C4—5082no—
352C2×C4—5082no—
353C2×C4—5082no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4D4×D12711yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—21no—
363C8—21no—
364C8—1272no—
365C8—2541no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—2541no—
372C22—2541no—
373C22—2541no—
374C22#1686123812yescomplemented
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—5081no—
381C4—5081no—
382C4—5081no—
383C4—2541no—
384C4—2541no—
385C4—2541no—
386C4—2542no—
387C4C2×D4×D12711yescharacteristic
388C4#1686125411yescharacteristic
389C2—22no—
390C2—81no—
391C2—5081no—
392C2#2255050811yescharacteristic
393C2C2×#1686123811yescharacteristic
394C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#448×C127C2
socleC22×C127C22×D4
prime radical#22550532C2
hypercenter#448D127
hypocenterC127#3270
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. C22×#14523238C2semidirect
3. Q8×C22⋊D127C2semidirect
5. #22550267C2semidirect
6. #22550133C2semidirect
7. #22550103C2semidirect
8. #22550096C2semidirect
9. #22550085C2semidirect
10. #22550035C2semidirect
11. #22550034C2semidirect
12. #22550033C2semidirect
13. #22549980C2semidirect
14. #22549944C2semidirect
15. #448×C127C2semidirect
83. #14523238D4semidirect
173. C22×C127#137semidirect
198. C127#3270semidirect
211. #448D127semidirect
374. C22#16861238semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C22×#145232382
Q8×C22⋊D1272
#225505322
#225502672
#225501332
#225501032
#225500962
#225500852
#225500352
#225500342
#225500332
#225499802
#225499442
#448×C1272
#3270127