Finite Group Databasebeta

← #30650317

Subgroups — #30650317

Order 16256.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C211yesmaximal, characteristic, complemented
2C2×#16861232C211yesmaximal, characteristic, complemented
3#22550548C211yesmaximal, characteristic, complemented
4#22550538C211yesmaximal, characteristic, complemented
5#22550265C211yesmaximal, characteristic, complemented
6#22550130C211yesmaximal, characteristic, complemented
7#22550106C211yesmaximal, characteristic, complemented
8#22550098C211yesmaximal, characteristic, complemented
9#22550076C211yesmaximal, characteristic, complemented
10#22550041C211yesmaximal, characteristic, complemented
11#22550024C211yesmaximal, characteristic, complemented
12#22550024C211yesmaximal, characteristic, complemented
13#22550016C211yesmaximal, characteristic, complemented
14#22549949C211yesmaximal, characteristic, complemented
15#441×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic
18C2×#14523249C2211yescharacteristic
19C2×Q8×D127C2211yescharacteristic
20#16861241—24no—
21#16861232—24no—
22C2×#14523238C2211yescharacteristic
23C2×#14523237C2211yescharacteristic
24C2×#14523237C2211yescharacteristic
25C2×#14523235C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523235C2211yescharacteristic
28C2×D1016C2211yescharacteristic
29C2×D1016C2211yescharacteristic
30C2×#14523231C2211yescharacteristic
31C2×C8×D127C2211yescharacteristic
32#16861206—21no—
33#16861206C2211yescharacteristic
34#16861197—21no—
35#16861197C2211yescharacteristic
36#16861189—21no—
37#16861189C2211yescharacteristic
38#16861184—21no—
39#16861184—21no—
40#16861184—21no—
41#16861184C2211yescharacteristic
42#16861184C2211yescharacteristic
43#16861184C2211yescharacteristic
44C4×D508—21no—
45C4×D508—21no—
46C4×D508C2211yescharacteristic
47C4⋊(C127⋊C8)C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861129C2211yescharacteristic
50#16861129C2211yescharacteristic
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—22no—
63#14523249—22no—
64#14523249—41no—
65#14523249—41no—
66#14523249—41no—
67C2×Dic254—21no—
68C2×Dic254C2311yescharacteristic
69Q8×D127—22no—
70Q8×D127—41no—
71C2×D508—21no—
72C2×D508—21no—
73C2×D508—21no—
74C2×D508—21no—
75C2×D508—21no—
76C2×D508—21no—
77C2×D508—21no—
78C2×D508—21no—
79C2×D508D411yescharacteristic
80C2×D508C2311yescharacteristic
81C2×D508C2311yescharacteristic
82C2×D508C2311yescharacteristic
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85C2×C4×D127—21no—
86C2×C4×D127—21no—
87C2×C4×D127C2311yescharacteristic
88C2×C4×D127C2311yescharacteristic
89#14523238—22no—
90#14523237—22no—
91#14523237—41no—
92#14523235—22no—
93#14523235—22no—
94#14523235—22no—
95D1016—22no—
96D1016—41no—
97#14523231—22no—
98C2×C127⋊C8C2311yescharacteristic
99C2×C127⋊C8C2311yescharacteristic
100C8×D127—22no—
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×D127—21no—
125C22×D127—21no—
126C22×D127C2×D411yescharacteristic
127C22×D127C2×D411yescharacteristic
128Dic254—21no—
129Dic254—21no—
130Dic254—42no—
131D508—21no—
132D508—21no—
133D508—21no—
134D508—21no—
135D508—21no—
136D508—21no—
137D508—41no—
138D508—42no—
139D508—42no—
140D508—42no—
141D508—42no—
142D508—42no—
143D508—81no—
144D508—81no—
145D508—81no—
146C2×Dic127—21no—
147C2×Dic127C2×D411yescharacteristic
148C2×Dic127C2×D411yescharacteristic
149C4×D127—21no—
150C4×D127—21no—
151C4×D127—21no—
152C4×D127—21no—
153C4×D127—42no—
154C4×D127—42no—
155C4×D127—42no—
156C4×D127—42no—
157C127⋊C8—21no—
158C127⋊C8—21no—
159Q8×C127—21no—
160Q8×C127—21no—
161Q8×C127C2×D412yes—
162Q8×C127C2×D412yes—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127C2×D412yes—
172C22×C127C22×D411yescharacteristic
173C2×D127—21no—
174C2×D127—21no—
175C2×D127—21no—
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C4×C127—22no—
189C4×C127—22no—
190C4×C127—21no—
191C4×C127—21no—
192C4×C127—41no—
193C4×C127—41no—
194C4×C127C22×D411yescharacteristic
195C4×C127H(5,𝔽2)11yescharacteristic
196Dic127—21no—
197Dic127—21no—
198Dic127—41no—
199C2×C127D4211yescharacteristic
200C2×C127#57511yescharacteristic
201C2×C127#57711yescharacteristic
202D127—41no—
203D127—41no—
204D127—81no—
205D127—81no—
206D127—81no—
207#3271—1271nomaximal, Sylow
208C127#327111yescharacteristic, complemented, Sylow
209C2×#137—1271no—
210C2×#135—1271no—
211#550—1271no—
212#548—1271no—
213#496—1271no—
214#469—1271no—
215#465—1271no—
216#464—1271no—
217#460—1271no—
218#452—1271no—
219#449—1271no—
220#449—1271no—
221#448—1271no—
222#441D12711yescharacteristic, complemented
223#435—1271no—
224C2×#41—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C22×Q8—1271no—
228#137—2544no—
229#135—2544no—
230C2×Q16—1271no—
231C2×Q16C2×D12711yescharacteristic
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16—1271no—
237C2×D8—1271no—
238C2×D8—1271no—
239C2×M16—1271no—
240C22×C8—1271no—
241#127—1271no—
242#127—2541no—
243#124—1271no—
244#124—2541no—
245C22⋊Q8—1271no—
246C22⋊Q8—2541no—
247#121—1271no—
248#121—1271no—
249#121—1271no—
250#121—2541no—
251#121—2541no—
252#121—2541no—
253C4×Q8C2×D12711yescharacteristic
254C4×Q8C2×D12711yescharacteristic
255C4×D4—1271no—
256C4×D4—2541no—
257C4×D4—2541no—
258#106C2×D12711yescharacteristic
259C4⋊C8—1271no—
260#103—1271no—
261#103C2×D12711yescharacteristic
262#103C2×D12711yescharacteristic
263#102—1271no—
264#102—1271no—
265#102—1271no—
266C22⋊C8—1271no—
267C22⋊C8—1271no—
268#97C2×D12711yescharacteristic
269#41—5081no—
270#41—5081no—
271#41—5081no—
272#41—2542no—
273#41—2542no—
274#41—2542no—
275C2×Q8—1271no—
276C2×Q8—5081no—
277C2×Q8—2541no—
278C2×Q8—2542no—
279C2×Q8C22×D12711yescharacteristic
280C2×Q8C22×D12711yescharacteristic
281C2×D4—1271no—
282C2×D4—1271no—
283C2×D4—1271no—
284C2×D4—1271no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4—2541no—
293C22×C4—1271no—
294C22×C4—1271no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4—2541no—
299Q16—2542no—
300Q16D50814yescomplemented
301SD16—5081no—
302SD16—2542no—
303SD16—2542no—
304SD16—2542no—
305SD16—2542no—
306D8—5081no—
307D8—2542no—
308M16—2542no—
309C2×C8—1271no—
310C2×C8—1271no—
311C2×C8—2542no—
312C2×C8C22×D12711yescharacteristic
313C2×C8C22×D12711yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—1271no—
317C4⋊C4—2541no—
318C4⋊C4C22×D12711yescharacteristic
319C4⋊C4C22×D12711yescharacteristic
320C22⋊C4—1271no—
321C22⋊C4—1271no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C42—21no—
329C42—21no—
330C42C22×D12711yescharacteristic
331C23—1271no—
332C23—1271no—
333C23—2541no—
334C23—2541no—
335C23—2541no—
336Q8—21no—
337Q8—21no—
338Q8—5082no—
339Q8—2541no—
340Q8—2541no—
341Q8C2×D50812yes—
342Q8C2×D50812yes—
343D4—10161no—
344D4—10161no—
345D4—10161no—
346D4—5082no—
347D4—5082no—
348D4—5082no—
349D4—5082no—
350D4—5081no—
351D4—5082no—
352D4—2541no—
353D4—2541no—
354D4—2541no—
355D4—2541no—
356D4—2541no—
357D4—2541no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—1271no—
363C2×C4—1271no—
364C2×C4—5082no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4D4×D12711yescharacteristic
374C2×C4D4×D12711yescharacteristic
375C2×C4C23×D12711yescharacteristic
376C8—21no—
377C8—2541no—
378C8—2541no—
379C8C2×D50812yes—
380C22—5081no—
381C22—5081no—
382C22—5081no—
383C22—5081no—
384C22—5081no—
385C22—5081no—
386C22—5081no—
387C22—5081no—
388C22—5081no—
389C22—2541no—
390C22—2541no—
391C22—2541no—
392C22—2541no—
393C22—2541no—
394C22—2541no—
395C22C2×D4×D12711yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—5081no—
403C4—2541no—
404C4—2541no—
405C4C22×D50811yescharacteristic
406C4#1686125511yescharacteristic
407C2—10161no—
408C2—10161no—
409C2—10161no—
410C2—5081no—
411C2—5081no—
412C2D4×D50811yescharacteristic
413C2#2255067511yescharacteristic
414C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#441×C127C2
socleC22×C127C22×D4
hypercenter#441D127
hypocenterC127#3271
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×#16861232C2semidirect
3. #22550548C2semidirect
4. #22550538C2semidirect
5. #22550265C2semidirect
6. #22550130C2semidirect
7. #22550106C2semidirect
8. #22550098C2semidirect
9. #22550076C2semidirect
10. #22550041C2semidirect
11. #22550024C2semidirect
12. #22550024C2semidirect
13. #22550016C2semidirect
14. #22549949C2semidirect
15. #441×C127C2semidirect
113. Q16×C127D4semidirect
208. C127#3271semidirect
222. #441D127semidirect
300. Q16D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×#168612322
#225505482
#225505382
#225502652
#225501302
#225501062
#225500982
#225500762
#225500412
#225500242
#225500162
#225499492
#441×C1272
#3271127