Finite Group Databasebeta

← #30650241

Subgroups — #30650241

Order 16256.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861240C211yesmaximal, characteristic, complemented
2C22×#14523235C211yesmaximal, characteristic, complemented
3#22550522C211yesmaximal, characteristic
4#22550268C211yesmaximal, characteristic, complemented
5#22550124C211yesmaximal, characteristic, complemented
6#22550115C211yesmaximal, characteristic, complemented
7#22550098C211yesmaximal, characteristic, complemented
8#22550078C211yesmaximal, characteristic, complemented
9#22550036C211yesmaximal, characteristic, complemented
10#22550032C211yesmaximal, characteristic, complemented
11#22550026C211yesmaximal, characteristic, complemented
12#22550015C211yesmaximal, characteristic, complemented
13#22549974C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15D4×Q8×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×D508C2211yescharacteristic
18#16861240—24no—
19C2×#14523238C2211yescharacteristic, complemented
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523235—24no—
23C2×#14523235C2211yescharacteristic
24C2×#14523235C2211yescharacteristic
25C2×#14523235C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523235C2211yescharacteristic
28C2×#14523235C2211yescharacteristic
29C2×#14523232C2211yescharacteristic, complemented
30C22×C127⋊C8C2211yescharacteristic
31#16861197C2211yescharacteristic
32#16861194—21no—
33#16861188C2211yescharacteristic
34#16861183C2211yescharacteristic
35#16861182—21no—
36#16861177—21no—
37C4×D508C2211yescharacteristic
38C22⋊C4×D127—21no—
39#16861142C2211yescharacteristic, complemented
40C4⋊(C127⋊C8)C2211yescharacteristic, complemented
41#16861133C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861133C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic
47C22⋊(C127⋊C8)C2211yescharacteristic
48C22⋊(C127⋊C8)C2211yescharacteristic, complemented
49#16861120C2211yescharacteristic, complemented
50C22×Q8×C127C2211yescharacteristic
51C22×Q8×C127C2211yescharacteristic, complemented
52C4⋊Q8×C127—21no—
53C4⋊Q8×C127C2211yescharacteristic, complemented
54C22⋊Q8×C127—21no—
55C22⋊Q8×C127—21no—
56C22⋊Q8×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic, complemented
58C4×Q8×C127C2211yescharacteristic, complemented
59C4×D4×C127—21no—
60C4×D4×C127C2211yescharacteristic
61C23×D127—21no—
62#14523248—22no—
63#14523248—41no—
64C2×Dic254C2311yescharacteristic
65C2×D508—22no—
66C2×D508—21no—
67C2×D508—41no—
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74#14523238—22no—
75#14523238—22no—
76#14523238—41no—
77#14523235—22no—
78#14523235—22no—
79#14523235—22no—
80#14523235—22no—
81#14523235—41no—
82#14523235D414yescomplemented
83#14523232—22no—
84C2×C127⋊C8—22no—
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C2×C127⋊C8C2311yescharacteristic
88C2×C127⋊C8C2311yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96C2×Q8×C127—21no—
97C2×Q8×C127—21no—
98C2×Q8×C127—22no—
99C2×Q8×C127—22no—
100C2×Q8×C127—41no—
101C2×Q8×C127—41no—
102C2×Q8×C127C2311yescharacteristic
103C2×Q8×C127C2311yescharacteristic
104C2×Q8×C127C2311yescharacteristic
105C2×D4×C127D411yescharacteristic
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127C2311yescharacteristic
109C22×C4×C127C2311yescharacteristic
110C4⋊C4×C127—21no—
111C4⋊C4×C127—21no—
112C4⋊C4×C127—21no—
113C4⋊C4×C127—21no—
114C4⋊C4×C127—21no—
115C4⋊C4×C127D411yescharacteristic
116C4⋊C4×C127C2311yescharacteristic
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127D411yescharacteristic
120C22⋊C4×C127D411yescharacteristic
121C42×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128Dic254—21no—
129Dic254—21no—
130D508—21no—
131D508—21no—
132D508—21no—
133D508—42no—
134D508C2×D412yes—
135C22⋊D127—42no—
136C22⋊D127—81no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C4×D127—42no—
140C4×D127—81no—
141C127⋊C8—21no—
142C127⋊C8—21no—
143C127⋊C8—21no—
144C127⋊C8C2×D412yes—
145C23×C127C2×D411yescharacteristic
146C23×C127C2×D411yescharacteristic
147Q8×C127—21no—
148Q8×C127—21no—
149Q8×C127—21no—
150Q8×C127—21no—
151Q8×C127—21no—
152Q8×C127—42no—
153Q8×C127—42no—
154Q8×C127C2×D412yes—
155D4×C127—22no—
156C2×C4×C127—21no—
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—21no—
164C2×C4×C127—42no—
165C2×C4×C127—42no—
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2×D411yescharacteristic
170C2×C4×C127C2411yescharacteristic
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—21no—
176C22×C127#13712yescomplemented
177C22×C127C22×D411yescharacteristic
178C2×D127—22no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—84no—
185C4×C127—21no—
186C4×C127—21no—
187C4×C127—22no—
188C4×C127—21no—
189C4×C127—21no—
190C4×C127—41no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127C22×D411yescharacteristic
194C4×C127H(5,𝔽2)11yescharacteristic
195Dic127—41no—
196Dic127—41no—
197C2×C127—22no—
198C2×C127—41no—
199C2×C127D4211yescharacteristic
200C2×C127C2×#13711yescharacteristic
201C2×C127#57711yescharacteristic
202D127—42no—
203D127—81no—
204#3266—1271nomaximal, Sylow
205C127#326611yescharacteristic, complemented, Sylow
206C2×#137—1271no—
207C22×SD16—1271no—
208D4×Q8D12711yescharacteristic, complemented
209#546—1271no—
210#497—1271no—
211#468—1271no—
212#467—1271no—
213#464—1271no—
214#461—1271no—
215#452—1271no—
216#451—1271no—
217#450—1271no—
218#448—1271no—
219#440—1271no—
220#435—1271no—
221C2×#41—1271no—
222C22×Q8C2×D12711yescharacteristic
223C22×Q8C2×D12711yescharacteristic, complemented
224C22×D4—1271no—
225#137—2544no—
226C2×Q16—1271no—
227C2×Q16—1271no—
228C2×Q16—1271no—
229C2×SD16—1271no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—2544no—
236C2×M16—1271no—
237C22×C8—1271no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D12711yescharacteristic, complemented
240#124—1271no—
241#123—2541no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1271no—
245C22⋊Q8C2×D12711yescharacteristic
246C22⋊Q8C2×D12711yescharacteristic, complemented
247#121—1271no—
248#121—2541no—
249#120—2541no—
250C4×Q8C2×D12711yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1271no—
253C4×D4C2×D12711yescharacteristic
254C2×C22⋊C4—2541no—
255#107—1271no—
256C4⋊C8—1271no—
257#103—1271no—
258#103—1271no—
259#103—1271no—
260#103—1271no—
261#102—1271no—
262#102—1271no—
263C22⋊C8—1271no—
264C22⋊C8—1271no—
265#97—1271no—
266C24—2541no—
267#41—5081no—
268#41—2542no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1271no—
276C2×Q8C22×D12711yescharacteristic
277C2×Q8C22×D12711yescharacteristic
278C2×Q8C22×D12711yescharacteristic
279C2×D4—1271no—
280C2×D4—1271no—
281C2×D4—5081no—
282C2×D4—2542no—
283C2×D4—2541no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4C22⋊D12711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2541no—
290C22×C4—2541no—
291C22×C4C22×D12711yescharacteristic
292C22×C4C22×D12711yescharacteristic
293Q16—5081no—
294Q16—2542no—
295Q16—2542no—
296SD16—1274no—
297SD16—5081no—
298SD16—2542no—
299SD16—2542no—
300SD16—2542no—
301SD16—2542no—
302M16—2542no—
303C2×C8—1271no—
304C2×C8—1271no—
305C2×C8—1271no—
306C2×C8—1271no—
307C2×C8—2542no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—1271no—
314C4⋊C4—2541no—
315C4⋊C4C22⋊D12711yescharacteristic
316C4⋊C4C22×D12711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4C22⋊D12711yescharacteristic
325C22⋊C4C22⋊D12711yescharacteristic
326C42—21no—
327C42C22×D12711yescharacteristic
328C23—10161no—
329C23—5082no—
330C23—5082no—
331C23—2541no—
332C23—2541no—
333C23C2×C22⋊D12711yescharacteristic
334C23C2×C22⋊D12711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—2541no—
343Q8—2541no—
344Q8D4×D12712yes—
345D4—22no—
346D4—1272no—
347D4—10161no—
348D4—5082no—
349D4—5082no—
350D4—2541no—
351D4—2541no—
352D4—2541no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—10161no—
364C2×C4—5082no—
365C2×C4—2541no—
366C2×C4—2541no—
367C2×C4C2×C22⋊D12711yescharacteristic
368C2×C4C2×C22⋊D12711yescharacteristic
369C2×C4C2×C22⋊D12711yescharacteristic
370C2×C4C2×C22⋊D12711yescharacteristic
371C2×C4C23×D12711yescharacteristic
372C8—1272no—
373C8—2541no—
374C8—2541no—
375C8—2541no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—10164no—
382C22—5081no—
383C22—5081no—
384C22—5081no—
385C22—5081no—
386C22—5081no—
387C22—2542no—
388C22#1686124012yescomplemented
389C22C22×C22⋊D12711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—5081no—
399C4—5081no—
400C4C2×D4×D12711yescharacteristic
401C4#1686125411yescharacteristic
402C2—22no—
403C2—41no—
404C2—10161no—
405C2—5082no—
406C2D4×C22⋊D12711yescharacteristic
407C2C2×#1686124011yescharacteristic
408C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD4×Q8×C127C2
socleC22×C127C22×D4
prime radical#22550522C2
hypercenterD4×Q8D127
hypocenterC127#3266
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
2. C22×#14523235C2semidirect
4. #22550268C2semidirect
5. #22550124C2semidirect
6. #22550115C2semidirect
7. #22550098C2semidirect
8. #22550078C2semidirect
9. #22550036C2semidirect
10. #22550032C2semidirect
11. #22550026C2semidirect
12. #22550015C2semidirect
13. #22549974C2semidirect
14. #22549948C2semidirect
15. D4×Q8×C127C2semidirect
19. C2×#14523238C22semidirect
20. C2×#14523238C22semidirect
21. C2×#14523238C22semidirect
29. C2×#14523232C22semidirect
39. #16861142C22semidirect
40. C4⋊(C127⋊C8)C22semidirect
41. #16861133C22semidirect
42. #16861133C22semidirect
43. #16861133C22semidirect
44. #16861132C22semidirect
48. C22⋊(C127⋊C8)C22semidirect
49. #16861120C22semidirect
51. C22×Q8×C127C22semidirect
53. C4⋊Q8×C127C22semidirect
57. C22⋊Q8×C127C22semidirect
58. C4×Q8×C127C22semidirect
82. #14523235D4semidirect
176. C22×C127#137semidirect
205. C127#3266semidirect
208. D4×Q8D127semidirect
223. C22×Q8C2×D127semidirect
239. C4⋊Q8C2×D127semidirect
246. C22⋊Q8C2×D127semidirect
250. C4×Q8C2×D127semidirect
388. C22#16861240semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C22×#145232352
#225505222
#225502682
#225501242
#225501152
#225500982
#225500782
#225500362
#225500322
#225500262
#225500152
#225499742
#225499482
D4×Q8×C1272
#3266127