Finite Group Databasebeta

← #30648181

Subgroups — #30648181

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×#16861229C211yesmaximal, characteristic, complemented
2#22550422C211yesmaximal, characteristic, complemented
3#41×Dic127C211yesmaximal, characteristic, complemented
4#22550100C211yesmaximal, characteristic, complemented
5#22550096C211yesmaximal, characteristic, complemented
6#22550093C211yesmaximal, characteristic, complemented
7#22550089C211yesmaximal, characteristic, complemented
8#22550084C211yesmaximal, characteristic, complemented
9#22550079C211yesmaximal, characteristic, complemented
10#22550074C211yesmaximal, characteristic, complemented
11#22550069C211yesmaximal, characteristic, complemented
12C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
13#22549873C211yesmaximal, characteristic, complemented
14#22549872C211yesmaximal, characteristic, complemented
15C2×#135×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17#16861229—24no—
18C2×#14523238C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21C2×#14523233C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861207C2211yescharacteristic
24#16861205C2211yescharacteristic
25#16861198C2211yescharacteristic
26#16861196C2211yescharacteristic
27#16861181—21no—
28#16861181—21no—
29Q8×Dic127C2211yescharacteristic
30Q8×Dic127C2211yescharacteristic
31D4×Dic127—21no—
32D4×Dic127—21no—
33D4×Dic127C2211yescharacteristic
34D4×Dic127C2211yescharacteristic
35C4×C22⋊D127—21no—
36C4×C22⋊D127—21no—
37#16861168—21no—
38#16861168C2211yescharacteristic
39C2×C4×Dic127—21no—
40C2×C4×Dic127C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43Dic127⋊C8C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45#16861134C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861133C2211yescharacteristic
48#16861133C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861130C2211yescharacteristic
52#16861130C2211yescharacteristic
53C2×#41×C127C2211yescharacteristic
54C2×#41×C127C2211yescharacteristic
55#135×C127—24no—
56C2×Q16×C127C2211yescharacteristic
57C2×SD16×C127C2211yescharacteristic
58C2×SD16×C127C2211yescharacteristic
59C2×D8×C127C2211yescharacteristic
60C22×C8×C127C2211yescharacteristic
61#14523248—22no—
62#14523248—22no—
63C2×Dic254C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C22×Dic127—21no—
69C22×Dic127—21no—
70C22×Dic127D411yescharacteristic
71C22×Dic127D411yescharacteristic
72C2×C4×D127—21no—
73#14523238—22no—
74#14523236—22no—
75#14523235—22no—
76#14523233—22no—
77C2×C127⋊C8—22no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C4⋊Dic127—21no—
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127D411yescharacteristic
97C4×Dic127D411yescharacteristic
98C4×Dic127C2311yescharacteristic
99C4×Dic127C2311yescharacteristic
100#41×C127—22no—
101#41×C127—22no—
102#41×C127—22no—
103#41×C127D414yescomplemented
104C2×Q8×C127C2311yescharacteristic
105C2×Q8×C127C2311yescharacteristic
106C2×D4×C127—21no—
107C2×D4×C127—21no—
108C2×D4×C127C2311yescharacteristic
109C2×D4×C127C2311yescharacteristic
110C22×C4×C127—21no—
111C22×C4×C127—21no—
112C22×C4×C127C2311yescharacteristic
113Q16×C127—22no—
114SD16×C127—22no—
115SD16×C127—22no—
116D8×C127—22no—
117C2×C8×C127—22no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C22×D127—21no—
121Dic254—21no—
122Dic254—21no—
123D508—21no—
124D508—21no—
125C22⋊D127—42no—
126C22⋊D127—42no—
127C22⋊D127—42no—
128C2×Dic127—22no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127—21no—
132C2×Dic127—41no—
133C2×Dic127—42no—
134C2×Dic127—42no—
135C2×Dic127C2×D411yescharacteristic
136C2×Dic127C2×D411yescharacteristic
137C2×Dic127C2×D411yescharacteristic
138C2×Dic127C2×D411yescharacteristic
139C4×D127—42no—
140C127⋊C8—21no—
141C127⋊C8—21no—
142C23×C127—21no—
143C23×C127—21no—
144C23×C127C2×D411yescharacteristic
145Q8×C127—21no—
146Q8×C127—21no—
147Q8×C127—21no—
148Q8×C127C2×D412yes—
149D4×C127—24no—
150D4×C127—21no—
151D4×C127—21no—
152D4×C127—21no—
153D4×C127—42no—
154D4×C127C2×D412yes—
155C2×C4×C127—21no—
156C2×C4×C127—24no—
157C2×C4×C127—21no—
158C2×C4×C127—42no—
159C2×C4×C127C2×D411yescharacteristic
160C2×C4×C127C2×D412yes—
161C2×C4×C127#4112yes—
162C2×C4×C127C2411yescharacteristic
163C8×C127—21no—
164C8×C127—21no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—22no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127#12112yes—
174C22×C127C22×D411yescharacteristic
175C2×D127—41no—
176C2×D127—41no—
177C2×D127—41no—
178C4×C127—22no—
179C4×C127—41no—
180C4×C127#12112yes—
181C4×C127C22×D411yescharacteristic
182C4×C127C2×#4111yescharacteristic
183Dic127—21no—
184Dic127—21no—
185Dic127—21no—
186Dic127—41no—
187Dic127—41no—
188Dic127—41no—
189Dic127#13512yescomplemented
190C2×C127—22no—
191C2×C127—42no—
192C2×C127—81no—
193C2×C127C2×#12111yescharacteristic
194C2×C127C2×#13511yescharacteristic
195C2×C127#57511yescharacteristic
196D127—81no—
197#3035—1271nomaximal, Sylow
198C127#303511yescharacteristic, complemented, Sylow
199C2×#135—1271no—
200C2×#135D12711yescharacteristic, complemented
201#532—1271no—
202C4×#41—1271no—
203#464—1271no—
204#464—1271no—
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×#41C2×D12711yescharacteristic
216C2×#41C2×D12711yescharacteristic
217#135—24no—
218#135—2544no—
219C2×Q16—1271no—
220C2×Q16C2×D12711yescharacteristic
221C2×SD16—1271no—
222C2×SD16—1271no—
223C2×SD16C2×D12711yescharacteristic
224C2×SD16C2×D12711yescharacteristic
225C2×D8—1271no—
226C2×D8C2×D12711yescharacteristic
227C22×C8—1271no—
228C22×C8C2×D12711yescharacteristic
229C4⋊Q8—1271no—
230#127—1271no—
231#124—1271no—
232#124—1271no—
233#121—2541no—
234#121—2541no—
235C4×Q8—1271no—
236C4×Q8—1271no—
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⋊C8—1271no—
251#103—1271no—
252#103—1271no—
253#103—1271no—
254#103—1271no—
255#102—1271no—
256#102—1271no—
257#102—1271no—
258#102—1271no—
259#41—22no—
260#41—22no—
261#41—22no—
262#41—2542no—
263#41—2542no—
264#41C22⋊D12714yescomplemented
265C2×Q8—1271no—
266C2×Q8C22×D12711yescharacteristic
267C2×Q8C22×D12711yescharacteristic
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—1271no—
271C2×D4—2541no—
272C2×D4—2541no—
273C2×D4—2541no—
274C2×D4C22×D12711yescharacteristic
275C2×D4C22×D12711yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—1271no—
279C22×C4—1271no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4C22×D12711yescharacteristic
284Q16—22no—
285Q16—2542no—
286SD16—22no—
287SD16—22no—
288SD16—2542no—
289SD16—2542no—
290D8—22no—
291D8—2542no—
292C2×C8—22no—
293C2×C8—1271no—
294C2×C8—1271no—
295C2×C8—2542no—
296C2×C8C22×D12711yescharacteristic
297C2×C8C22×D12711yescharacteristic
298C4⋊C4—1271no—
299C4⋊C4—1271no—
300C4⋊C4—2541no—
301C4⋊C4—2541no—
302C4⋊C4—2541no—
303C4⋊C4—2541no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C22⋊C4—2541no—
307C22⋊C4—2541no—
308C22⋊C4—2541no—
309C22⋊C4—2541no—
310C22⋊C4—2541no—
311C42—1271no—
312C42—1271no—
313C42—1271no—
314C42—1271no—
315C42—2541no—
316C42—2541no—
317C42—2541no—
318C23—21no—
319C23—21no—
320C23—2541no—
321C23D4×D12711yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—2541no—
326Q8—2541no—
327Q8C2×C22⋊D12712yes—
328D4—24no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—5082no—
334D4—5082no—
335D4—5082no—
336D4—2541no—
337D4—2541no—
338D4C2×C22⋊D12712yes—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1271no—
344C2×C4—1271no—
345C2×C4—1271no—
346C2×C4—1271no—
347C2×C4—5082no—
348C2×C4—5082no—
349C2×C4—5082no—
350C2×C4—5081no—
351C2×C4—2542no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4D4×D12711yescharacteristic
356C2×C4C2×C22⋊D12712yes—
357C2×C4#1452324712yes—
358C2×C4C23×D12711yescharacteristic
359C8—21no—
360C8—21no—
361C8—2541no—
362C8—2541no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—5081no—
372C22—5081no—
373C22—5081no—
374C22#1686118112yes—
375C22C2×D4×D12711yescharacteristic
376C4—22no—
377C4—41no—
378C4—1272no—
379C4—5081no—
380C4—5081no—
381C4—5081no—
382C4—2541no—
383C4—2541no—
384C4—2541no—
385C4#1686118112yes—
386C4C22×C22⋊D12711yescharacteristic
387C4C2×#1452324711yescharacteristic
388C2—22no—
389C2—42no—
390C2—81no—
391C2—10161no—
392C2C2×#1686118111yescharacteristic
393C2#135×D12711yescharacteristic
394C2#2255067311yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#135×C127C2
socleC22×C127C22×D4
hypercenterC2×#135D127
hypocenterC127#3035
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. #22550422C2semidirect
3. #41×Dic127C2semidirect
4. #22550100C2semidirect
5. #22550096C2semidirect
6. #22550093C2semidirect
7. #22550089C2semidirect
8. #22550084C2semidirect
9. #22550079C2semidirect
10. #22550074C2semidirect
11. #22550069C2semidirect
12. C2×Dic127⋊C8C2semidirect
13. #22549873C2semidirect
14. #22549872C2semidirect
15. C2×#135×C127C2semidirect
103. #41×C127D4semidirect
189. Dic127#135semidirect
198. C127#3035semidirect
200. C2×#135D127semidirect
264. #41C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
#225504222
#41×Dic1272
#225501002
#225500962
#225500932
#225500892
#225500842
#225500792
#225500742
#225500692
C2×Dic127⋊C82
#225498732
#225498722
C2×#135×C1272
#3035127