Finite Group Databasebeta

← #34080612

Subgroups — #34080612

Order 19328.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012767C211yesmaximal, characteristic, complemented
2C2×SD16×D151C211yesmaximal, characteristic, complemented
3#23752872C211yesmaximal, characteristic, complemented
4#23752846C211yesmaximal, characteristic, complemented
5#23752598C211yesmaximal, characteristic, complemented
6#23752449C211yesmaximal, characteristic, complemented
7#23752430C211yesmaximal, characteristic, complemented
8#23752411C211yesmaximal, characteristic, complemented
9#23752362C211yesmaximal, characteristic, complemented
10#23752355C211yesmaximal, characteristic, complemented
11#23752352C211yesmaximal, characteristic, complemented
12#23752347C211yesmaximal, characteristic, complemented
13#23752311C211yesmaximal, characteristic, complemented
14#23752281C211yesmaximal, characteristic, complemented
15#468×C151C211yesmaximal, characteristic, complemented
16C2×#14559004C2211yescharacteristic, complemented
17C2×Q8×D151C2211yescharacteristic, complemented
18C22×D604C2211yescharacteristic
19C2×D4×D151—21no—
20C2×D4×D151C2211yescharacteristic
21#18012767—24no—
22C2×#14558993C2211yescharacteristic
23C2×#14558993C2211yescharacteristic, complemented
24C2×#14558992C2211yescharacteristic, complemented
25C2×#14558991C2211yescharacteristic
26C2×#14558991C2211yescharacteristic, complemented
27SD16×D151—24no—
28C2×D1208C2211yescharacteristic
29C2×D1208C2211yescharacteristic, complemented
30C2×#14558989C2211yescharacteristic, complemented
31C2×C8×D151C2211yescharacteristic, complemented
32#18012735C2211yescharacteristic
33#18012728C2211yescharacteristic, complemented
34#18012725—21no—
35#18012717—21no—
36#18012714C2211yescharacteristic
37#18012714C2211yescharacteristic, complemented
38#18012712—21no—
39#18012712—21no—
40#18012709—21no—
41#18012707—21no—
42#18012704C2211yescharacteristic, complemented
43#18012701—21no—
44#18012701C2211yescharacteristic
45C4×C22⋊D151C2211yescharacteristic
46Dic151⋊C8C2211yescharacteristic, complemented
47#18012662C2211yescharacteristic, complemented
48#18012659C2211yescharacteristic
49#18012659C2211yescharacteristic, complemented
50#18012659C2211yescharacteristic, complemented
51C22⋊(C151⋊C8)C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53#18012649C2211yescharacteristic, complemented
54C2×SD16×C151C2211yescharacteristic, complemented
55C2×M16×C151C2211yescharacteristic, complemented
56C22⋊Q8×C151C2211yescharacteristic, complemented
57#121×C151C2211yescharacteristic
58#107×C151C2211yescharacteristic, complemented
59#103×C151C2211yescharacteristic, complemented
60#102×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63#14559004—22no—
64#14559004—41no—
65C2×Dic302—21no—
66C2×Dic302C2311yescharacteristic
67Q8×D151—22no—
68Q8×D151—41no—
69C2×D604—22no—
70C2×D604—21no—
71C2×D604—21no—
72C2×D604—21no—
73C2×D604—41no—
74C2×D604C2311yescharacteristic
75C2×D604C2311yescharacteristic
76C2×D604C2311yescharacteristic
77C2×C22⋊D151—21no—
78C2×C22⋊D151—21no—
79C2×C22⋊D151—21no—
80C2×C22⋊D151—21no—
81C2×C22⋊D151D411yescharacteristic
82D4×D151—22no—
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#14558993D414yescomplemented
91#14558992—22no—
92#14558991—22no—
93#14558991—41no—
94D1208—22no—
95D1208—22no—
96#14558989—41no—
97C2×C151⋊C8C2311yescharacteristic
98C2×C151⋊C8C2311yescharacteristic
99C8×D151—22no—
100Dic151⋊C4—21no—
101Dic151⋊C4D411yescharacteristic
102C22⋊Dic151D411yescharacteristic
103(C2×D151)⋊C4—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4—21no—
108(C2×D151)⋊C4D411yescharacteristic
109C4×Dic151—21no—
110C4×Dic151C2311yescharacteristic
111C2×Q8×C151C2311yescharacteristic
112C2×D4×C151—21no—
113C2×D4×C151C2311yescharacteristic
114C22×C4×C151C2311yescharacteristic
115SD16×C151—22no—
116M16×C151—22no—
117C2×C8×C151C2311yescharacteristic
118C2×C8×C151C2311yescharacteristic
119C4⋊C4×C151—21no—
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—81no—
132C22×D151—81no—
133C22×D151C2×D411yescharacteristic
134Dic302—21no—
135Dic302—42no—
136Dic302C2×D412yescomplemented
137D604—21no—
138D604—21no—
139D604—21no—
140D604—21no—
141D604—21no—
142D604—42no—
143D604—42no—
144D604—42no—
145D604C2×D412yes—
146C22⋊D151—22no—
147C22⋊D151—42no—
148C22⋊D151—42no—
149C22⋊D151—42no—
150C22⋊D151—81no—
151C2×Dic151—21no—
152C2×Dic151C2×D411yescharacteristic
153C2×Dic151C2×D411yescharacteristic
154C2×Dic151C2×D411yescharacteristic
155C4×D151—21no—
156C4×D151—21no—
157C4×D151—42no—
158C4×D151—42no—
159C4×D151—42no—
160C151⋊C8—21no—
161C151⋊C8—21no—
162C23×C151—21no—
163C23×C151C2×D411yescharacteristic
164Q8×C151—21no—
165Q8×C151—21no—
166D4×C151—21no—
167D4×C151—21no—
168D4×C151—42no—
169C2×C4×C151—21no—
170C2×C4×C151—21no—
171C2×C4×C151—21no—
172C2×C4×C151—21no—
173C2×C4×C151—21no—
174C2×C4×C151C2×D411yescharacteristic
175C2×C4×C151C2411yescharacteristic
176C8×C151—21no—
177C8×C151C2×D412yescomplemented
178C22×C151—21no—
179C22×C151—21no—
180C22×C151—21no—
181C22×C151—41no—
182C22×C151—41no—
183C22×C151—41no—
184C22×C151C22×D411yescharacteristic
185C2×D151—21no—
186C2×D151—21no—
187C2×D151—22no—
188C2×D151—41no—
189C2×D151—41no—
190C2×D151—41no—
191C2×D151—41no—
192C2×D151—41no—
193C2×D151—41no—
194C2×D151—41no—
195C2×D151—41no—
196C2×D151—84no—
197C2×D151—84no—
198C2×D151AGL(1,ℤ/8)12yescomplemented
199C4×C151—21no—
200C4×C151—41no—
201C4×C151—41no—
202C4×C151—41no—
203C4×C151C22×D411yescharacteristic
204C4×C151H(5,𝔽2)11yescharacteristic
205Dic151—21no—
206Dic151—22no—
207Dic151—21no—
208Dic151—21no—
209C2×C151—41no—
210C2×C151—81no—
211C2×C151D4211yescharacteristic
212C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C151#57711yescharacteristic
214D151—22no—
215D151—42no—
216D151—81no—
217D151—81no—
218C151#326511yescharacteristic, complemented, Sylow
219#3265—1511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1511no—
221C22×SD16—1511no—
222#548—1511no—
223D42—1511no—
224#496—1511no—
225#468D15111yescharacteristic, complemented
226#466—1511no—
227#463—1511no—
228#460—1511no—
229#450—1511no—
230#449—1511no—
231#448—1511no—
232#447—1511no—
233#440—1511no—
234#435—1511no—
235C2×#41—1511no—
236C22×Q8—1511no—
237C22×D4—3021no—
238C22×D4—1511no—
239C22×D4—1511no—
240AGL(1,ℤ/8)—3024no—
241C2×SD16—3024no—
242C2×SD16—1511no—
243C2×SD16—1511no—
244C2×SD16—1511no—
245C2×SD16—1511no—
246C2×SD16—1511no—
247C2×SD16C2×D15111yescharacteristic, complemented
248C2×D8—1511no—
249C2×D8—1511no—
250C2×D8—1511no—
251C2×M16C2×D15111yescharacteristic, complemented
252C22×C8—1511no—
253#127—1511no—
254#124—3021no—
255#124—1511no—
256C22⋊Q8—3021no—
257C22⋊Q8C2×D15111yescharacteristic, complemented
258#121—3021no—
259#121—3021no—
260#121—1511no—
261#121—1511no—
262#121C2×D15111yescharacteristic
263#120—3021no—
264#120—3021no—
265C4×Q8—1511no—
266C4×D4—3021no—
267C4×D4—1511no—
268C4×D4—1511no—
269#107C2×D15111yescharacteristic, complemented
270C4⋊C8—1511no—
271#103—1511no—
272#103C2×D15111yescharacteristic, complemented
273#102—1511no—
274#102—1511no—
275#102—1511no—
276#102C2×D15111yescharacteristic, complemented
277C22⋊C8—1511no—
278C22⋊C8—1511no—
279#97—1511no—
280C24—3021no—
281C24—3021no—
282#41—3022no—
283#41—6041no—
284C2×Q8—3021no—
285C2×Q8—3022no—
286C2×Q8—6041no—
287C2×Q8—1511no—
288C2×Q8C22×D15111yescharacteristic
289C2×D4—21no—
290C2×D4—3022no—
291C2×D4—3021no—
292C2×D4—3021no—
293C2×D4—3022no—
294C2×D4—3021no—
295C2×D4—3021no—
296C2×D4—3021no—
297C2×D4—3021no—
298C2×D4—3021no—
299C2×D4—6041no—
300C2×D4—6041no—
301C2×D4—6044no—
302C2×D4—1511no—
303C2×D4—1511no—
304C2×D4—1511no—
305C2×D4—1511no—
306C2×D4C22×D15111yescharacteristic
307C22×C4—3021no—
308C22×C4—3021no—
309C22×C4—3021no—
310C22×C4—1511no—
311C22×C4C22×D15111yescharacteristic
312SD16—22no—
313SD16—3022no—
314SD16—3022no—
315SD16—3022no—
316SD16—6041no—
317SD16—1514no—
318D8—3022no—
319D8—3022no—
320D8—6041no—
321M16—22no—
322C2×C8—3022no—
323C2×C8—1511no—
324C2×C8—1511no—
325C2×C8C22×D15111yescharacteristic
326C2×C8C22×D15111yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—3021no—
329C4⋊C4—1511no—
330C4⋊C4C22×D15111yescharacteristic
331C4⋊C4C22×D15111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—3021no—
335C22⋊C4—3021no—
336C22⋊C4—3021no—
337C22⋊C4—3021no—
338C22⋊C4—3021no—
339C22⋊C4—1511no—
340C22⋊C4—1511no—
341C42—3021no—
342C42—1511no—
343C23—21no—
344C23—3021no—
345C23—3021no—
346C23—3021no—
347C23—6042no—
348C23—6042no—
349C23—6042no—
350C23—6042no—
351C23—1511no—
352C23—12081no—
353C23—12081no—
354C23D4×D15111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3021no—
358Q8—6042no—
359Q8—1512no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—3022no—
364D4—3021no—
365D4—3021no—
366D4—3021no—
367D4—3021no—
368D4—3021no—
369D4—6042no—
370D4—6042no—
371D4—6042no—
372D4—6042no—
373D4—6042no—
374D4—6042no—
375D4—1512no—
376D4—12081no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—3021no—
383C2×C4—3021no—
384C2×C4—3021no—
385C2×C4—6042no—
386C2×C4—6042no—
387C2×C4—6042no—
388C2×C4—1511no—
389C2×C4—1511no—
390C2×C4—1511no—
391C2×C4D4×D15111yescharacteristic
392C2×C4C23×D15111yescharacteristic
393C8—21no—
394C8—3021no—
395C8—3021no—
396C8D4×D15112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—3021no—
404C22—3021no—
405C22—3022no—
406C22—6041no—
407C22—6041no—
408C22—6041no—
409C22—6041no—
410C22—6041no—
411C22—6041no—
412C22—6041no—
413C22—6041no—
414C22—1512no—
415C22—12084no—
416C22—12084no—
417C22C2×D4×D15111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—3021no—
423C4—3021no—
424C4—3021no—
425C4—3022no—
426C4C2×D4×D15111yescharacteristic
427C4#1801278511yescharacteristic
428C2—41no—
429C2—81no—
430C2—3022no—
431C2—6042no—
432C2—12081no—
433C2—12081no—
434C2#2375284611yescharacteristic
435C2AGL(1,ℤ/8)×D15111yescharacteristic
436C2#2375302111yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#468×C151C2
socleC22×C151C22×D4
hypercenter#468D151
hypocenterC151#3265
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#18012767C2semidirect
2. C2×SD16×D151C2semidirect
3. #23752872C2semidirect
4. #23752846C2semidirect
5. #23752598C2semidirect
6. #23752449C2semidirect
7. #23752430C2semidirect
8. #23752411C2semidirect
9. #23752362C2semidirect
10. #23752355C2semidirect
11. #23752352C2semidirect
12. #23752347C2semidirect
13. #23752311C2semidirect
14. #23752281C2semidirect
15. #468×C151C2semidirect
16. C2×#14559004C22semidirect
17. C2×Q8×D151C22semidirect
23. C2×#14558993C22semidirect
24. C2×#14558992C22semidirect
26. C2×#14558991C22semidirect
29. C2×D1208C22semidirect
30. C2×#14558989C22semidirect
31. C2×C8×D151C22semidirect
33. #18012728C22semidirect
37. #18012714C22semidirect
42. #18012704C22semidirect
46. Dic151⋊C8C22semidirect
47. #18012662C22semidirect
49. #18012659C22semidirect
50. #18012659C22semidirect
51. C22⋊(C151⋊C8)C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. #18012649C22semidirect
54. C2×SD16×C151C22semidirect
55. C2×M16×C151C22semidirect
56. C22⋊Q8×C151C22semidirect
58. #107×C151C22semidirect
59. #103×C151C22semidirect
60. #102×C151C22semidirect
90. #14558993D4semidirect
136. Dic302C2×D4semidirect
177. C8×C151C2×D4semidirect
198. C2×D151AGL(1,ℤ/8)semidirect
218. C151#3265semidirect
225. #468D151semidirect
247. C2×SD16C2×D151semidirect
251. C2×M16C2×D151semidirect
257. C22⋊Q8C2×D151semidirect
269. #107C2×D151semidirect
272. #103C2×D151semidirect
276. #102C2×D151semidirect
396. 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×SD16×D1512
#237528722
#237528462
#237525982
#237524492
#237524302
#237524112
#237523622
#237523552
#237523522
#237523472
#237523112
#237522812
#468×C1512
#3265151