Finite Group Databasebeta

← #456×D127 #30648069

Subgroups — #456×D127 #30648069

Order 16256.

708 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 135 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
12−1+4×D127C211yesmaximal, characteristic
2#137×D127C212yesmaximal, complemented
3C4⋊Q8×D127C211yesmaximal, characteristic, complemented
4#22550061C211yesmaximal, characteristic, complemented
5#22550060C212yesmaximal, complemented
6#22550059C211yesmaximal, characteristic, complemented
7#22550058C211yesmaximal, characteristic, complemented
8#22550057C212yesmaximal, complemented
9#104×D127C212yesmaximal, complemented
10#101×D127C211yesmaximal, characteristic, complemented
11#456×C127C211yesmaximal, characteristic, complemented
12#16861257—22no—
13#16861257C2211yescharacteristic
14#16861256—22no—
15#16861256—22no—
16#16861256C2212yes—
17#41×D127—22no—
18#41×D127—22no—
19#41×D127C2212yes—
20C2×Q8×D127—22no—
21C2×Q8×D127C2211yescharacteristic
22C2×Q8×D127C2211yescharacteristic
23#16861242C2212yescomplemented
24#16861241—22no—
25#16861240C2212yescomplemented
26#16861239C2212yescomplemented
27#16861238—22no—
28Q16×D127—22no—
29SD16×D127—22no—
30M16×D127C2212yes—
31#16861209C2211yescharacteristic, complemented
32#16861208—22no—
33#16861207C2211yescharacteristic, complemented
34#16861207C2211yescharacteristic, complemented
35C4⋊C4×D127—22no—
36C42×D127C2211yescharacteristic
37#16861137C2212yescomplemented
38#16861136C2212yescomplemented
39#16861135C2212yescomplemented
40#16861128C2211yescharacteristic, complemented
41#16861127C2212yescomplemented
422−1+4×C127C2211yescharacteristic
43#137×C127C2212yescomplemented
44C4⋊Q8×C127C2211yescharacteristic, complemented
45#104×C127C2212yescomplemented
46#101×C127C2211yescharacteristic, complemented
47#14523249—22no—
48#14523249—22no—
49#14523249D412yes—
50#14523248—24no—
51#14523248—24no—
52#14523248—22no—
53#14523248—22no—
54#14523248—22no—
55#14523248C2312yes—
56#14523247—24no—
57#14523247—22no—
58#14523247—24no—
59#14523247—22no—
60#14523247—22no—
61#14523247D412yes—
62C2×Dic254—24no—
63C2×Dic254—22no—
64C2×Dic254—21no—
65C2×Dic254D412yes—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—24no—
70Q8×D127—22no—
71Q8×D127—22no—
72Q8×D127—22no—
73Q8×D127—22no—
74Q8×D127—22no—
75Q8×D127—22no—
76Q8×D127D412yes—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127D412yes—
80C2×C4×D127—24no—
81C2×C4×D127—21no—
82C2×C4×D127—22no—
83C2×C4×D127—21no—
84C2×C4×D127D412yes—
85C2×C4×D127C2311yescharacteristic
86Dic508—22no—
87#14523238—22no—
88#14523238—22no—
89#14523237—22no—
90#14523236—22no—
91#14523235—22no—
92#14523232C2312yes—
93#14523231—22no—
94C8×D127—22no—
95C4⋊Dic127—22no—
96Dic127⋊C4—22no—
97Dic127⋊C4—22no—
98C4×Dic127—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—22no—
112C22×D127C2×D411yescharacteristic
113Dic254—22no—
114Dic254—24no—
115Dic254—22no—
116Dic254—24no—
117Dic254—22no—
118Dic254—22no—
119Dic254—42no—
120Dic254C2×D412yes—
121D508—22no—
122D508—22no—
123D508C2×D412yes—
124C22⋊D127—24no—
125C22⋊D127—22no—
126C22⋊D127—22no—
127C22⋊D127—22no—
128C2×Dic127—24no—
129C2×Dic127—21no—
130C2×Dic127—22no—
131C2×Dic127—21no—
132C2×Dic127C2×D412yes—
133C2×Dic127C2×D411yescharacteristic
134C4×D127—24no—
135C4×D127—24no—
136C4×D127—24no—
137C4×D127—22no—
138C4×D127—22no—
139C4×D127—22no—
140C4×D127—22no—
141C4×D127—22no—
142C4×D127—41no—
143C4×D127—41no—
144C4×D127C2×D412yes—
145C4×D127C2×D412yes—
146C127⋊C8—22no—
147Q8×C127—22no—
148Q8×C127—22no—
149Q8×C127—22no—
150Q8×C127C2×D412yes—
151D4×C127—22no—
152D4×C127—22no—
153D4×C127C2×D412yes—
154C2×C4×C127—24no—
155C2×C4×C127—21no—
156C2×C4×C127—22no—
157C2×C4×C127—21no—
158C2×C4×C127C2×D412yes—
159C2×C4×C127C2411yescharacteristic
160C8×C127—22no—
161C22×C127—22no—
162C22×C127C22×D411yescharacteristic
163C2×D127—22no—
164C2×D127—22no—
165C2×D127—44no—
166C2×D127#12011yescharacteristic
167C2×D127#12011yescharacteristic
168C4×C127—22no—
169C4×C127—22no—
170C4×C127—22no—
171C4×C127—41no—
172C4×C127C22×D412yes—
173Dic127—22no—
174Dic127—22no—
175Dic127—22no—
176Dic127—41no—
177Dic127#12012yes—
178C2×C127—21no—
179C2×C127—42no—
180C2×C127C2×#12011yescharacteristic
181D127—21no—
182D127—42no—
183D127#45612yesdirect product factor
184C2×#456—1271nomaximal, Sylow
185C127C2×#45611yescharacteristic, complemented, Sylow
186C2×2−1+4—1271no—
187C2×#137—1272no—
188C2×C4⋊Q8—1271no—
189#456—1271no—
190#456—1272no—
191#456—1271no—
192#456—1272no—
193#456—1271no—
194#456D12711yescharacteristic, direct product factor
195C2×#104—1272no—
196C2×#101—1271no—
1972−1+4—1272no—
1982−1+4—1271no—
1992−1+4—2542no—
2002−1+4—2542no—
2012−1+4—2542no—
2022−1+4C2×D12711yescharacteristic
203C2×#41—1272no—
204C2×#41—2542no—
205C2×#41—2542no—
206C22×Q8—1271no—
207C22×Q8—1271no—
208C22×Q8—2542no—
209#137—1272no—
210#137—1272no—
211#137—1272no—
212#137—2542no—
213#137—2542no—
214#137C2×D12712yescomplemented
215C2×Q16—2542no—
216C2×SD16—2542no—
217C2×M16—1272no—
218C4⋊Q8—1271no—
219C4⋊Q8—1271no—
220C4⋊Q8—1271no—
221C4⋊Q8—2542no—
222C4⋊Q8C2×D12711yescharacteristic, complemented
223C2×C4⋊C4—2542no—
224C2×C42—1271no—
225#104—1272no—
226#104—1272no—
227#104—1272no—
228#104C2×D12712yescomplemented
229#101—1271no—
230#101—1272no—
231#101C2×D12711yescharacteristic, complemented
232#41—22no—
233#41—22no—
234#41—1272no—
235#41—1272no—
236#41—1272no—
237#41—2542no—
238#41—2542no—
239#41—2544no—
240#41—2544no—
241#41—2542no—
242#41—2544no—
243#41—2542no—
244#41—2544no—
245#41—2542no—
246#41—2542no—
247#41—2542no—
248#41—2542no—
249#41C22×D12712yes—
250C2×Q8—22no—
251C2×Q8—1272no—
252C2×Q8—1271no—
253C2×Q8—1272no—
254C2×Q8—1271no—
255C2×Q8—2541no—
256C2×Q8—2542no—
257C2×Q8—2544no—
258C2×Q8—2542no—
259C2×Q8—2542no—
260C2×Q8—2542no—
261C2×Q8—2542no—
262C2×Q8—2542no—
263C2×Q8—2542no—
264C2×Q8—2544no—
265C2×Q8—2542no—
266C2×Q8C22×D12711yescharacteristic
267C2×Q8C22×D12711yescharacteristic
268C2×D4—1272no—
269C2×D4—2542no—
270C2×D4—2542no—
271C22×C4—1271no—
272C22×C4—1272no—
273C22×C4—2542no—
274C22×C4—2541no—
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—2542no—
291C4⋊C4—2542no—
292C4⋊C4—2542no—
293C42—1271no—
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—1271no—
297C23—2542no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—1272no—
302Q8—5082no—
303Q8—2542no—
304Q8—2544no—
305Q8—2544no—
306Q8—2542no—
307Q8—2542no—
308Q8—2542no—
309Q8D4×D12712yes—
310D4—22no—
311D4—22no—
312D4—1272no—
313D4—2542no—
314D4—2542no—
315D4—2542no—
316D4—2542no—
317D4—2544no—
318D4—2542no—
319D4D4×D12712yes—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—1271no—
325C2×C4—1272no—
326C2×C4—1272no—
327C2×C4—1272no—
328C2×C4—5081no—
329C2×C4—5081no—
330C2×C4—2542no—
331C2×C4—2542no—
332C2×C4—2542no—
333C2×C4—2542no—
334C2×C4—2544no—
335C2×C4—2544no—
336C2×C4—2544no—
337C2×C4—2542no—
338C2×C4—2541no—
339C2×C4—2544no—
340C2×C4—2541no—
341C2×C4—2542no—
342C2×C4D4×D12712yes—
343C2×C4C23×D12711yescharacteristic
344C8—22no—
345C8—2542no—
346C22—22no—
347C22—1271no—
348C22—1271no—
349C22—5084no—
350C22—2542no—
351C22—2542no—
352C22C2×D4×D12711yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—1272no—
358C4—5081no—
359C4—2542no—
360C4—2542no—
361C4—2542no—
362C4C2×D4×D12712yes—
363C2—21no—
364C2—42no—
365C2—1272no—
366C2—5082no—
367C2—2541no—
368C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#456×C127C2
socleC2×C127C2×#120
prime radical2−1+4×D127C2
hypercenter#456D127
hypocenterC127C2×#456
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. C4⋊Q8×D127C2semidirect
4. #22550061C2semidirect
5. #22550060C2semidirect
6. #22550059C2semidirect
7. #22550058C2semidirect
8. #22550057C2semidirect
9. #104×D127C2semidirect
10. #101×D127C2semidirect
11. #456×C127C2semidirect
23. #16861242C22semidirect
25. #16861240C22semidirect
26. #16861239C22semidirect
31. #16861209C22semidirect
33. #16861207C22semidirect
34. #16861207C22semidirect
37. #16861137C22semidirect
38. #16861136C22semidirect
39. #16861135C22semidirect
40. #16861128C22semidirect
41. #16861127C22semidirect
43. #137×C127C22semidirect
44. C4⋊Q8×C127C22semidirect
45. #104×C127C22semidirect
46. #101×C127C22semidirect
183. D127#456direct (N × H)
185. C127C2×#456semidirect
194. #456D127direct (N × H)
214. #137C2×D127semidirect
222. C4⋊Q8C2×D127semidirect
228. #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
2−1+4×D1272
#137×D1272
C4⋊Q8×D1272
#225500612
#225500602
#225500592
#225500582
#225500572
#104×D1272
#101×D1272
#456×C1272
C2×#456127