Finite Group Databasebeta

← #30650369

Subgroups — #30650369

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×#16861241C211yesmaximal, characteristic, complemented
2C2×#16861240C211yesmaximal, characteristic, complemented
3Q8×C22⋊D127C211yesmaximal, characteristic, complemented
4#22550528C211yesmaximal, characteristic
5#22550254C211yesmaximal, characteristic, complemented
6#22550133C211yesmaximal, characteristic, complemented
7#22550132C211yesmaximal, characteristic, complemented
8#22550084C211yesmaximal, characteristic, complemented
9#22550080C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550037C211yesmaximal, characteristic, complemented
12#22550033C211yesmaximal, characteristic, complemented
13#22549961C211yesmaximal, characteristic, complemented
14#22549951C211yesmaximal, characteristic, complemented
15#451×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861241—24no—
20#16861240—24no—
21C2×Dic508C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25C2×#14523237C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523235C2211yescharacteristic
28C2×#14523232C2211yescharacteristic
29C2×#14523231C2211yescharacteristic
30#16861207—21no—
31#16861207C2211yescharacteristic
32#16861207C2211yescharacteristic
33#16861194—21no—
34#16861191—21no—
35#16861189C2211yescharacteristic
36#16861188—21no—
37#16861188C2211yescharacteristic
38#16861185—21no—
39#16861185C2211yescharacteristic
40#16861182—21no—
41Q8×Dic127C2211yescharacteristic
42#16861171C2211yescharacteristic
43C4×C22⋊D127—21no—
44C4×C22⋊D127C2211yescharacteristic
45C4⋊C4×D127—21no—
46#16861140C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861133C2211yescharacteristic
50#16861132C2211yescharacteristic
51#16861131C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53C8×Dic127C2211yescharacteristic
54C22×Q8×C127C2211yescharacteristic
55C2×Q16×C127C2211yescharacteristic
56C2×Q16×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—41no—
63#14523248—22no—
64#14523248—41no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—41no—
70C2×D508—21no—
71C2×D508C2311yescharacteristic
72C2×C22⋊D127—21no—
73C2×C22⋊D127D411yescharacteristic
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127C2311yescharacteristic
79Dic508—41no—
80#14523238—22no—
81#14523238—22no—
82#14523238—22no—
83#14523237—22no—
84#14523235—22no—
85#14523235D414yescomplemented
86#14523232—22no—
87#14523231—22no—
88C2×C127⋊C8C2311yescharacteristic
89C2×C127⋊C8C2311yescharacteristic
90C4⋊Dic127—21no—
91C4⋊Dic127—21no—
92C4⋊Dic127C2311yescharacteristic
93Dic127⋊C4—21no—
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4D411yescharacteristic
100C22⋊Dic127—21no—
101C22⋊Dic127D411yescharacteristic
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4D411yescharacteristic
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108C2×Q8×C127—21no—
109C2×Q8×C127—22no—
110C2×Q8×C127—41no—
111C2×Q8×C127C2311yescharacteristic
112C2×Q8×C127C2311yescharacteristic
113C2×Q8×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116Q16×C127—22no—
117Q16×C127—41no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127—21no—
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—42no—
130D508—21no—
131D508—42no—
132D508C2×D412yes—
133C22⋊D127—22no—
134C22⋊D127—42no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127C2×D411yescharacteristic
139C2×Dic127C2×D411yescharacteristic
140C2×Dic127C2×D411yescharacteristic
141C4×D127—21no—
142C4×D127—21no—
143C4×D127—42no—
144C4×D127—42no—
145C4×D127—42no—
146C4×D127—42no—
147C127⋊C8—21no—
148C127⋊C8C2×D412yes—
149C23×C127C2×D411yescharacteristic
150Q8×C127—21no—
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127—21no—
155Q8×C127—42no—
156Q8×C127C2×D412yes—
157C2×C4×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—42no—
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—21no—
167C8×C127—21no—
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—22no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127C22×D411yescharacteristic
184C4×C127H(5,𝔽2)11yescharacteristic
185Dic127—21no—
186Dic127—21no—
187Dic127—41no—
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#470—1271no—
206#462—1271no—
207#461—1271no—
208#452—1271no—
209#452—1271no—
210#451—1271no—
211#451D12711yescharacteristic, complemented
212C4×SD16—1271no—
213#436—1271no—
214C2×#41—1271no—
215C2×#41—1271no—
216C22×Q8—1271no—
217C22×Q8C2×D12711yescharacteristic
218#137—2544no—
219#137—2544no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×Q16—1271no—
223C2×Q16—1271no—
224C2×Q16C2×D12711yescharacteristic
225C2×Q16C2×D12711yescharacteristic
226C2×SD16—1271no—
227C2×SD16—1271no—
228C2×SD16—1271no—
229C2×M16—1271no—
230C2×M16—1271no—
231C4⋊Q8—1271no—
232C4⋊Q8—1271no—
233C4⋊Q8—2541no—
234#123—2541no—
235#123—2541no—
236C22⋊Q8—1271no—
237C22⋊Q8—1271no—
238C22⋊Q8—1271no—
239C22⋊Q8—2541no—
240C22⋊Q8—2541no—
241C22⋊Q8C2×D12711yescharacteristic
242#121—2541no—
243C4×Q8—1271no—
244C4×D4—1271no—
245C4×D4—1271no—
246C4×D4—2541no—
247C2×C4⋊C4—2541no—
248#106—1271no—
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⋊C8C2×D12711yescharacteristic
258C4×C8—1271no—
259#41—5081no—
260#41—5081no—
261#41—2542no—
262#41—2542no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—1271no—
267C2×Q8—1271no—
268C2×Q8—5081no—
269C2×Q8—2541no—
270C2×Q8—2542no—
271C2×Q8C22×D12711yescharacteristic
272C2×Q8C22×D12711yescharacteristic
273C2×Q8C22×D12711yescharacteristic
274C2×D4—1271no—
275C2×D4—1271no—
276C2×D4—2541no—
277C2×D4—2541no—
278C22×C4—21no—
279C22×C4—1271no—
280C22×C4—2541no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284C22×C4C22×D12711yescharacteristic
285Q16—22no—
286Q16—41no—
287Q16—5081no—
288Q16—2542no—
289Q16—2542no—
290Q16—2542no—
291SD16—1274no—
292SD16—2542no—
293SD16—2542no—
294M16—2542no—
295M16—2542no—
296C2×C8—1271no—
297C2×C8—1271no—
298C2×C8C22×D12711yescharacteristic
299C2×C8C22×D12711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—1271no—
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⋊C4—2541no—
311C4⋊C4C22×D12711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—1271no—
314C22⋊C4—1271no—
315C22⋊C4—2541no—
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—21no—
329Q8—42no—
330Q8—5082no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8—2541no—
335Q8D4×D12712yes—
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—42no—
348C2×C4—1271no—
349C2×C4—1271no—
350C2×C4—1271no—
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—21no—
364C8—1272no—
365C8—2541no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—5081no—
370C22—5081no—
371C22—2541no—
372C22—2541no—
373C22—2541no—
374C22—2542no—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1272no—
382C4—5081no—
383C4—5081no—
384C4—5081no—
385C4—2541no—
386C4—2541no—
387C4C2×D4×D12711yescharacteristic
388C4#1686125411yescharacteristic
389C2—41no—
390C2—5082no—
391C2—5081no—
392C2#2255050811yescharacteristic
393C2#137×D12711yescharacteristic
394C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#451×C127C2
socleC22×C127C22×D4
prime radical#22550528C2
hypercenter#451D127
hypocenterC127#3276
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
2. C2×#16861240C2semidirect
3. Q8×C22⋊D127C2semidirect
5. #22550254C2semidirect
6. #22550133C2semidirect
7. #22550132C2semidirect
8. #22550084C2semidirect
9. #22550080C2semidirect
10. #22550040C2semidirect
11. #22550037C2semidirect
12. #22550033C2semidirect
13. #22549961C2semidirect
14. #22549951C2semidirect
15. #451×C127C2semidirect
85. #14523235D4semidirect
190. Dic127#137semidirect
198. C127#3276semidirect
211. #451D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×#168612402
Q8×C22⋊D1272
#225505282
#225502542
#225501332
#225501322
#225500842
#225500802
#225500402
#225500372
#225500332
#225499612
#225499512
#451×C1272
#3276127