Finite Group Databasebeta

← #34080589

Subgroups — #34080589

Order 19328.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012766C211yesmaximal, characteristic, complemented
2C2×D8×D151C211yesmaximal, characteristic, complemented
3#23752855C211yesmaximal, characteristic, complemented
4#23752847C211yesmaximal, characteristic, complemented
5#23752462C211yesmaximal, characteristic, complemented
6#23752438C211yesmaximal, characteristic, complemented
7#23752424C211yesmaximal, characteristic, complemented
8#23752404C211yesmaximal, characteristic, complemented
9#23752361C211yesmaximal, characteristic, complemented
10#23752356C211yesmaximal, characteristic, complemented
11#23752344C211yesmaximal, characteristic, complemented
12#23752344C211yesmaximal, characteristic, complemented
13#23752325C211yesmaximal, characteristic, complemented
14#23752285C211yesmaximal, characteristic, complemented
15#496×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic, complemented
17C2×D4×D151—21no—
18C2×D4×D151C2211yescharacteristic
19C2×D4×D151C2211yescharacteristic, complemented
20C2×D4×D151C2211yescharacteristic, complemented
21#18012766—24no—
22C2×#14558991C2211yescharacteristic, complemented
23C2×D1208C2211yescharacteristic
24C2×#14558989C2211yescharacteristic
25C2×#14558989C2211yescharacteristic, complemented
26C2×#14558989C2211yescharacteristic, complemented
27D8×D151—24no—
28C2×#14558987C2211yescharacteristic, complemented
29C2×C8×D151C2211yescharacteristic, complemented
30#18012721—21no—
31#18012715C2211yescharacteristic, complemented
32#18012712—21no—
33#18012710—21no—
34#18012710C2211yescharacteristic
35#18012710C2211yescharacteristic, complemented
36#18012710C2211yescharacteristic, complemented
37#18012709—21no—
38#18012707—21no—
39#18012707—21no—
40C4×D604C2211yescharacteristic
41C4×D604C2211yescharacteristic, complemented
42C4×D604C2211yescharacteristic, complemented
43C22⋊C4×D151—21no—
44#18012671C2211yescharacteristic, complemented
45C4⋊(C151⋊C8)C2211yescharacteristic, complemented
46#18012663C2211yescharacteristic, complemented
47#18012661C2211yescharacteristic, complemented
48#18012661C2211yescharacteristic, complemented
49#18012660C2211yescharacteristic, complemented
50#18012660C2211yescharacteristic, complemented
51#18012660C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53(C2×D151)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C151C2211yescharacteristic, complemented
55C2×SD16×C151C2211yescharacteristic, complemented
56C2×D8×C151C2211yescharacteristic, complemented
57C2×D8×C151C2211yescharacteristic, complemented
58#127×C151C2211yescharacteristic, complemented
59#124×C151C2211yescharacteristic, complemented
60#97×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63C23×D151—21no—
64#14559005—22no—
65#14559005—41no—
66C2×D604—21no—
67C2×D604D411yescharacteristic
68C2×D604C2311yescharacteristic
69C2×D604C2311yescharacteristic
70C2×C22⋊D151—21no—
71C2×C22⋊D151—21no—
72C2×C22⋊D151—21no—
73C2×C22⋊D151—21no—
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76D4×D151—22no—
77D4×D151—22no—
78D4×D151—22no—
79D4×D151—41no—
80D4×D151—41no—
81D4×D151—41no—
82D4×D151—44no—
83C2×C4×D151—21no—
84C2×C4×D151—21no—
85C2×C4×D151—21no—
86C2×C4×D151C2311yescharacteristic
87C2×C4×D151C2311yescharacteristic
88#14558991—22no—
89D1208D414yescomplemented
90#14558989—22no—
91#14558989—22no—
92#14558989—22no—
93#14558987—22no—
94C2×C151⋊C8C2311yescharacteristic
95C2×C151⋊C8C2311yescharacteristic
96C8×D151—22no—
97C4⋊Dic151D411yescharacteristic
98C4⋊Dic151C2311yescharacteristic
99C4⋊Dic151C2311yescharacteristic
100Dic151⋊C4—21no—
101C22⋊Dic151—21no—
102C22⋊Dic151—21no—
103C22⋊Dic151—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4D411yescharacteristic
108(C2×D151)⋊C4D411yescharacteristic
109C2×Q8×C151C2311yescharacteristic
110C2×D4×C151—21no—
111C2×D4×C151—21no—
112C2×D4×C151C2311yescharacteristic
113C2×D4×C151C2311yescharacteristic
114C2×D4×C151C2311yescharacteristic
115SD16×C151—22no—
116SD16×C151—41no—
117D8×C151—22no—
118D8×C151—41no—
119C2×C8×C151C2311yescharacteristic
120C2×C8×C151C2311yescharacteristic
121C22⋊C4×C151—21no—
122C22⋊C4×C151—21no—
123C42×C151C2311yescharacteristic
124C22×D151—21no—
125C22×D151—21no—
126C22×D151—42no—
127C22×D151—42no—
128C22×D151—42no—
129C22×D151—42no—
130C22×D151—42no—
131C22×D151—42no—
132C22×D151—81no—
133C22×D151—81no—
134C22×D151—81no—
135C22×D151C2×D411yescharacteristic
136C22×D151C2×D411yescharacteristic
137D604—22no—
138D604—21no—
139D604—21no—
140D604—42no—
141D604C2×D412yes—
142D604C2×D412yescomplemented
143C22⋊D151—42no—
144C22⋊D151—42no—
145C22⋊D151—42no—
146C22⋊D151—42no—
147C22⋊D151—42no—
148C22⋊D151—81no—
149C2×Dic151—21no—
150C2×Dic151—21no—
151C2×Dic151C2×D411yescharacteristic
152C2×Dic151C2×D411yescharacteristic
153C4×D151—21no—
154C4×D151—21no—
155C4×D151—21no—
156C4×D151—21no—
157C4×D151—42no—
158C4×D151—42no—
159C4×D151—81no—
160C151⋊C8—21no—
161C151⋊C8—21no—
162C23×C151—21no—
163C23×C151—21no—
164C23×C151—21no—
165Q8×C151—21no—
166Q8×C151—21no—
167D4×C151—21no—
168D4×C151—21no—
169D4×C151—21no—
170D4×C151—21no—
171D4×C151—21no—
172D4×C151—21no—
173D4×C151—42no—
174D4×C151—81no—
175C2×C4×C151—21no—
176C2×C4×C151C2×D411yescharacteristic
177C2×C4×C151C2×D411yescharacteristic
178C2×C4×C151C2411yescharacteristic
179C8×C151—21no—
180C8×C151C2×D412yescomplemented
181C22×C151—41no—
182C22×C151—41no—
183C22×C151—41no—
184C22×C151—41no—
185C22×C151—41no—
186C22×C151—41no—
187C22×C151—41no—
188C22×C151—41no—
189C22×C151—41no—
190C22×C151C22×D411yescharacteristic
191C2×D151—21no—
192C2×D151—21no—
193C2×D151—21no—
194C2×D151—22no—
195C2×D151—22no—
196C2×D151—21no—
197C2×D151—21no—
198C2×D151—41no—
199C2×D151—41no—
200C2×D151—41no—
201C2×D151—41no—
202C2×D151—84no—
203C2×D151—84no—
204C2×D151—84no—
205C2×D151AGL(1,ℤ/8)12yescomplemented
206C4×C151—21no—
207C4×C151—21no—
208C4×C151—41no—
209C4×C151C22×D411yescharacteristic
210C4×C151H(5,𝔽2)11yescharacteristic
211Dic151—21no—
212Dic151—21no—
213Dic151—41no—
214Dic151—41no—
215C2×C151—81no—
216C2×C151—81no—
217C2×C151—81no—
218C2×C151D4211yescharacteristic
219C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C151#57611yescharacteristic
221D151—22no—
222D151—41no—
223D151—42no—
224D151—42no—
225C151#326411yescharacteristic, complemented, Sylow
226#3264—1511nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1511no—
228C22×D8—1511no—
229#546—1511no—
230D42—1511no—
231#496D15111yescharacteristic, complemented
232#469—1511no—
233#465—1511no—
234#463—1511no—
235#459—1511no—
236#450—1511no—
237#449—1511no—
238#447—1511no—
239#447—1511no—
240#442—1511no—
241#435—1511no—
242C2×#41—1511no—
243C22×D4—3021no—
244C22×D4—1511no—
245C22×D4—1511no—
246C22×D4—1511no—
247AGL(1,ℤ/8)—3024no—
248C2×SD16—1511no—
249C2×SD16C2×D15111yescharacteristic, complemented
250C2×SD16C2×D15111yescharacteristic, complemented
251C2×D8—3024no—
252C2×D8—1511no—
253C2×D8—1511no—
254C2×D8—1511no—
255C2×D8—1511no—
256C2×D8C2×D15111yescharacteristic, complemented
257C2×D8C2×D15111yescharacteristic, complemented
258C2×M16—1511no—
259C22×C8—1511no—
260#127C2×D15111yescharacteristic, complemented
261#124C2×D15111yescharacteristic, complemented
262#123—3021no—
263C22⋊Q8—1511no—
264#121—3021no—
265#121—3021no—
266#121—1511no—
267#121—1511no—
268#121—1511no—
269#120—3021no—
270#120—3021no—
271#120—3021no—
272C4×D4—1511no—
273C4×D4—1511no—
274C4×D4—1511no—
275C2×C22⋊C4—3021no—
276#106—1511no—
277C4⋊C8—1511no—
278#103—1511no—
279#102—1511no—
280#102—1511no—
281#102—1511no—
282#102—1511no—
283#102—1511no—
284C22⋊C8—1511no—
285C22⋊C8—1511no—
286#97C2×D15111yescharacteristic, complemented
287C24—3021no—
288C24—3021no—
289C24—3021no—
290#41—3022no—
291#41—6041no—
292C2×Q8C22×D15111yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3021no—
296C2×D4—3022no—
297C2×D4—3022no—
298C2×D4—3022no—
299C2×D4—3021no—
300C2×D4—3021no—
301C2×D4—3021no—
302C2×D4—3021no—
303C2×D4—3021no—
304C2×D4—3021no—
305C2×D4—6041no—
306C2×D4—6041no—
307C2×D4—6041no—
308C2×D4—6044no—
309C2×D4—1511no—
310C2×D4—1511no—
311C2×D4—1511no—
312C2×D4C22×D15111yescharacteristic
313C2×D4C22×D15111yescharacteristic
314C2×D4C22×D15111yescharacteristic
315C22×C4—3021no—
316C22×C4—3021no—
317C22×C4—3021no—
318C22×C4—1511no—
319C22×C4—1511no—
320SD16—22no—
321SD16—41no—
322SD16—3022no—
323D8—22no—
324D8—41no—
325D8—3022no—
326D8—3022no—
327D8—3022no—
328D8—1514no—
329M16—3022no—
330C2×C8—3022no—
331C2×C8—1511no—
332C2×C8—1511no—
333C2×C8C22×D15111yescharacteristic
334C2×C8C22×D15111yescharacteristic
335C4⋊C4—3021no—
336C4⋊C4—1511no—
337C4⋊C4—1511no—
338C4⋊C4—1511no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—3021no—
342C22⋊C4—3021no—
343C22⋊C4—3021no—
344C22⋊C4—3021no—
345C22⋊C4—3021no—
346C22⋊C4—3021no—
347C22⋊C4—1511no—
348C22⋊C4—1511no—
349C42C22×D15111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—3021no—
354C23—3021no—
355C23—6042no—
356C23—6042no—
357C23—6042no—
358C23—6042no—
359C23—6042no—
360C23—6042no—
361C23—1511no—
362C23—1511no—
363C23—12081no—
364C23—12081no—
365C23—12081no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—3022no—
377D4—3021no—
378D4—3021no—
379D4—6042no—
380D4—6042no—
381D4—6042no—
382D4—6042no—
383D4—6042no—
384D4—6042no—
385D4—1512no—
386D4—1512no—
387D4—12081no—
388C2×C4—21no—
389C2×C4—3021no—
390C2×C4—3021no—
391C2×C4—3021no—
392C2×C4—3021no—
393C2×C4—3021no—
394C2×C4—3021no—
395C2×C4—6042no—
396C2×C4—6042no—
397C2×C4—1511no—
398C2×C4—1511no—
399C2×C4—12081no—
400C2×C4D4×D15111yescharacteristic
401C2×C4D4×D15111yescharacteristic
402C2×C4C23×D15111yescharacteristic
403C8—21no—
404C8—3021no—
405C8—3021no—
406C8D4×D15112yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3021no—
417C22—3021no—
418C22—3021no—
419C22—3021no—
420C22—3021no—
421C22—3022no—
422C22—3022no—
423C22—6041no—
424C22—6041no—
425C22—6041no—
426C22—6041no—
427C22—1512no—
428C22—12084no—
429C22—12084no—
430C22—12084no—
431C22C2×D4×D15111yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—3021no—
436C4—3021no—
437C4—6041no—
438C4—6041no—
439C4C2×D4×D15111yescharacteristic
440C4#1801278411yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—3022no—
445C2—6041no—
446C2—6042no—
447C2—6042no—
448C2#2375284711yescharacteristic
449C2AGL(1,ℤ/8)×D15111yescharacteristic
450C2#2375301611yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#496×C151C2
socleC22×C151C22×D4
hypercenter#496D151
hypocenterC151#3264
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#18012766C2semidirect
2. C2×D8×D151C2semidirect
3. #23752855C2semidirect
4. #23752847C2semidirect
5. #23752462C2semidirect
6. #23752438C2semidirect
7. #23752424C2semidirect
8. #23752404C2semidirect
9. #23752361C2semidirect
10. #23752356C2semidirect
11. #23752344C2semidirect
12. #23752344C2semidirect
13. #23752325C2semidirect
14. #23752285C2semidirect
15. #496×C151C2semidirect
16. C2×#14559005C22semidirect
19. C2×D4×D151C22semidirect
20. C2×D4×D151C22semidirect
22. C2×#14558991C22semidirect
25. C2×#14558989C22semidirect
26. C2×#14558989C22semidirect
28. C2×#14558987C22semidirect
29. C2×C8×D151C22semidirect
31. #18012715C22semidirect
35. #18012710C22semidirect
36. #18012710C22semidirect
41. C4×D604C22semidirect
42. C4×D604C22semidirect
44. #18012671C22semidirect
45. C4⋊(C151⋊C8)C22semidirect
46. #18012663C22semidirect
47. #18012661C22semidirect
48. #18012661C22semidirect
49. #18012660C22semidirect
50. #18012660C22semidirect
51. #18012660C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. (C2×D151)⋊C8C22semidirect
54. C2×SD16×C151C22semidirect
55. C2×SD16×C151C22semidirect
56. C2×D8×C151C22semidirect
57. C2×D8×C151C22semidirect
58. #127×C151C22semidirect
59. #124×C151C22semidirect
60. #97×C151C22semidirect
89. D1208D4semidirect
142. D604C2×D4semidirect
180. C8×C151C2×D4semidirect
205. C2×D151AGL(1,ℤ/8)semidirect
225. C151#3264semidirect
231. #496D151semidirect
249. C2×SD16C2×D151semidirect
250. C2×SD16C2×D151semidirect
256. C2×D8C2×D151semidirect
257. C2×D8C2×D151semidirect
260. #127C2×D151semidirect
261. #124C2×D151semidirect
286. #97C2×D151semidirect
406. C8D4×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127662
C2×D8×D1512
#237528552
#237528472
#237524622
#237524382
#237524242
#237524042
#237523612
#237523562
#237523442
#237523252
#237522852
#496×C1512
#3264151