Finite Group Databasebeta

← #30647944

Subgroups — #30647944

Order 16256.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D127C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3#22550383C211yesmaximal, characteristic, complemented
4#22550037C211yesmaximal, characteristic, complemented
5#22550035C211yesmaximal, characteristic, complemented
6#22550030C211yesmaximal, characteristic, complemented
7#22550025C211yesmaximal, characteristic, complemented
8#22550020C211yesmaximal, characteristic, complemented
9#22550018C211yesmaximal, characteristic, complemented
10#22550013C211yesmaximal, characteristic, complemented
11#22550008C211yesmaximal, characteristic, complemented
12#22549873C211yesmaximal, characteristic, complemented
13#22549872C211yesmaximal, characteristic, complemented
14C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
15C2×#135×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248—21no—
18C2×#14523248C2211yescharacteristic
19C2×#14523247—21no—
20C2×#14523247C2211yescharacteristic
21#41×D127—24no—
22#41×D127—24no—
23C2×Q8×D127C2211yescharacteristic
24C2×D4×D127—21no—
25C2×D4×D127C2211yescharacteristic
26C22×C4×D127—21no—
27C22×C4×D127C2211yescharacteristic
28#16861229—24no—
29C2×#14523238C2211yescharacteristic, complemented
30C2×#14523236C2211yescharacteristic, complemented
31C2×#14523235C2211yescharacteristic, complemented
32C2×#14523233C2211yescharacteristic, complemented
33C22×C127⋊C8C2211yescharacteristic, complemented
34#16861190—21no—
35#16861185C2211yescharacteristic
36#16861185C2211yescharacteristic, complemented
37#16861180C2211yescharacteristic
38#16861180C2211yescharacteristic, complemented
39#16861177—21no—
40C4×C22⋊D127—21no—
41C4×C22⋊D127—21no—
42#16861168C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861133C2211yescharacteristic, complemented
46#16861133C2211yescharacteristic, complemented
47#16861131C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53(C2×D127)⋊C8C2211yescharacteristic, complemented
54(C2×D127)⋊C8C2211yescharacteristic, complemented
55C2×#41×C127C2211yescharacteristic
56C2×#41×C127C2211yescharacteristic, complemented
57#135×C127—24no—
58C2×Q16×C127C2211yescharacteristic, complemented
59C2×SD16×C127C2211yescharacteristic, complemented
60C2×SD16×C127C2211yescharacteristic, complemented
61C2×D8×C127C2211yescharacteristic, complemented
62C22×C8×C127C2211yescharacteristic, complemented
63C23×D127—21no—
64C23×D127D411yescharacteristic
65#14523249—22no—
66#14523249—22no—
67#14523248—28no—
68#14523248—22no—
69#14523248D414yescomplemented
70#14523247—28no—
71#14523247—22no—
72#14523247—22no—
73C2×Dic254—21no—
74C2×Dic254C2311yescharacteristic
75Q8×D127—22no—
76Q8×D127—22no—
77C2×D508—21no—
78C2×D508C2311yescharacteristic
79C2×C22⋊D127—21no—
80C2×C22⋊D127—21no—
81C2×C22⋊D127—21no—
82C2×C22⋊D127—21no—
83C2×C22⋊D127—21no—
84D4×D127—28no—
85D4×D127—22no—
86D4×D127—22no—
87C22×Dic127—21no—
88C22×Dic127D411yescharacteristic
89C2×C4×D127—28no—
90C2×C4×D127—21no—
91C2×C4×D127—21no—
92C2×C4×D127—21no—
93C2×C4×D127—22no—
94C2×C4×D127—22no—
95C2×C4×D127D411yescharacteristic
96C2×C4×D127D411yescharacteristic
97C2×C4×D127C2311yescharacteristic
98C2×C4×D127C2311yescharacteristic
99#14523238—22no—
100#14523236—22no—
101#14523235—22no—
102#14523233—22no—
103C2×C127⋊C8—22no—
104C2×C127⋊C8C2311yescharacteristic
105C2×C127⋊C8C2311yescharacteristic
106C4⋊Dic127C2311yescharacteristic
107C4⋊Dic127C2311yescharacteristic
108Dic127⋊C4—21no—
109Dic127⋊C4—21no—
110C22⋊Dic127—21no—
111C22⋊Dic127—21no—
112C22⋊Dic127—21no—
113(C2×D127)⋊C4—21no—
114(C2×D127)⋊C4—21no—
115C4×Dic127—21no—
116#41×C127—22no—
117#41×C127—22no—
118#41×C127—22no—
119#41×C127D414yescomplemented
120C2×Q8×C127C2311yescharacteristic
121C2×Q8×C127C2311yescharacteristic
122C2×D4×C127—21no—
123C2×D4×C127—21no—
124C2×D4×C127C2311yescharacteristic
125C2×D4×C127C2311yescharacteristic
126C22×C4×C127—21no—
127C22×C4×C127—21no—
128C22×C4×C127C2311yescharacteristic
129Q16×C127—22no—
130SD16×C127—22no—
131SD16×C127—22no—
132D8×C127—22no—
133C2×C8×C127—22no—
134C2×C8×C127C2311yescharacteristic
135C2×C8×C127C2311yescharacteristic
136C22×D127—24no—
137C22×D127—22no—
138C22×D127—21no—
139C22×D127—22no—
140C22×D127—41no—
141C22×D127—42no—
142C22×D127—42no—
143C22×D127C2×D411yescharacteristic
144C22×D127C2×D411yescharacteristic
145Dic254—24no—
146Dic254—21no—
147Dic254C2×D412yes—
148D508—24no—
149D508—21no—
150D508C2×D412yes—
151C22⋊D127—24no—
152C22⋊D127—24no—
153C22⋊D127—24no—
154C22⋊D127—42no—
155C22⋊D127—42no—
156C2×Dic127—24no—
157C2×Dic127—22no—
158C2×Dic127—21no—
159C2×Dic127—21no—
160C2×Dic127—21no—
161C2×Dic127C2×D411yescharacteristic
162C2×Dic127C2×D411yescharacteristic
163C4×D127—24no—
164C4×D127—24no—
165C4×D127—22no—
166C4×D127—24no—
167C4×D127—22no—
168C4×D127—21no—
169C4×D127—21no—
170C4×D127—21no—
171C4×D127—21no—
172C127⋊C8—21no—
173C127⋊C8—21no—
174C23×C127—21no—
175C23×C127—21no—
176C23×C127C2×D411yescharacteristic
177Q8×C127—21no—
178Q8×C127—21no—
179Q8×C127—21no—
180Q8×C127C2×D412yes—
181D4×C127—24no—
182D4×C127—21no—
183D4×C127—21no—
184D4×C127—21no—
185D4×C127—42no—
186D4×C127C2×D412yes—
187C2×C4×C127—21no—
188C2×C4×C127—24no—
189C2×C4×C127—21no—
190C2×C4×C127—42no—
191C2×C4×C127C2×D412yes—
192C2×C4×C127C2×D411yescharacteristic
193C2×C4×C127C2×D412yes—
194C2×C4×C127C2411yescharacteristic
195C8×C127—21no—
196C8×C127—21no—
197C22×C127—21no—
198C22×C127—21no—
199C22×C127—22no—
200C22×C127—41no—
201C22×C127—41no—
202C22×C127—41no—
203C22×C127—41no—
204C22×C127—41no—
205C22×C127#12012yes—
206C22×C127C22×D411yescharacteristic
207C2×D127—21no—
208C2×D127—21no—
209C2×D127—21no—
210C2×D127—22no—
211C2×D127—21no—
212C2×D127—21no—
213C2×D127—44no—
214C2×D127—48no—
215C2×D127—41no—
216C2×D127—41no—
217C2×D127#13512yescomplemented
218C4×C127—22no—
219C4×C127—41no—
220C4×C127#12012yes—
221C4×C127C22×D411yescharacteristic
222C4×C127C22×D411yescharacteristic
223Dic127—21no—
224Dic127—22no—
225Dic127—21no—
226Dic127—41no—
227Dic127—41no—
228C2×C127—22no—
229C2×C127—42no—
230C2×C127—81no—
231C2×C127C2×#12011yescharacteristic
232C2×C127C2×#13511yescharacteristic
233C2×C127#57511yescharacteristic
234D127—22no—
235D127—41no—
236D127—42no—
237#3002—1271nomaximal, Sylow
238C127#300211yescharacteristic, complemented, Sylow
239C22×#41—1271no—
240C2×#135—1271no—
241C2×#135D12711yescharacteristic, complemented
242#525—1271no—
243#452—1271no—
244#452—1271no—
245#451—1271no—
246#450—1271no—
247#449—1271no—
248#449—1271no—
249#448—1271no—
250#447—1271no—
251#416—1271no—
252#416—1271no—
253C2×C22⋊C8—1271no—
254C2×#41—1271no—
255C2×#41—1271no—
256C2×#41—1271no—
257C2×#41—2541no—
258C2×#41—2541no—
259C2×#41—2544no—
260C2×#41—2544no—
261C2×#41C2×D12711yescharacteristic
262C2×#41C2×D12711yescharacteristic, complemented
263C22×Q8—1271no—
264C22×D4—1271no—
265C22×D4—2541no—
266C23×C4—1271no—
267C23×C4—2541no—
268#135—24no—
269#135—2544no—
270C2×Q16—1271no—
271C2×Q16C2×D12711yescharacteristic, complemented
272C2×SD16—1271no—
273C2×SD16—1271no—
274C2×SD16C2×D12711yescharacteristic, complemented
275C2×SD16C2×D12711yescharacteristic, complemented
276C2×D8—1271no—
277C2×D8C2×D12711yescharacteristic, complemented
278C22×C8—1271no—
279C22×C8C2×D12711yescharacteristic, complemented
280#123—2541no—
281C22⋊Q8—1271no—
282C22⋊Q8—1271no—
283#121—1271no—
284#121—1271no—
285#120—2541no—
286C4×D4—2541no—
287C4×D4—2541no—
288#117—1271no—
289#103—1271no—
290#103—1271no—
291#103—1271no—
292#103—1271no—
293#102—1271no—
294#102—1271no—
295#102—1271no—
296#102—1271no—
297C22⋊C8—1271no—
298C22⋊C8—1271no—
299C22⋊C8—1271no—
300C22⋊C8—1271no—
301C24—1271no—
302C24—2541no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—1274no—
307#41—2548no—
308#41—2542no—
309#41—2548no—
310#41—2542no—
311#41—2542no—
312#41—2542no—
313#41—2542no—
314#41C22⋊D12714yescomplemented
315C2×Q8—1271no—
316C2×Q8—2541no—
317C2×Q8—2542no—
318C2×Q8—2542no—
319C2×Q8C22×D12711yescharacteristic
320C2×Q8C22×D12711yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—1271no—
324C2×D4—2541no—
325C2×D4—2541no—
326C2×D4—2542no—
327C2×D4—2548no—
328C2×D4—2541no—
329C2×D4—2541no—
330C2×D4—2541no—
331C2×D4—2541no—
332C2×D4—2542no—
333C2×D4C22×D12711yescharacteristic
334C2×D4C22×D12711yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—1271no—
338C22×C4—1271no—
339C22×C4—1271no—
340C22×C4—1271no—
341C22×C4—1271no—
342C22×C4—2541no—
343C22×C4—2541no—
344C22×C4—2548no—
345C22×C4—2541no—
346C22×C4—2542no—
347C22×C4—2541no—
348C22×C4—2542no—
349C22×C4C22×D12711yescharacteristic
350Q16—22no—
351Q16—2542no—
352SD16—22no—
353SD16—22no—
354SD16—2542no—
355SD16—2542no—
356D8—22no—
357D8—2542no—
358C2×C8—22no—
359C2×C8—1271no—
360C2×C8—1271no—
361C2×C8—2542no—
362C2×C8C22×D12711yescharacteristic
363C2×C8C22×D12711yescharacteristic
364C4⋊C4—1271no—
365C4⋊C4—1271no—
366C4⋊C4—2541no—
367C4⋊C4—2541no—
368C22⋊C4—2541no—
369C22⋊C4—2541no—
370C22⋊C4—2541no—
371C22⋊C4—2541no—
372C22⋊C4—2541no—
373C42—2541no—
374C23—21no—
375C23—21no—
376C23—1271no—
377C23—1271no—
378C23—5081no—
379C23—5082no—
380C23—5082no—
381C23—2542no—
382C23—2544no—
383C23—2541no—
384C23—2542no—
385C23D4×D12711yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—1272no—
390Q8—2544no—
391Q8—2541no—
392Q8C2×C22⋊D12712yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—1272no—
399D4—5082no—
400D4—5082no—
401D4—2544no—
402D4—2544no—
403D4—2544no—
404D4—2544no—
405D4—2541no—
406D4C2×C22⋊D12712yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—1271no—
412C2×C4—1271no—
413C2×C4—2544no—
414C2×C4—2544no—
415C2×C4—2541no—
416C2×C4—2541no—
417C2×C4—2544no—
418C2×C4—2541no—
419C2×C4—2542no—
420C2×C4—2541no—
421C2×C4—2541no—
422C2×C4—2541no—
423C2×C4—2541no—
424C2×C4—2542no—
425C2×C4—2542no—
426C2×C4—2544no—
427C2×C4D4×D12712yes—
428C2×C4D4×D12711yescharacteristic
429C2×C4C2×C22⋊D12712yes—
430C2×C4C23×D12711yescharacteristic
431C8—21no—
432C8—21no—
433C8—2541no—
434C8—2541no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—1272no—
444C22—5084no—
445C22—5088no—
446C22—5081no—
447C22—5081no—
448C22—2542no—
449C22—2541no—
450C22—2541no—
451C22—2541no—
452C22—2541no—
453C22—2541no—
454C22#1686117712yes—
455C22C2×D4×D12711yescharacteristic
456C4—22no—
457C4—41no—
458C4—5081no—
459C4—5081no—
460C4—2541no—
461C4—2541no—
462C4—2542no—
463C4#1686117712yes—
464C4C2×D4×D12711yescharacteristic
465C4C22×C22⋊D12711yescharacteristic
466C2—22no—
467C2—42no—
468C2—81no—
469C2—5082no—
470C2—5081no—
471C2—2542no—
472C2C2×#1686117711yescharacteristic
473C2#135×D12711yescharacteristic
474C2#2255067311yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#135×C127C2
socleC22×C127C22×D4
hypercenterC2×#135D127
hypocenterC127#3002
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#41×D127C2semidirect
2. C2×#16861229C2semidirect
3. #22550383C2semidirect
4. #22550037C2semidirect
5. #22550035C2semidirect
6. #22550030C2semidirect
7. #22550025C2semidirect
8. #22550020C2semidirect
9. #22550018C2semidirect
10. #22550013C2semidirect
11. #22550008C2semidirect
12. #22549873C2semidirect
13. #22549872C2semidirect
14. C2×(C2×D127)⋊C8C2semidirect
15. C2×#135×C127C2semidirect
29. C2×#14523238C22semidirect
30. C2×#14523236C22semidirect
31. C2×#14523235C22semidirect
32. C2×#14523233C22semidirect
33. C22×C127⋊C8C22semidirect
36. #16861185C22semidirect
38. #16861180C22semidirect
42. #16861168C22semidirect
43. #16861134C22semidirect
44. #16861134C22semidirect
45. #16861133C22semidirect
46. #16861133C22semidirect
47. #16861131C22semidirect
48. #16861131C22semidirect
49. #16861130C22semidirect
50. #16861130C22semidirect
53. (C2×D127)⋊C8C22semidirect
54. (C2×D127)⋊C8C22semidirect
56. C2×#41×C127C22semidirect
58. C2×Q16×C127C22semidirect
59. C2×SD16×C127C22semidirect
60. C2×SD16×C127C22semidirect
61. C2×D8×C127C22semidirect
62. C22×C8×C127C22semidirect
69. #14523248D4semidirect
119. #41×C127D4semidirect
217. C2×D127#135semidirect
238. C127#3002semidirect
241. C2×#135D127semidirect
262. C2×#41C2×D127semidirect
271. C2×Q16C2×D127semidirect
274. C2×SD16C2×D127semidirect
275. C2×SD16C2×D127semidirect
277. C2×D8C2×D127semidirect
279. C22×C8C2×D127semidirect
314. #41C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1272
C2×#168612292
#225503832
#225500372
#225500352
#225500302
#225500252
#225500202
#225500182
#225500132
#225500082
#225498732
#225498722
C2×(C2×D127)⋊C82
C2×#135×C1272
#3002127