Finite Group Databasebeta

← #30648200

Subgroups — #30648200

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C211yesmaximal, characteristic, complemented
2C2×#16861238C211yesmaximal, characteristic, complemented
3#22550425C211yesmaximal, characteristic
4C4×Q8×D127C211yesmaximal, characteristic, complemented
5#22550099C211yesmaximal, characteristic, complemented
6#22550098C211yesmaximal, characteristic, complemented
7#22550095C211yesmaximal, characteristic, complemented
8#22550085C211yesmaximal, characteristic, complemented
9#22550084C211yesmaximal, characteristic, complemented
10#22550075C211yesmaximal, characteristic, complemented
11#22550074C211yesmaximal, characteristic, complemented
12#22549898C211yesmaximal, characteristic, complemented
13#22549876C211yesmaximal, characteristic, complemented
14#22549874C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861241—24no—
20#16861238—24no—
21C2×Dic508C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523237C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×#14523231C2211yescharacteristic
29#16861207C2211yescharacteristic
30#16861205C2211yescharacteristic
31#16861197C2211yescharacteristic
32#16861182—21no—
33#16861182—21no—
34Q8×Dic127C2211yescharacteristic
35C4×Dic254—21no—
36C4×Dic254C2211yescharacteristic
37#16861174—21no—
38#16861174C2211yescharacteristic
39(C22⋊D127)⋊C4—21no—
40(C22⋊D127)⋊C4—21no—
41C4⋊C4×D127—21no—
42C4⋊C4×D127C2211yescharacteristic
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#14523247—22no—
64#14523247—22no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127D414yescomplemented
70C2×D508—21no—
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×D127D411yescharacteristic
80C2×C4×D127D411yescharacteristic
81C2×C4×D127C2311yescharacteristic
82Dic508—22no—
83#14523238—22no—
84#14523238—22no—
85#14523237—22no—
86#14523236—22no—
87#14523235—22no—
88#14523231—22no—
89#14523231—22no—
90C2×C127⋊C8C2311yescharacteristic
91C2×C127⋊C8C2311yescharacteristic
92C4⋊Dic127—21no—
93C4⋊Dic127C2311yescharacteristic
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101C4×Dic127—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×D127C2×D411yescharacteristic
122Dic254—21no—
123Dic254—24no—
124Dic254—21no—
125Dic254—21no—
126Dic254C2×D412yes—
127D508—21no—
128D508—21no—
129D508—42no—
130C22⋊D127—42no—
131C22⋊D127—42no—
132C22⋊D127—42no—
133C2×Dic127—21no—
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×D127C2×D412yes—
146C4×D127#4112yes—
147C127⋊C8—21no—
148C127⋊C8—21no—
149C23×C127—21no—
150Q8×C127—21no—
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127C2×D412yes—
154D4×C127—21no—
155D4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127C2×D411yescharacteristic
160C2×C4×C127C2×D411yescharacteristic
161C2×C4×C127C2411yescharacteristic
162C8×C127—21no—
163C8×C127—21no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127C22×D411yescharacteristic
168C2×D127—21no—
169C2×D127—21no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127#12112yes—
174C4×C127—22no—
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127C22×D411yescharacteristic
180C4×C127C2×#4111yescharacteristic
181Dic127—22no—
182Dic127—21no—
183Dic127—41no—
184Dic127—41no—
185Dic127#12112yes—
186Dic127#13712yescomplemented
187C2×C127—81no—
188C2×C127C2×#12111yescharacteristic
189C2×C127C2×#13711yescharacteristic
190C2×C127#57511yescharacteristic
191D127—22no—
192D127—81no—
193#3037—1271nomaximal, Sylow
194C127#303711yescharacteristic, complemented, Sylow
195C2×#137—1271no—
196C2×#137—1271no—
197#532—1271no—
198C2×C4×Q8—1271no—
199#464—1271no—
200#464—1271no—
201#464—1271no—
202#464D12711yescharacteristic, complemented
203#462—1271no—
204#462—1271no—
205#460—1271no—
206#460—1271no—
207#423—1271no—
208#417—1271no—
209#417—1271no—
210C2×#41—1271no—
211C2×#41—1271no—
212C22×Q8—1271no—
213#137—2544no—
214#137—2544no—
215C2×Q16—1271no—
216C2×Q16—1271no—
217C2×Q16—1271no—
218C2×Q16C2×D12711yescharacteristic
219C2×SD16—1271no—
220C2×SD16—1271no—
221C2×SD16—1271no—
222C2×SD16C2×D12711yescharacteristic
223C2×M16—1271no—
224C2×M16—1271no—
225C4⋊Q8—1271no—
226#127—1271no—
227#124—1271no—
228#124C2×D12711yescharacteristic
229#121—2541no—
230#121—2541no—
231C4×Q8—1271no—
232C4×Q8—1271no—
233C4×Q8—1271no—
234C4×Q8—2541no—
235C4×Q8—2541no—
236C4×Q8C2×D12711yescharacteristic
237C4×D4—2541no—
238C4×D4—2541no—
239C2×C4⋊C4—1271no—
240C2×C4⋊C4—2541no—
241C2×C42—1271no—
242C2×C42—2541no—
243C4⋊C8—1271no—
244C4⋊C8—1271no—
245C4⋊C8—1271no—
246C4⋊C8C2×D12711yescharacteristic
247#103—1271no—
248#103—1271no—
249#103—1271no—
250#103C2×D12711yescharacteristic
251#102—1271no—
252#102—1271no—
253#102—1271no—
254#102C2×D12711yescharacteristic
255#41—2542no—
256#41—2542no—
257#41—2542no—
258#41—2542no—
259C2×Q8—1274no—
260C2×Q8—1271no—
261C2×Q8—1271no—
262C2×Q8—2541no—
263C2×Q8—2542no—
264C2×Q8C22×D12711yescharacteristic
265C2×Q8C22×D12711yescharacteristic
266C2×D4—1271no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4—2541no—
271C2×D4C22×D12711yescharacteristic
272C22×C4—1271no—
273C22×C4—1271no—
274C22×C4—1271no—
275C22×C4—2541no—
276C22×C4—2541no—
277C22×C4—2541no—
278C22×C4—2541no—
279Q16—22no—
280Q16—2542no—
281Q16—2542no—
282Q16—2542no—
283SD16—22no—
284SD16—2542no—
285SD16—2542no—
286SD16—2542no—
287M16—2542no—
288M16—2542no—
289C2×C8—1271no—
290C2×C8—1271no—
291C2×C8C22×D12711yescharacteristic
292C2×C8C22×D12711yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—1271no—
295C4⋊C4—2541no—
296C4⋊C4—2541no—
297C4⋊C4—2541no—
298C4⋊C4—2541no—
299C4⋊C4—2541no—
300C4⋊C4—2541no—
301C4⋊C4C22×D12711yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C42—21no—
307C42—1271no—
308C42—1271no—
309C42—1271no—
310C42—2541no—
311C42—2541no—
312C42—2541no—
313C42C22×D12711yescharacteristic
314C23—21no—
315C23—1271no—
316C23—2541no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—1272no—
321Q8—2544no—
322Q8—2541no—
323Q8—2541no—
324Q8—2541no—
325Q8D4×D12712yes—
326D4—21no—
327D4—21no—
328D4—5082no—
329D4—5082no—
330D4—5082no—
331D4—5082no—
332D4—2541no—
333D4—2541no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—1271no—
338C2×C4—1271no—
339C2×C4—1272no—
340C2×C4—1272no—
341C2×C4—1271no—
342C2×C4—5081no—
343C2×C4—5082no—
344C2×C4—5082no—
345C2×C4—5082no—
346C2×C4—2542no—
347C2×C4—2544no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4D4×D12711yescharacteristic
352C2×C4D4×D12711yescharacteristic
353C2×C4C23×D12711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2541no—
357C8—2541no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—1272no—
362C22—5081no—
363C22—5081no—
364C22—5081no—
365C22—2541no—
366C22—2541no—
367C22C2×D4×D12711yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—1272no—
374C4—1272no—
375C4—5081no—
376C4—5081no—
377C4—2542no—
378C4—2541no—
379C4C2×D4×D12711yescharacteristic
380C4#41×D12711yescharacteristic
381C2—81no—
382C2—10161no—
383C2—2542no—
384C2#121×D12711yescharacteristic
385C2#137×D12711yescharacteristic
386C2#2255067311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
prime radical#22550425C2
hypercenter#464D127
hypocenterC127#3037
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×#16861241C2semidirect
2. C2×#16861238C2semidirect
4. C4×Q8×D127C2semidirect
5. #22550099C2semidirect
6. #22550098C2semidirect
7. #22550095C2semidirect
8. #22550085C2semidirect
9. #22550084C2semidirect
10. #22550075C2semidirect
11. #22550074C2semidirect
12. #22549898C2semidirect
13. #22549876C2semidirect
14. #22549874C2semidirect
15. #464×C127C2semidirect
69. Q8×D127D4semidirect
186. Dic127#137semidirect
194. C127#3037semidirect
202. #464D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×#168612382
#225504252
C4×Q8×D1272
#225500992
#225500982
#225500952
#225500852
#225500842
#225500752
#225500742
#225498982
#225498762
#225498742
#464×C1272
#3037127