Finite Group Databasebeta

← #30650373

Subgroups — #30650373

Order 16256.

472 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×#16861242C211yesmaximal, characteristic, complemented
2C2×#16861241C211yesmaximal, characteristic, complemented
3#22550539C211yesmaximal, characteristic, complemented
4#22550528C211yesmaximal, characteristic
5#22550254C211yesmaximal, characteristic, complemented
6#22550135C211yesmaximal, characteristic, complemented
7#22550085C211yesmaximal, characteristic, complemented
8#22550080C211yesmaximal, characteristic, complemented
9#22550041C211yesmaximal, characteristic, complemented
10#22550036C211yesmaximal, characteristic, complemented
11#22550033C211yesmaximal, characteristic, complemented
12#22550031C211yesmaximal, characteristic, complemented
13#22549963C211yesmaximal, characteristic, complemented
14#22549952C211yesmaximal, characteristic, complemented
15#470×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C22×Dic254C2211yescharacteristic
19C2×Q8×D127C2211yescharacteristic
20#16861242—24no—
21#16861241—24no—
22C2×Dic508C2211yescharacteristic
23C2×Dic508C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25C2×#14523238C2211yescharacteristic
26C2×#14523238C2211yescharacteristic
27C2×#14523237C2211yescharacteristic
28C2×#14523237C2211yescharacteristic
29C2×#14523235C2211yescharacteristic
30C2×#14523231C2211yescharacteristic
31#16861208—21no—
32#16861207C2211yescharacteristic
33#16861207C2211yescharacteristic
34#16861194—21no—
35#16861191—21no—
36#16861189C2211yescharacteristic
37#16861189C2211yescharacteristic
38#16861187—21no—
39#16861186—21no—
40#16861182—21no—
41#16861174C2211yescharacteristic
42#16861171C2211yescharacteristic
43C4×C22⋊D127C2211yescharacteristic
44(C22⋊D127)⋊C4—21no—
45C4⋊C4×D127—21no—
46Dic127⋊C8C2211yescharacteristic
47#16861132C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861132C2211yescharacteristic
50#16861129C2211yescharacteristic
51C22⋊(C127⋊C8)C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53C8×Dic127C2211yescharacteristic
54C2×Q16×C127C2211yescharacteristic
55C2×M16×C127C2211yescharacteristic
56C22⋊Q8×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic
58#106×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#103×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—41no—
63#14523248—22no—
64#14523248—41no—
65C2×Dic254—21no—
66C2×Dic254—22no—
67C2×Dic254—21no—
68C2×Dic254—41no—
69C2×Dic254C2311yescharacteristic
70C2×Dic254C2311yescharacteristic
71C2×Dic254C2311yescharacteristic
72Q8×D127—22no—
73Q8×D127—41no—
74C2×D508—21no—
75C2×D508C2311yescharacteristic
76C2×C22⋊D127—21no—
77C2×C22⋊D127D411yescharacteristic
78C22×Dic127—21no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127—21no—
82C2×C4×D127—21no—
83C2×C4×D127C2311yescharacteristic
84Dic508—22no—
85Dic508—22no—
86#14523238—22no—
87#14523238—41no—
88#14523238—41no—
89#14523237—22no—
90#14523237D414yescomplemented
91#14523235—22no—
92#14523231—22no—
93C2×C127⋊C8C2311yescharacteristic
94C2×C127⋊C8C2311yescharacteristic
95C4⋊Dic127—21no—
96C4⋊Dic127—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101Dic127⋊C4—21no—
102Dic127⋊C4D411yescharacteristic
103C22⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108C4×Dic127—21no—
109C4×Dic127C2311yescharacteristic
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C22×C4×C127C2311yescharacteristic
113Q16×C127—22no—
114M16×C127—22no—
115C2×C8×C127C2311yescharacteristic
116C2×C8×C127C2311yescharacteristic
117C4⋊C4×C127—21no—
118C4⋊C4×C127—21no—
119C4⋊C4×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—21no—
130Dic254—42no—
131Dic254—42no—
132Dic254C2×D412yes—
133D508—21no—
134D508—42no—
135D508C2×D412yes—
136C22⋊D127—22no—
137C22⋊D127—42no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127—42no—
142C2×Dic127C2×D411yescharacteristic
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C4×D127—21no—
146C4×D127—21no—
147C4×D127—42no—
148C4×D127—42no—
149C4×D127—42no—
150C4×D127—42no—
151C127⋊C8—21no—
152C127⋊C8—21no—
153C23×C127C2×D411yescharacteristic
154Q8×C127—21no—
155Q8×C127—21no—
156Q8×C127—21no—
157Q8×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—21no—
167C8×C127C2×D412yes—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127C22×D411yescharacteristic
172C2×D127—21no—
173C2×D127—22no—
174C2×D127—21no—
175C2×D127—21no—
176C2×D127—41no—
177C2×D127—41no—
178C4×C127—21no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127C22×D411yescharacteristic
184C4×C127H(5,𝔽2)11yescharacteristic
185Dic127—21no—
186Dic127—22no—
187Dic127—21no—
188Dic127—41no—
189Dic127—41no—
190Dic127#13712yescomplemented
191C2×C127—41no—
192C2×C127D4211yescharacteristic
193C2×C127C2×#13711yescharacteristic
194C2×C127#57811yescharacteristic
195D127—41no—
196D127—42no—
197#3276—1271nomaximal, Sylow
198C127#327611yescharacteristic, complemented, Sylow
199C2×#137—1271no—
200C2×#137—1271no—
201D4×Q8—1271no—
202#547—1271no—
203#494—1271no—
204#470—1271no—
205#470D12711yescharacteristic, complemented
206#462—1271no—
207#461—1271no—
208#452—1271no—
209#452—1271no—
210#451—1271no—
211#451—1271no—
212C4×SD16—1271no—
213#436—1271no—
214C2×#41—1271no—
215C2×#41—1271no—
216C22×Q8—1271no—
217C22×Q8—1271no—
218#137—2544no—
219#137—2544no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×Q16—1271no—
223C2×Q16—1271no—
224C2×Q16—1271no—
225C2×Q16C2×D12711yescharacteristic
226C2×SD16—1271no—
227C2×SD16—1271no—
228C2×SD16—1271no—
229C2×M16—1271no—
230C2×M16C2×D12711yescharacteristic
231C4⋊Q8—1271no—
232C4⋊Q8—1271no—
233C4⋊Q8—2541no—
234#123—2541no—
235#123—2541no—
236C22⋊Q8—1271no—
237C22⋊Q8—1271no—
238C22⋊Q8—2541no—
239C22⋊Q8—2541no—
240C22⋊Q8C2×D12711yescharacteristic
241C22⋊Q8C2×D12711yescharacteristic
242#121—2541no—
243C4×Q8—1271no—
244C4×D4—1271no—
245C4×D4—1271no—
246C4×D4—2541no—
247C2×C4⋊C4—2541no—
248#106C2×D12711yescharacteristic
249C4⋊C8—1271no—
250#103—1271no—
251#103—1271no—
252#103—1271no—
253#103C2×D12711yescharacteristic
254#103C2×D12711yescharacteristic
255#102—1271no—
256C22⋊C8—1271no—
257C22⋊C8—1271no—
258C4×C8—1271no—
259#41—5081no—
260#41—5081no—
261#41—2542no—
262#41—2542no—
263C2×Q8—1271no—
264C2×Q8—1271no—
265C2×Q8—1271no—
266C2×Q8—5081no—
267C2×Q8—5081no—
268C2×Q8—2541no—
269C2×Q8—2542no—
270C2×Q8—2542no—
271C2×Q8—2541no—
272C2×Q8C22×D12711yescharacteristic
273C2×Q8C22×D12711yescharacteristic
274C2×D4—1271no—
275C2×D4—1271no—
276C2×D4—2541no—
277C2×D4—2541no—
278C22×C4—1271no—
279C22×C4—2541no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284C22×C4C22×D12711yescharacteristic
285Q16—22no—
286Q16—5081no—
287Q16—5081no—
288Q16—2542no—
289Q16—2542no—
290Q16—2542no—
291SD16—1274no—
292SD16—2542no—
293SD16—2542no—
294M16—22no—
295M16—2542no—
296C2×C8—1271no—
297C2×C8—1271no—
298C2×C8C22×D12711yescharacteristic
299C2×C8C22×D12711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—1271no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4—2541no—
308C4⋊C4—2541no—
309C4⋊C4—2541no—
310C4⋊C4C22×D12711yescharacteristic
311C4⋊C4C22×D12711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—1271no—
315C22⋊C4—1271no—
316C22⋊C4—2541no—
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C42—1271no—
320C42—2541no—
321C23—1271no—
322C23—2541no—
323C23D4×D12711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—1272no—
329Q8—5082no—
330Q8—5082no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8—2541no—
335Q8—2541no—
336D4—1272no—
337D4—5082no—
338D4—5082no—
339D4—2542no—
340D4—2541no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—1271no—
348C2×C4—1271no—
349C2×C4—1271no—
350C2×C4—5082no—
351C2×C4—5082no—
352C2×C4—5082no—
353C2×C4—5082no—
354C2×C4—5082no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4D4×D12711yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—21no—
363C8—2541no—
364C8—2541no—
365C8D4×D12712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—5081no—
370C22—5081no—
371C22—2541no—
372C22—2541no—
373C22—2541no—
374C22—2542no—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1272no—
382C4—5081no—
383C4—5081no—
384C4—2541no—
385C4—2541no—
386C4—2542no—
387C4C2×D4×D12711yescharacteristic
388C4#1686125511yescharacteristic
389C2—41no—
390C2—5082no—
391C2—5081no—
392C2#2255051011yescharacteristic
393C2#137×D12711yescharacteristic
394C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#470×C127C2
socleC22×C127C22×D4
prime radical#22550528C2
hypercenter#470D127
hypocenterC127#3276
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C2×#16861241C2semidirect
3. #22550539C2semidirect
5. #22550254C2semidirect
6. #22550135C2semidirect
7. #22550085C2semidirect
8. #22550080C2semidirect
9. #22550041C2semidirect
10. #22550036C2semidirect
11. #22550033C2semidirect
12. #22550031C2semidirect
13. #22549963C2semidirect
14. #22549952C2semidirect
15. #470×C127C2semidirect
90. #14523237D4semidirect
190. Dic127#137semidirect
198. C127#3276semidirect
205. #470D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612412
#225505392
#225505282
#225502542
#225501352
#225500852
#225500802
#225500412
#225500362
#225500332
#225500312
#225499632
#225499522
#470×C1272
#3276127