Finite Group Databasebeta

← #34080597

Subgroups — #34080597

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×#18012767C211yesmaximal, characteristic, complemented
2C2×D8×D151C211yesmaximal, characteristic, complemented
3#23752859C211yesmaximal, characteristic, complemented
4#23752846C211yesmaximal, characteristic, complemented
5#23752600C211yesmaximal, characteristic, complemented
6#23752442C211yesmaximal, characteristic, complemented
7#23752426C211yesmaximal, characteristic, complemented
8#23752406C211yesmaximal, characteristic, complemented
9#23752364C211yesmaximal, characteristic, complemented
10#23752355C211yesmaximal, characteristic, complemented
11#23752347C211yesmaximal, characteristic, complemented
12#23752345C211yesmaximal, characteristic, complemented
13#23752323C211yesmaximal, characteristic, complemented
14#23752281C211yesmaximal, characteristic, complemented
15#469×C151C211yesmaximal, characteristic, complemented
16C2×#14559004C2211yescharacteristic, complemented
17C22×D604C2211yescharacteristic, complemented
18C2×D4×D151—21no—
19C2×D4×D151C2211yescharacteristic
20C2×D4×D151C2211yescharacteristic, complemented
21#18012767—24no—
22C2×#14558993C2211yescharacteristic, complemented
23C2×#14558993C2211yescharacteristic, complemented
24C2×#14558992C2211yescharacteristic, complemented
25C2×D1208C2211yescharacteristic
26C2×D1208C2211yescharacteristic, complemented
27C2×#14558989C2211yescharacteristic
28C2×#14558989C2211yescharacteristic, complemented
29C2×#14558989C2211yescharacteristic, complemented
30D8×D151—24no—
31C2×C8×D151C2211yescharacteristic, complemented
32#18012735C2211yescharacteristic, complemented
33#18012726C2211yescharacteristic, complemented
34#18012721—21no—
35#18012719C2211yescharacteristic, complemented
36#18012714C2211yescharacteristic
37#18012712—21no—
38#18012710—21no—
39#18012709—21no—
40#18012707—21no—
41#18012707—21no—
42#18012701C2211yescharacteristic, complemented
43#18012701C2211yescharacteristic, complemented
44C4×C22⋊D151C2211yescharacteristic
45C22⋊C4×D151—21no—
46Dic151⋊C8C2211yescharacteristic, complemented
47#18012662C2211yescharacteristic, complemented
48#18012659C2211yescharacteristic, complemented
49#18012659C2211yescharacteristic, complemented
50#18012659C2211yescharacteristic, complemented
51C22⋊(C151⋊C8)C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53#18012649C2211yescharacteristic, complemented
54C2×D8×C151C2211yescharacteristic, complemented
55C2×M16×C151C2211yescharacteristic, complemented
56#121×C151C2211yescharacteristic, complemented
57#121×C151C2211yescharacteristic, complemented
58#106×C151C2211yescharacteristic, complemented
59#102×C151C2211yescharacteristic, complemented
60#102×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63C23×D151—21no—
64#14559004—22no—
65#14559004—41no—
66C2×Dic302C2311yescharacteristic
67C2×D604—22no—
68C2×D604—21no—
69C2×D604—21no—
70C2×D604—41no—
71C2×D604C2311yescharacteristic
72C2×D604C2311yescharacteristic
73C2×D604C2311yescharacteristic
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76C2×C22⋊D151—21no—
77C2×C22⋊D151—21no—
78C2×C22⋊D151—21no—
79C2×C22⋊D151D411yescharacteristic
80D4×D151—22no—
81D4×D151—22no—
82D4×D151—41no—
83D4×D151—41no—
84D4×D151—44no—
85C2×C4×D151—21no—
86C2×C4×D151—21no—
87C2×C4×D151—21no—
88C2×C4×D151C2311yescharacteristic
89#14558993—22no—
90#14558993—22no—
91#14558992—41no—
92D1208—22no—
93D1208D414yescomplemented
94#14558989—22no—
95#14558989—22no—
96#14558989—41no—
97C2×C151⋊C8C2311yescharacteristic
98C2×C151⋊C8C2311yescharacteristic
99C8×D151—22no—
100C4⋊Dic151—21no—
101Dic151⋊C4D411yescharacteristic
102C22⋊Dic151—21no—
103C22⋊Dic151—21no—
104C22⋊Dic151D411yescharacteristic
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4—21no—
108(C2×D151)⋊C4—21no—
109(C2×D151)⋊C4D411yescharacteristic
110C4×Dic151C2311yescharacteristic
111C2×D4×C151—21no—
112C2×D4×C151—21no—
113C2×D4×C151C2311yescharacteristic
114C2×D4×C151C2311yescharacteristic
115C22×C4×C151C2311yescharacteristic
116D8×C151—22no—
117M16×C151—22no—
118C2×C8×C151C2311yescharacteristic
119C2×C8×C151C2311yescharacteristic
120C4⋊C4×C151C2311yescharacteristic
121C4⋊C4×C151C2311yescharacteristic
122C22⋊C4×C151—21no—
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—42no—
132C22×D151—42no—
133C22×D151—81no—
134C22×D151—81no—
135C22×D151—81no—
136C22×D151C2×D411yescharacteristic
137Dic302—21no—
138Dic302—21no—
139D604—21no—
140D604—21no—
141D604—21no—
142D604—21no—
143D604—42no—
144D604—42no—
145D604C2×D412yes—
146D604C2×D412yescomplemented
147C22⋊D151—22no—
148C22⋊D151—42no—
149C22⋊D151—42no—
150C22⋊D151—42no—
151C22⋊D151—42no—
152C22⋊D151—81no—
153C2×Dic151—21no—
154C2×Dic151C2×D411yescharacteristic
155C2×Dic151C2×D411yescharacteristic
156C2×Dic151C2×D411yescharacteristic
157C4×D151—21no—
158C4×D151—21no—
159C4×D151—42no—
160C4×D151—42no—
161C4×D151—81no—
162C151⋊C8—21no—
163C151⋊C8—21no—
164C23×C151—21no—
165C23×C151—21no—
166C23×C151C2×D411yescharacteristic
167D4×C151—21no—
168D4×C151—21no—
169D4×C151—21no—
170D4×C151—21no—
171D4×C151—42no—
172D4×C151—81no—
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×C151C2×D412yescomplemented
181C22×C151—21no—
182C22×C151—21no—
183C22×C151—21no—
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—22no—
194C2×D151—22no—
195C2×D151—41no—
196C2×D151—41no—
197C2×D151—41no—
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—41no—
208C4×C151—41no—
209C4×C151C22×D411yescharacteristic
210C4×C151H(5,𝔽2)11yescharacteristic
211Dic151—21no—
212Dic151—21no—
213Dic151—21no—
214Dic151—41no—
215C2×C151—41no—
216C2×C151—81no—
217C2×C151—81no—
218C2×C151D4211yescharacteristic
219C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C151#57611yescharacteristic
221D151—22no—
222D151—42no—
223D151—42no—
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#469D15111yescharacteristic, complemented
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×SD16—1511no—
250C2×SD16—1511no—
251C2×D8—3024no—
252C2×D8—1511no—
253C2×D8—1511no—
254C2×D8—1511no—
255C2×D8—1511no—
256C2×D8—1511no—
257C2×D8C2×D15111yescharacteristic, complemented
258C2×M16C2×D15111yescharacteristic, complemented
259C22×C8—1511no—
260#127—1511no—
261#124—1511no—
262#123—3021no—
263C22⋊Q8—1511no—
264#121—3021no—
265#121—3021no—
266#121—1511no—
267#121C2×D15111yescharacteristic, complemented
268#121C2×D15111yescharacteristic, complemented
269#120—3021no—
270#120—3021no—
271#120—3021no—
272C4×D4—1511no—
273C4×D4—1511no—
274C4×D4—1511no—
275C2×C22⋊C4—3021no—
276#106C2×D15111yescharacteristic, complemented
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⋊C8—1511no—
286#97—1511no—
287C24—3021no—
288C24—3021no—
289C24—3021no—
290#41—3022no—
291#41—6041no—
292C2×Q8—1511no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3022no—
296C2×D4—3021no—
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×D4—1511no—
313C2×D4C22×D15111yescharacteristic
314C2×D4C22×D15111yescharacteristic
315C22×C4—3021no—
316C22×C4—3021no—
317C22×C4—3021no—
318C22×C4—1511no—
319C22×C4C22×D15111yescharacteristic
320SD16—3022no—
321SD16—3022no—
322SD16—6041no—
323D8—22no—
324D8—3022no—
325D8—3022no—
326D8—3022no—
327D8—6041no—
328D8—1514no—
329M16—22no—
330C2×C8—3022no—
331C2×C8—1511no—
332C2×C8—1511no—
333C2×C8C22×D15111yescharacteristic
334C2×C8C22×D15111yescharacteristic
335C4⋊C4—3021no—
336C4⋊C4—1511no—
337C4⋊C4C22×D15111yescharacteristic
338C4⋊C4C22×D15111yescharacteristic
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—
349C42—1511no—
350C23—21no—
351C23—21no—
352C23—3021no—
353C23—3021no—
354C23—3021no—
355C23—6042no—
356C23—6042no—
357C23—6042no—
358C23—6042no—
359C23—6042no—
360C23—6042no—
361C23—1511no—
362C23—12081no—
363C23—12081no—
364C23—12081no—
365C23D4×D15111yescharacteristic
366Q8—3021no—
367Q8—3021no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—3022no—
375D4—3021no—
376D4—3021no—
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—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—3021no—
405C8—3021no—
406C8D4×D15112yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3021no—
417C22—3021no—
418C22—3022no—
419C22—3022no—
420C22—6041no—
421C22—6041no—
422C22—6041no—
423C22—6041no—
424C22—6041no—
425C22—6041no—
426C22—6041no—
427C22—1512no—
428C22—12084no—
429C22—12084no—
430C22—12084no—
431C22C2×D4×D15111yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3021no—
436C4—3021no—
437C4—3021no—
438C4—6041no—
439C4C2×D4×D15111yescharacteristic
440C4#1801278511yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—3022no—
445C2—6042no—
446C2—6042no—
447C2—12081no—
448C2#2375284611yescharacteristic
449C2AGL(1,ℤ/8)×D15111yescharacteristic
450C2#2375301611yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#469×C151C2
socleC22×C151C22×D4
hypercenter#469D151
hypocenterC151#3264
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#18012767C2semidirect
2. C2×D8×D151C2semidirect
3. #23752859C2semidirect
4. #23752846C2semidirect
5. #23752600C2semidirect
6. #23752442C2semidirect
7. #23752426C2semidirect
8. #23752406C2semidirect
9. #23752364C2semidirect
10. #23752355C2semidirect
11. #23752347C2semidirect
12. #23752345C2semidirect
13. #23752323C2semidirect
14. #23752281C2semidirect
15. #469×C151C2semidirect
16. C2×#14559004C22semidirect
17. C22×D604C22semidirect
20. C2×D4×D151C22semidirect
22. C2×#14558993C22semidirect
23. C2×#14558993C22semidirect
24. C2×#14558992C22semidirect
26. C2×D1208C22semidirect
28. C2×#14558989C22semidirect
29. C2×#14558989C22semidirect
31. C2×C8×D151C22semidirect
32. #18012735C22semidirect
33. #18012726C22semidirect
35. #18012719C22semidirect
42. #18012701C22semidirect
43. #18012701C22semidirect
46. Dic151⋊C8C22semidirect
47. #18012662C22semidirect
48. #18012659C22semidirect
49. #18012659C22semidirect
50. #18012659C22semidirect
51. C22⋊(C151⋊C8)C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. #18012649C22semidirect
54. C2×D8×C151C22semidirect
55. C2×M16×C151C22semidirect
56. #121×C151C22semidirect
57. #121×C151C22semidirect
58. #106×C151C22semidirect
59. #102×C151C22semidirect
60. #102×C151C22semidirect
93. D1208D4semidirect
146. D604C2×D4semidirect
180. C8×C151C2×D4semidirect
205. C2×D151AGL(1,ℤ/8)semidirect
225. C151#3264semidirect
232. #469D151semidirect
257. C2×D8C2×D151semidirect
258. C2×M16C2×D151semidirect
267. #121C2×D151semidirect
268. #121C2×D151semidirect
276. #106C2×D151semidirect
282. #102C2×D151semidirect
283. #102C2×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×#180127672
C2×D8×D1512
#237528592
#237528462
#237526002
#237524422
#237524262
#237524062
#237523642
#237523552
#237523472
#237523452
#237523232
#237522812
#469×C1512
#3264151