Finite Group Databasebeta

← #30648107

Subgroups — #30648107

Order 16256.

704 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550713C211yesmaximal, characteristic
2#137×D127C212yesmaximal, complemented
3#22550451C211yesmaximal, characteristic, complemented
4#22550061C211yesmaximal, characteristic, complemented
5#22550059C211yesmaximal, characteristic, complemented
6#22550058C211yesmaximal, characteristic, complemented
7#22550055C212yesmaximal, complemented
8#22550052C212yesmaximal, complemented
9#22549891C212yesmaximal, complemented
10#22549859C211yesmaximal, characteristic, complemented
11#456×C127C211yesmaximal, characteristic, complemented
12#16861257—22no—
13#16861257C2211yescharacteristic
14#16861255—22no—
15#16861255—22no—
16#16861255C2212yes—
17C2×#14523249—22no—
18C2×#14523249C2211yescharacteristic
19#41×D127—22no—
20#41×D127—22no—
21#41×D127C2212yes—
22C2×Q8×D127C2211yescharacteristic, complemented
23#16861242C2212yescomplemented
24#16861241—22no—
25#16861240C2212yescomplemented
26#16861239C2212yescomplemented
27#16861238—22no—
28Q16×D127—22no—
29SD16×D127—22no—
30M16×D127C2212yescomplemented
31#16861207C2211yescharacteristic, complemented
32#16861198C2211yescharacteristic, complemented
33#16861197C2211yescharacteristic, complemented
34#16861194—22no—
35#16861187—22no—
36#16861166C2211yescharacteristic, complemented
37#16861137C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861128C2211yescharacteristic, complemented
41#16861125C2212yescomplemented
422−1+4×C127C2211yescharacteristic
43#137×C127C2212yescomplemented
44C4⋊Q8×C127C2211yescharacteristic, complemented
45#104×C127C2212yescomplemented
46#101×C127C2211yescharacteristic, complemented
47#14523249—22no—
48#14523249—24no—
49#14523249—22no—
50#14523249—22no—
51#14523249—22no—
52#14523249—22no—
53#14523249D412yes—
54#14523248—24no—
55#14523248—24no—
56#14523248—22no—
57#14523248—22no—
58#14523248—22no—
59#14523248C2312yes—
60#14523247—22no—
61#14523247—22no—
62#14523247D412yes—
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65Q8×D127—22no—
66Q8×D127—22no—
67Q8×D127—22no—
68Q8×D127—22no—
69Q8×D127D412yes—
70C2×D508—24no—
71C2×D508—22no—
72C2×D508D412yes—
73C2×D508C2311yescharacteristic
74D4×D127—24no—
75D4×D127—22no—
76D4×D127—24no—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127D412yes—
80C2×C4×D127—24no—
81C2×C4×D127—22no—
82C2×C4×D127—21no—
83C2×C4×D127D412yes—
84C2×C4×D127C2311yescharacteristic
85Dic508—22no—
86#14523238—22no—
87#14523238—22no—
88#14523237—22no—
89#14523236—22no—
90#14523235—22no—
91#14523232C2312yes—
92#14523231—22no—
93C8×D127—22no—
94Dic127⋊C4—21no—
95Dic127⋊C4—22no—
96(C2×D127)⋊C4—22no—
97(C2×D127)⋊C4—22no—
98(C2×D127)⋊C4—21no—
99C4×Dic127C2311yescharacteristic
100#41×C127—22no—
101#41×C127—22no—
102#41×C127C2312yes—
103C2×Q8×C127—22no—
104C2×Q8×C127C2311yescharacteristic
105C2×Q8×C127C2311yescharacteristic
106Q16×C127—22no—
107SD16×C127—22no—
108M16×C127C2312yes—
109C4⋊C4×C127—22no—
110C42×C127C2311yescharacteristic
111C22×D127—24no—
112C22×D127—22no—
113C22×D127C2×D412yes—
114C22×D127C2×D411yescharacteristic
115Dic254—22no—
116Dic254—22no—
117Dic254—22no—
118Dic254—41no—
119Dic254—41no—
120Dic254C2×D412yes—
121D508—24no—
122D508—22no—
123D508—22no—
124D508—24no—
125D508—22no—
126D508C2×D412yes—
127C22⋊D127—24no—
128C22⋊D127—22no—
129C22⋊D127—22no—
130C22⋊D127—22no—
131C2×Dic127—22no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127C2×D411yescharacteristic
135C4×D127—24no—
136C4×D127—24no—
137C4×D127—22no—
138C4×D127—22no—
139C4×D127—22no—
140C4×D127—22no—
141C4×D127—22no—
142C4×D127—42no—
143C4×D127C2×D412yes—
144C4×D127C2×D412yes—
145C127⋊C8—22no—
146Q8×C127—22no—
147Q8×C127—22no—
148Q8×C127—22no—
149Q8×C127C2×D412yes—
150D4×C127—22no—
151D4×C127—22no—
152D4×C127C2×D412yes—
153C2×C4×C127—24no—
154C2×C4×C127—22no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127C2×D412yes—
158C2×C4×C127C2411yescharacteristic
159C8×C127—22no—
160C22×C127—22no—
161C22×C127C22×D411yescharacteristic
162C2×D127—22no—
163C2×D127—22no—
164C2×D127—41no—
165C2×D127—42no—
166C2×D127—82no—
167C2×D127—84no—
168C2×D127#12012yes—
169C4×C127—22no—
170C4×C127—22no—
171C4×C127—41no—
172C4×C127—41no—
173C4×C127C22×D412yes—
174Dic127—22no—
175Dic127—41no—
176Dic127—41no—
177Dic127#12011yescharacteristic
178Dic127#12011yescharacteristic
179C2×C127—21no—
180C2×C127—42no—
181C2×C127C2×#12011yescharacteristic
182D127—22no—
183D127—42no—
184D127—42no—
185#3022—1271nomaximal, Sylow
186C127#302211yescharacteristic, complemented, Sylow
187#585—1271no—
188C2×#137—1272no—
189#536—1271no—
190#456—1271no—
191#456—1271no—
192#456—1271no—
193#456D12711yescharacteristic, complemented
194#455—1272no—
195#455—1272no—
196#421—1272no—
197#413—1271no—
1982−1+4—1271no—
1992−1+4—2542no—
2002−1+4C2×D12711yescharacteristic
201H(5,𝔽2)—1272no—
202H(5,𝔽2)—2542no—
203H(5,𝔽2)—2542no—
204C2×#41—1271no—
205C2×#41—1272no—
206C2×#41—2542no—
207C2×#41—2542no—
208C2×#41—2542no—
209C22×Q8—1271no—
210#137—1272no—
211#137—1272no—
212#137—1272no—
213#137—2542no—
214#137—2542no—
215#137C2×D12712yescomplemented
216C2×Q16—2542no—
217C2×SD16—2542no—
218C2×M16—1272no—
219C4⋊Q8—1271no—
220C4⋊Q8C2×D12711yescharacteristic, complemented
221#124—1271no—
222#124—1271no—
223#123—2542no—
224C22⋊Q8—2542no—
225#117—1271no—
226#104—1272no—
227#104—1272no—
228#104—1272no—
229#104C2×D12712yescomplemented
230#101—1271no—
231#101C2×D12711yescharacteristic, complemented
232#100—1272no—
233#41—22no—
234#41—22no—
235#41—1272no—
236#41—1272no—
237#41—1272no—
238#41—2542no—
239#41—2542no—
240#41—2544no—
241#41—2544no—
242#41—2542no—
243#41—2542no—
244#41—2542no—
245#41—2542no—
246#41—2542no—
247#41—2542no—
248#41—2544no—
249#41—2542no—
250#41—2542no—
251#41C22×D12712yes—
252C2×Q8—22no—
253C2×Q8—1272no—
254C2×Q8—1271no—
255C2×Q8—2541no—
256C2×Q8—2542no—
257C2×Q8—2542no—
258C2×Q8—2542no—
259C2×Q8—2542no—
260C2×Q8C22×D12711yescharacteristic
261C2×Q8C22×D12711yescharacteristic
262C2×D4—1272no—
263C2×D4—1272no—
264C2×D4—1271no—
265C2×D4—2542no—
266C2×D4—2544no—
267C2×D4—2542no—
268C2×D4—2544no—
269C2×D4—2542no—
270C2×D4—2544no—
271C2×D4—2542no—
272C22×C4—1271no—
273C22×C4—1272no—
274C22×C4—2542no—
275C22×C4—2541no—
276C22×C4—2544no—
277Q16—22no—
278Q16—2542no—
279Q16—2542no—
280Q16—2542no—
281SD16—22no—
282SD16—2542no—
283SD16—2542no—
284SD16—2542no—
285M16—1272no—
286M16—2542no—
287M16C22×D12712yes—
288C2×C8—2542no—
289C4⋊C4—22no—
290C4⋊C4—2541no—
291C4⋊C4—2542no—
292C22⋊C4—2541no—
293C22⋊C4—2542no—
294C22⋊C4—2542no—
295C42—1271no—
296C42C22×D12711yescharacteristic
297C23—1271no—
298C23—1272no—
299C23—2542no—
300C23—2544no—
301Q8—22no—
302Q8—22no—
303Q8—22no—
304Q8—1272no—
305Q8—5081no—
306Q8—5081no—
307Q8—2542no—
308Q8—2542no—
309Q8—2542no—
310Q8D4×D12712yes—
311D4—22no—
312D4—22no—
313D4—1272no—
314D4—2544no—
315D4—2542no—
316D4—2544no—
317D4—2542no—
318D4—2542no—
319D4—2542no—
320D4—2544no—
321D4—2542no—
322D4—2542no—
323D4D4×D12712yes—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—1271no—
329C2×C4—1272no—
330C2×C4—1272no—
331C2×C4—5082no—
332C2×C4—2542no—
333C2×C4—2542no—
334C2×C4—2542no—
335C2×C4—2542no—
336C2×C4—2544no—
337C2×C4—2544no—
338C2×C4—2542no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4—2542no—
342C2×C4D4×D12712yes—
343C2×C4C23×D12711yescharacteristic
344C8—22no—
345C8—2542no—
346C22—22no—
347C22—1272no—
348C22—10164no—
349C22—10162no—
350C22—5082no—
351C22—5081no—
352C22—2542no—
353C22—2542no—
354C22C2×D4×D12711yescharacteristic
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—1271no—
360C4—1271no—
361C4—5081no—
362C4—5081no—
363C4—2542no—
364C4C2×D4×D12712yes—
365C2—21no—
366C2—42no—
367C2—5082no—
368C2—5082no—
369C2—2542no—
370C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#456×C127C2
socleC2×C127C2×#120
prime radical#22550713C2
hypercenter#456D127
hypocenterC127#3022
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D127C2semidirect
3. #22550451C2semidirect
4. #22550061C2semidirect
5. #22550059C2semidirect
6. #22550058C2semidirect
7. #22550055C2semidirect
8. #22550052C2semidirect
9. #22549891C2semidirect
10. #22549859C2semidirect
11. #456×C127C2semidirect
22. C2×Q8×D127C22semidirect
23. #16861242C22semidirect
25. #16861240C22semidirect
26. #16861239C22semidirect
30. M16×D127C22semidirect
31. #16861207C22semidirect
32. #16861198C22semidirect
33. #16861197C22semidirect
36. #16861166C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861128C22semidirect
41. #16861125C22semidirect
43. #137×C127C22semidirect
44. C4⋊Q8×C127C22semidirect
45. #104×C127C22semidirect
46. #101×C127C22semidirect
186. C127#3022semidirect
193. #456D127semidirect
215. #137C2×D127semidirect
220. C4⋊Q8C2×D127semidirect
229. #104C2×D127semidirect
231. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507132
#137×D1272
#225504512
#225500612
#225500592
#225500582
#225500552
#225500522
#225498912
#225498592
#456×C1272
#3022127