Finite Group Databasebeta

← #34080594

Subgroups — #34080594

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
2C22×#14558989C211yesmaximal, characteristic, complemented
3#23752853C211yesmaximal, characteristic, complemented
4D4×C22⋊D151C211yesmaximal, characteristic, complemented
5#23752598C211yesmaximal, characteristic, complemented
6#23752464C211yesmaximal, characteristic, complemented
7#23752443C211yesmaximal, characteristic, complemented
8#23752429C211yesmaximal, characteristic, complemented
9#23752406C211yesmaximal, characteristic, complemented
10#23752356C211yesmaximal, characteristic, complemented
11#23752348C211yesmaximal, characteristic, complemented
12#23752347C211yesmaximal, characteristic, complemented
13#23752321C211yesmaximal, characteristic, complemented
14#23752280C211yesmaximal, characteristic, complemented
15#450×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic, complemented
17C22×D604C2211yescharacteristic, complemented
18C22×C22⋊D151—21no—
19C2×D4×D151C2211yescharacteristic, complemented
20#18012766—24no—
21C2×#14558991C2211yescharacteristic, complemented
22C2×D1208C2211yescharacteristic, complemented
23C2×D1208C2211yescharacteristic, complemented
24C2×#14558989—24no—
25C2×#14558989C2211yescharacteristic
26C2×#14558989C2211yescharacteristic
27C2×#14558989C2211yescharacteristic, complemented
28C2×#14558989C2211yescharacteristic, complemented
29C2×#14558987C2211yescharacteristic, complemented
30C22×C151⋊C8C2211yescharacteristic, complemented
31#18012735C2211yescharacteristic, complemented
32#18012728C2211yescharacteristic, complemented
33#18012724—21no—
34#18012714—21no—
35#18012714C2211yescharacteristic, complemented
36#18012713C2211yescharacteristic
37#18012711—21no—
38#18012710C2211yescharacteristic, complemented
39#18012709—21no—
40#18012708—21no—
41#18012707—21no—
42D4×Dic151C2211yescharacteristic, complemented
43#18012701C2211yescharacteristic, complemented
44C4×C22⋊D151C2211yescharacteristic
45C22⋊C4×D151—21no—
46#18012670C2211yescharacteristic, complemented
47Dic151⋊C8C2211yescharacteristic, complemented
48#18012663C2211yescharacteristic, complemented
49#18012661C2211yescharacteristic, complemented
50#18012661C2211yescharacteristic, complemented
51#18012659C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53#18012649C2211yescharacteristic, complemented
54C22×D4×C151C2211yescharacteristic
55C2×SD16×C151C2211yescharacteristic, complemented
56C2×SD16×C151C2211yescharacteristic, complemented
57C22⋊Q8×C151C2211yescharacteristic, complemented
58#102×C151C2211yescharacteristic, complemented
59#102×C151C2211yescharacteristic, complemented
60C22⋊C8×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63#14559005—22no—
64#14559005—41no—
65C2×D604—22no—
66C2×D604—21no—
67C2×D604—21no—
68C2×D604—21no—
69C2×D604—41no—
70C2×D604C2311yescharacteristic
71C2×D604C2311yescharacteristic
72C2×D604C2311yescharacteristic
73C2×C22⋊D151—21no—
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76C2×C22⋊D151—21no—
77C2×C22⋊D151—21no—
78C2×C22⋊D151—44no—
79C2×C22⋊D151D411yescharacteristic
80D4×D151—22no—
81D4×D151—41no—
82C22×Dic151—21no—
83C2×C4×D151—21no—
84C2×C4×D151—21no—
85C2×C4×D151C2311yescharacteristic
86#14558991—22no—
87D1208—22no—
88D1208—41no—
89#14558989—22no—
90#14558989—22no—
91#14558989—22no—
92#14558989D414yescomplemented
93#14558987—22no—
94C2×C151⋊C8—22no—
95C2×C151⋊C8C2311yescharacteristic
96C2×C151⋊C8C2311yescharacteristic
97C4⋊Dic151C2311yescharacteristic
98Dic151⋊C4D411yescharacteristic
99C22⋊Dic151—21no—
100C22⋊Dic151—21no—
101C22⋊Dic151D411yescharacteristic
102(C2×D151)⋊C4—21no—
103(C2×D151)⋊C4—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4D411yescharacteristic
108C4×Dic151C2311yescharacteristic
109C24×C151—21no—
110C2×Q8×C151C2311yescharacteristic
111C2×D4×C151—22no—
112C2×D4×C151—21no—
113C2×D4×C151—41no—
114C2×D4×C151C2311yescharacteristic
115C2×D4×C151C2311yescharacteristic
116C22×C4×C151C2311yescharacteristic
117SD16×C151—22no—
118SD16×C151—41no—
119C2×C8×C151C2311yescharacteristic
120C2×C8×C151C2311yescharacteristic
121C4⋊C4×C151—21no—
122C4⋊C4×C151C2311yescharacteristic
123C22⋊C4×C151—21no—
124C22×D151—21no—
125C22×D151—21no—
126C22×D151—21no—
127C22×D151—42no—
128C22×D151—42no—
129C22×D151—42no—
130C22×D151—42no—
131C22×D151—81no—
132C22×D151—81no—
133C22×D151C2×D411yescharacteristic
134D604—21no—
135D604—21no—
136D604—21no—
137D604—21no—
138D604—21no—
139D604—42no—
140D604—42no—
141D604—81no—
142D604C2×D412yescomplemented
143C22⋊D151—22no—
144C22⋊D151—42no—
145C22⋊D151—42no—
146C22⋊D151—42no—
147C22⋊D151—42no—
148C22⋊D151—81no—
149C2×Dic151—21no—
150C2×Dic151—42no—
151C2×Dic151C2×D411yescharacteristic
152C2×Dic151C2×D411yescharacteristic
153C2×Dic151C2×D411yescharacteristic
154C4×D151—21no—
155C4×D151—21no—
156C4×D151—42no—
157C4×D151—81no—
158C151⋊C8—21no—
159C151⋊C8C2×D412yescomplemented
160C23×C151—21no—
161C23×C151—21no—
162C23×C151—42no—
163C23×C151—42no—
164C23×C151—81no—
165C23×C151C2×D411yescharacteristic
166Q8×C151—21no—
167Q8×C151—21no—
168D4×C151—21no—
169D4×C151—21no—
170D4×C151—21no—
171D4×C151—42no—
172D4×C151C2×D412yes—
173C2×C4×C151—21no—
174C2×C4×C151—21no—
175C2×C4×C151—21no—
176C2×C4×C151—21no—
177C2×C4×C151C2×D411yescharacteristic
178C2×C4×C151C2411yescharacteristic
179C8×C151—21no—
180C8×C151—21no—
181C22×C151—21no—
182C22×C151—21no—
183C22×C151—22no—
184C22×C151—41no—
185C22×C151—41no—
186C22×C151—41no—
187C22×C151—41no—
188C22×C151—41no—
189C22×C151—84no—
190C22×C151AGL(1,ℤ/8)12yescomplemented
191C22×C151C22×D411yescharacteristic
192C2×D151—21no—
193C2×D151—22no—
194C2×D151—21no—
195C2×D151—21no—
196C2×D151—41no—
197C2×D151—41no—
198C2×D151—41no—
199C2×D151—41no—
200C2×D151—41no—
201C2×D151—41no—
202C2×D151—41no—
203C2×D151—41no—
204C2×D151—84no—
205C2×D151—84no—
206C4×C151—21no—
207C4×C151—41no—
208C4×C151—41no—
209C4×C151C22×D411yescharacteristic
210C4×C151H(5,𝔽2)11yescharacteristic
211Dic151—21no—
212Dic151—21no—
213Dic151—21no—
214Dic151—41no—
215C2×C151—22no—
216C2×C151—42no—
217C2×C151—81no—
218C2×C151D4211yescharacteristic
219C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C151#57611yescharacteristic
221D151—41no—
222D151—42no—
223D151—81no—
224D151—81no—
225C151#326411yescharacteristic, complemented, Sylow
226#3264—1511nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1511no—
228C22×D8—1511no—
229#546—1511no—
230D42—1511no—
231#496—1511no—
232#469—1511no—
233#465—1511no—
234#463—1511no—
235#459—1511no—
236#450D15111yescharacteristic, complemented
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×D4C2×D15111yescharacteristic
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×D8—1511no—
257C2×D8—1511no—
258C2×M16—1511no—
259C22×C8—1511no—
260#127—1511no—
261#124—1511no—
262#123—3021no—
263C22⋊Q8C2×D15111yescharacteristic, complemented
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#102C2×D15111yescharacteristic, complemented
283#102C2×D15111yescharacteristic, complemented
284C22⋊C8—1511no—
285C22⋊C8C2×D15111yescharacteristic, complemented
286#97—1511no—
287C24—21no—
288C24—3021no—
289C24—3021no—
290#41—3022no—
291#41—6041no—
292C2×Q8C22×D15111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—3022no—
297C2×D4—3021no—
298C2×D4—3021no—
299C2×D4—3022no—
300C2×D4—3021no—
301C2×D4—3021no—
302C2×D4—3021no—
303C2×D4—3021no—
304C2×D4—3021no—
305C2×D4—3021no—
306C2×D4—6041no—
307C2×D4—6041no—
308C2×D4—6044no—
309C2×D4—1511no—
310C2×D4—1511no—
311C2×D4—1511no—
312C2×D4—1511no—
313C2×D4C22×D15111yescharacteristic
314C2×D4C22×D15111yescharacteristic
315C22×C4—3021no—
316C22×C4—3021no—
317C22×C4—3021no—
318C22×C4—1511no—
319C22×C4C22×D15111yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—3022no—
323D8—3022no—
324D8—3022no—
325D8—3022no—
326D8—3022no—
327D8—6041no—
328D8—1514no—
329M16—3022no—
330C2×C8—3022no—
331C2×C8—1511no—
332C2×C8—1511no—
333C2×C8C22×D15111yescharacteristic
334C2×C8C22×D15111yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—1511no—
337C4⋊C4—1511no—
338C4⋊C4C22×D15111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—3021no—
341C22⋊C4—3021no—
342C22⋊C4—3021no—
343C22⋊C4—3021no—
344C22⋊C4—3021no—
345C22⋊C4—3021no—
346C22⋊C4—3021no—
347C22⋊C4—1511no—
348C22⋊C4—1511no—
349C42—1511no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—3021no—
356C23—3021no—
357C23—3021no—
358C23—6042no—
359C23—6042no—
360C23—6042no—
361C23—6042no—
362C23—1511no—
363C23—12081no—
364C23—12081no—
365C23D4×D15111yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—3022no—
373D4—3021no—
374D4—3021no—
375D4—3021no—
376D4—3021no—
377D4—3021no—
378D4—6042no—
379D4—6042no—
380D4—6042no—
381D4—6042no—
382D4—6042no—
383D4—6042no—
384D4—1512no—
385D4—12081no—
386D4—12081no—
387D4D4×D15112yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—3021no—
393C2×C4—3021no—
394C2×C4—3021no—
395C2×C4—6042no—
396C2×C4—6042no—
397C2×C4—1511no—
398C2×C4—1511no—
399C2×C4—1511no—
400C2×C4—12081no—
401C2×C4D4×D15111yescharacteristic
402C2×C4C23×D15111yescharacteristic
403C8—21no—
404C8—21no—
405C8—3021no—
406C8—1512no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—3021no—
417C22—3021no—
418C22—3021no—
419C22—3022no—
420C22—6041no—
421C22—6041no—
422C22—6041no—
423C22—6041no—
424C22—6041no—
425C22—6041no—
426C22—6041no—
427C22—6041no—
428C22—12084no—
429C22—12084no—
430C22#1801276612yescomplemented
431C22C2×D4×D15111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3021no—
436C4—3021no—
437C4—3021no—
438C4—6041no—
439C4C2×D4×D15111yescharacteristic
440C4#1801278411yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—6041no—
445C2—6042no—
446C2—12081no—
447C2—12081no—
448C2#2375284411yescharacteristic
449C2C2×#1801276611yescharacteristic
450C2#2375301611yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#450×C151C2
socleC22×C151C22×D4
hypercenter#450D151
hypocenterC151#3264
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#18012766C2semidirect
2. C22×#14558989C2semidirect
3. #23752853C2semidirect
4. D4×C22⋊D151C2semidirect
5. #23752598C2semidirect
6. #23752464C2semidirect
7. #23752443C2semidirect
8. #23752429C2semidirect
9. #23752406C2semidirect
10. #23752356C2semidirect
11. #23752348C2semidirect
12. #23752347C2semidirect
13. #23752321C2semidirect
14. #23752280C2semidirect
15. #450×C151C2semidirect
16. C2×#14559005C22semidirect
17. C22×D604C22semidirect
19. C2×D4×D151C22semidirect
21. C2×#14558991C22semidirect
22. C2×D1208C22semidirect
23. C2×D1208C22semidirect
27. C2×#14558989C22semidirect
28. C2×#14558989C22semidirect
29. C2×#14558987C22semidirect
30. C22×C151⋊C8C22semidirect
31. #18012735C22semidirect
32. #18012728C22semidirect
35. #18012714C22semidirect
38. #18012710C22semidirect
42. D4×Dic151C22semidirect
43. #18012701C22semidirect
46. #18012670C22semidirect
47. Dic151⋊C8C22semidirect
48. #18012663C22semidirect
49. #18012661C22semidirect
50. #18012661C22semidirect
51. #18012659C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. #18012649C22semidirect
55. C2×SD16×C151C22semidirect
56. C2×SD16×C151C22semidirect
57. C22⋊Q8×C151C22semidirect
58. #102×C151C22semidirect
59. #102×C151C22semidirect
60. C22⋊C8×C151C22semidirect
92. #14558989D4semidirect
142. D604C2×D4semidirect
159. C151⋊C8C2×D4semidirect
190. C22×C151AGL(1,ℤ/8)semidirect
225. C151#3264semidirect
236. #450D151semidirect
249. C2×SD16C2×D151semidirect
250. C2×SD16C2×D151semidirect
263. C22⋊Q8C2×D151semidirect
282. #102C2×D151semidirect
283. #102C2×D151semidirect
285. C22⋊C8C2×D151semidirect
430. C22#18012766semidirect

Maximal subgroups

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

subgroupindex
C2×#180127662
C22×#145589892
#237528532
D4×C22⋊D1512
#237525982
#237524642
#237524432
#237524292
#237524062
#237523562
#237523482
#237523472
#237523212
#237522802
#450×C1512
#3264151