Finite Group Databasebeta

← #30648184

Subgroups — #30648184

Order 16256.

496 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×#16861232C211yesmaximal, characteristic, complemented
2C2×#16861230C211yesmaximal, characteristic, complemented
3#22550425C211yesmaximal, characteristic, complemented
4C4×#14523249C211yesmaximal, characteristic, complemented
5#22550098C211yesmaximal, characteristic, complemented
6#22550092C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550084C211yesmaximal, characteristic, complemented
9#22550080C211yesmaximal, characteristic, complemented
10#22550074C211yesmaximal, characteristic, complemented
11#22550070C211yesmaximal, characteristic, complemented
12C4⋊C8×D127C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic
18C2×#14523247C2211yescharacteristic
19#16861232—24no—
20#16861230—24no—
21C2×#14523238C2211yescharacteristic
22C2×#14523237C2211yescharacteristic
23C2×#14523235C2211yescharacteristic
24C2×#14523235C2211yescharacteristic
25C2×D1016C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×C8×D127C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861207C2211yescharacteristic
30#16861205C2211yescharacteristic
31#16861197C2211yescharacteristic
32#16861182—21no—
33#16861182—21no—
34Q8×Dic127C2211yescharacteristic
35C4×D508—21no—
36C4×D508C2211yescharacteristic
37#16861171—21no—
38#16861171C2211yescharacteristic
39(C22⋊D127)⋊C4—21no—
40(C22⋊D127)⋊C4—21no—
41#16861167—21no—
42#16861167C2211yescharacteristic
43C42×D127—21no—
44C42×D127C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic
46Dic127⋊C8C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861133C2211yescharacteristic
50#16861132C2211yescharacteristic
51#16861131C2211yescharacteristic
52#16861130C2211yescharacteristic
53#16861129C2211yescharacteristic
54C2×Q16×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic
57C4×Q8×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—22no—
63#14523249—22no—
64#14523249D414yescomplemented
65#14523247—22no—
66#14523247—22no—
67C2×Dic254C2311yescharacteristic
68C2×D508—21no—
69C2×D508—21no—
70C2×D508C2311yescharacteristic
71C2×D508C2311yescharacteristic
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C22×Dic127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80C2×C4×D127D411yescharacteristic
81C2×C4×D127D411yescharacteristic
82C2×C4×D127C2311yescharacteristic
83#14523238—22no—
84#14523237—22no—
85#14523235—22no—
86#14523235—22no—
87D1016—22no—
88#14523233—22no—
89C2×C127⋊C8C2311yescharacteristic
90C2×C127⋊C8C2311yescharacteristic
91C8×D127—22no—
92C8×D127—22no—
93C4⋊Dic127—21no—
94C4⋊Dic127C2311yescharacteristic
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4—21no—
102C4×Dic127—21no—
103C4×Dic127—21no—
104C4×Dic127D411yescharacteristic
105C4×Dic127D411yescharacteristic
106C4×Dic127C2311yescharacteristic
107C2×Q8×C127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110Q16×C127—22no—
111SD16×C127—22no—
112C2×C8×C127C2311yescharacteristic
113C2×C8×C127C2311yescharacteristic
114C4⋊C4×C127—21no—
115C4⋊C4×C127C2311yescharacteristic
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C42×C127—21no—
119C42×C127C2311yescharacteristic
120C22×D127—21no—
121C22×D127—21no—
122C22×D127C2×D411yescharacteristic
123Dic254—21no—
124Dic254—21no—
125D508—24no—
126D508—21no—
127D508—21no—
128D508—21no—
129D508—42no—
130D508C2×D412yes—
131C22⋊D127—42no—
132C22⋊D127—42no—
133C22⋊D127—42no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—42no—
137C2×Dic127C2×D411yescharacteristic
138C2×Dic127C2×D411yescharacteristic
139C2×Dic127C2×D411yescharacteristic
140C4×D127—24no—
141C4×D127—22no—
142C4×D127—41no—
143C4×D127—42no—
144C4×D127—42no—
145C4×D127—42no—
146C4×D127C2×D412yes—
147C4×D127#4112yes—
148C127⋊C8—21no—
149C127⋊C8—21no—
150C23×C127—21no—
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127C2×D412yes—
155D4×C127—21no—
156D4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2×D411yescharacteristic
162C2×C4×C127C2411yescharacteristic
163C8×C127—21no—
164C8×C127—21no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127C22×D411yescharacteristic
169C2×D127—21no—
170C2×D127—21no—
171C2×D127—22no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C2×D127—41no—
177C2×D127#12112yes—
178C4×C127—22no—
179C4×C127—21no—
180C4×C127—21no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127C22×D411yescharacteristic
184C4×C127C2×#4111yescharacteristic
185Dic127—21no—
186Dic127—41no—
187Dic127—41no—
188Dic127#12112yes—
189Dic127#13512yescomplemented
190C2×C127—81no—
191C2×C127C2×#12111yescharacteristic
192C2×C127C2×#13511yescharacteristic
193C2×C127#57511yescharacteristic
194D127—22no—
195D127—42no—
196D127—81no—
197#3035—1271nomaximal, Sylow
198C127#303511yescharacteristic, complemented, Sylow
199C2×#135—1271no—
200C2×#135—1271no—
201#532—1271no—
202C4×#41—1271no—
203#464—1271no—
204#464D12711yescharacteristic, complemented
205#463—1271no—
206#463—1271no—
207#462—1271no—
208#461—1271no—
209#460—1271no—
210#459—1271no—
211C2×C4⋊C8—1271no—
212#416—1271no—
213#416—1271no—
214C2×#41—1271no—
215C2×#41—1271no—
216C2×#41—1271no—
217#135—2544no—
218#135—2544no—
219C2×Q16—1271no—
220C2×Q16C2×D12711yescharacteristic
221C2×SD16—1271no—
222C2×SD16—1271no—
223C2×SD16—1271no—
224C2×SD16C2×D12711yescharacteristic
225C2×D8—1271no—
226C2×D8—1271no—
227C22×C8—1271no—
228C22×C8—1271no—
229C4⋊Q8—1271no—
230#127—1271no—
231#124—1271no—
232#124C2×D12711yescharacteristic
233#121—2541no—
234#121—2541no—
235C4×Q8—1271no—
236C4×Q8C2×D12711yescharacteristic
237C4×D4—1271no—
238C4×D4—1271no—
239C4×D4—2541no—
240C4×D4—2541no—
241C4×D4—2541no—
242C4×D4—2541no—
243#117—1271no—
244#117—2541no—
245C2×C42—1271no—
246C2×C42—2541no—
247C4⋊C8—1271no—
248C4⋊C8—1271no—
249C4⋊C8—1271no—
250C4⋊C8C2×D12711yescharacteristic
251#103—1271no—
252#103—1271no—
253#103—1271no—
254#103C2×D12711yescharacteristic
255#102—1271no—
256#102—1271no—
257#102—1271no—
258#102C2×D12711yescharacteristic
259#41—1274no—
260#41—2542no—
261#41—2542no—
262#41—2542no—
263#41—2542no—
264#41—2542no—
265C2×Q8—1271no—
266C2×Q8C22×D12711yescharacteristic
267C2×Q8C22×D12711yescharacteristic
268C2×D4—1271no—
269C2×D4—1271no—
270C2×D4—2541no—
271C2×D4—2541no—
272C2×D4—2541no—
273C2×D4—2541no—
274C2×D4—2541no—
275C2×D4C22×D12711yescharacteristic
276C22×C4—1271no—
277C22×C4—1271no—
278C22×C4—1271no—
279C22×C4—2541no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284Q16—22no—
285Q16—2542no—
286SD16—22no—
287SD16—2542no—
288SD16—2542no—
289SD16—2542no—
290D8—2542no—
291D8—2542no—
292C2×C8—1271no—
293C2×C8—1271no—
294C2×C8—2542no—
295C2×C8—2542no—
296C2×C8C22×D12711yescharacteristic
297C2×C8C22×D12711yescharacteristic
298C4⋊C4—21no—
299C4⋊C4—1271no—
300C4⋊C4—2541no—
301C4⋊C4—2541no—
302C4⋊C4—2541no—
303C4⋊C4C22×D12711yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—2541no—
307C22⋊C4—2541no—
308C22⋊C4—2541no—
309C22⋊C4—2541no—
310C22⋊C4—2541no—
311C42—21no—
312C42—1271no—
313C42—1271no—
314C42—1271no—
315C42—2541no—
316C42—2541no—
317C42C22×D12711yescharacteristic
318C23—21no—
319C23—1271no—
320C23—2541no—
321C23—2541no—
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—2541no—
326Q8—2541no—
327Q8D4×D12712yes—
328D4—21no—
329D4—21no—
330D4—1272no—
331D4—5082no—
332D4—5082no—
333D4—5082no—
334D4—5082no—
335D4—2544no—
336D4—2541no—
337D4—2541no—
338D4—2541no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—1271no—
343C2×C4—1271no—
344C2×C4—1272no—
345C2×C4—1272no—
346C2×C4—1271no—
347C2×C4—5082no—
348C2×C4—5081no—
349C2×C4—5082no—
350C2×C4—5082no—
351C2×C4—5082no—
352C2×C4—2542no—
353C2×C4—2544no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4D4×D12711yescharacteristic
357C2×C4D4×D12711yescharacteristic
358C2×C4C23×D12711yescharacteristic
359C8—21no—
360C8—21no—
361C8—2541no—
362C8—2541no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—1272no—
367C22—5081no—
368C22—5081no—
369C22—5081no—
370C22—5081no—
371C22—5081no—
372C22—2542no—
373C22—2541no—
374C22—2541no—
375C22C2×D4×D12711yescharacteristic
376C4—22no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—1272no—
382C4—1272no—
383C4—5081no—
384C4—5081no—
385C4—2541no—
386C4C2×D4×D12711yescharacteristic
387C4#41×D12711yescharacteristic
388C2—81no—
389C2—10161no—
390C2—5082no—
391C2—2542no—
392C2#121×D12711yescharacteristic
393C2#135×D12711yescharacteristic
394C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
hypercenter#464D127
hypocenterC127#3035
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. C2×#16861230C2semidirect
3. #22550425C2semidirect
4. C4×#14523249C2semidirect
5. #22550098C2semidirect
6. #22550092C2semidirect
7. #22550088C2semidirect
8. #22550084C2semidirect
9. #22550080C2semidirect
10. #22550074C2semidirect
11. #22550070C2semidirect
12. C4⋊C8×D127C2semidirect
13. #22549871C2semidirect
14. #22549869C2semidirect
15. #464×C127C2semidirect
64. #14523249D4semidirect
189. Dic127#135semidirect
198. C127#3035semidirect
204. #464D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612322
C2×#168612302
#225504252
C4×#145232492
#225500982
#225500922
#225500882
#225500842
#225500802
#225500742
#225500702
C4⋊C8×D1272
#225498712
#225498692
#464×C1272
#3035127