Finite Group Databasebeta

← #34080604

Subgroups — #34080604

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×#18012763C211yesmaximal, characteristic, complemented
2C2×SD16×D151C211yesmaximal, characteristic, complemented
3#23752870C211yesmaximal, characteristic, complemented
4#23752847C211yesmaximal, characteristic, complemented
5#23752455C211yesmaximal, characteristic, complemented
6#23752445C211yesmaximal, characteristic, complemented
7#23752424C211yesmaximal, characteristic, complemented
8#23752409C211yesmaximal, characteristic, complemented
9#23752361C211yesmaximal, characteristic, complemented
10#23752354C211yesmaximal, characteristic, complemented
11#23752349C211yesmaximal, characteristic, complemented
12#23752344C211yesmaximal, characteristic, complemented
13#23752314C211yesmaximal, characteristic, complemented
14#23752285C211yesmaximal, characteristic, complemented
15#496×C151C211yesmaximal, characteristic, complemented
16C2×#14559003C2211yescharacteristic, complemented
17C2×Q8×D151C2211yescharacteristic, complemented
18C2×D4×D151—21no—
19C2×D4×D151C2211yescharacteristic
20C2×D4×D151C2211yescharacteristic
21#18012763—24no—
22C2×#14558993C2211yescharacteristic
23C2×#14558992C2211yescharacteristic, complemented
24C2×#14558992C2211yescharacteristic, complemented
25C2×#14558991C2211yescharacteristic
26SD16×D151—24no—
27C2×#14558989C2211yescharacteristic
28C2×#14558987C2211yescharacteristic, complemented
29C2×C8×D151C2211yescharacteristic, complemented
30#18012725—21no—
31#18012716—21no—
32#18012715C2211yescharacteristic, complemented
33#18012712—21no—
34#18012710—21no—
35#18012710C2211yescharacteristic
36#18012710C2211yescharacteristic
37#18012710C2211yescharacteristic, complemented
38#18012707—21no—
39#18012707—21no—
40C4×Dic302C2211yescharacteristic, complemented
41C4×D604C2211yescharacteristic
42C4×D604C2211yescharacteristic
43(C22⋊D151)⋊C4—21no—
44#18012673C2211yescharacteristic, complemented
45C4⋊(C151⋊C8)C2211yescharacteristic, complemented
46#18012664C2211yescharacteristic, complemented
47#18012663C2211yescharacteristic, complemented
48#18012661C2211yescharacteristic
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
59#124×C151C2211yescharacteristic, complemented
60#97×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63#14559003—22no—
64#14559003—41no—
65C2×Dic302—21no—
66C2×Dic302C2311yescharacteristic
67Q8×D151—22no—
68Q8×D151—41no—
69C2×D604D411yescharacteristic
70C2×D604C2311yescharacteristic
71C2×C22⋊D151—21no—
72C2×C22⋊D151—21no—
73C2×C22⋊D151—21no—
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76C2×C22⋊D151—21no—
77D4×D151—22no—
78D4×D151—22no—
79D4×D151—41no—
80D4×D151—41no—
81D4×D151—44no—
82C22×Dic151—21no—
83C2×C4×D151—21no—
84C2×C4×D151—21no—
85C2×C4×D151C2311yescharacteristic
86C2×C4×D151C2311yescharacteristic
87#14558993D414yescomplemented
88#14558992—22no—
89#14558992—22no—
90#14558991—22no—
91#14558989—22no—
92#14558987—22no—
93C2×C151⋊C8C2311yescharacteristic
94C2×C151⋊C8C2311yescharacteristic
95C8×D151—22no—
96C4⋊Dic151D411yescharacteristic
97C4⋊Dic151C2311yescharacteristic
98C4⋊Dic151C2311yescharacteristic
99Dic151⋊C4—21no—
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)⋊C4D411yescharacteristic
107(C2×D151)⋊C4D411yescharacteristic
108C4×Dic151—21no—
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—42no—
126C22×D151—42no—
127C22×D151—42no—
128C22×D151—42no—
129C22×D151—81no—
130C22×D151—81no—
131C22×D151C2×D411yescharacteristic
132C22×D151C2×D411yescharacteristic
133Dic302—21no—
134Dic302—42no—
135Dic302C2×D412yescomplemented
136D604—22no—
137D604—21no—
138D604C2×D412yes—
139C22⋊D151—42no—
140C22⋊D151—42no—
141C22⋊D151—42no—
142C22⋊D151—42no—
143C22⋊D151—42no—
144C22⋊D151—42no—
145C2×Dic151—21no—
146C2×Dic151—21no—
147C2×Dic151—21no—
148C2×Dic151—42no—
149C2×Dic151C2×D411yescharacteristic
150C2×Dic151C2×D411yescharacteristic
151C4×D151—21no—
152C4×D151—21no—
153C4×D151—21no—
154C4×D151—21no—
155C4×D151—42no—
156C4×D151—42no—
157C151⋊C8—21no—
158C151⋊C8—21no—
159C23×C151—21no—
160C23×C151—21no—
161C23×C151—21no—
162Q8×C151—21no—
163Q8×C151—21no—
164D4×C151—21no—
165D4×C151—21no—
166D4×C151—21no—
167D4×C151—21no—
168D4×C151—21no—
169D4×C151—21no—
170D4×C151—42no—
171D4×C151—81no—
172C2×C4×C151—21no—
173C2×C4×C151C2×D411yescharacteristic
174C2×C4×C151C2×D411yescharacteristic
175C2×C4×C151C2411yescharacteristic
176C8×C151—21no—
177C8×C151C2×D412yescomplemented
178C22×C151—41no—
179C22×C151—41no—
180C22×C151—41no—
181C22×C151—41no—
182C22×C151—41no—
183C22×C151—41no—
184C22×C151—41no—
185C22×C151—41no—
186C22×C151—41no—
187C22×C151C22×D411yescharacteristic
188C2×D151—21no—
189C2×D151—21no—
190C2×D151—21no—
191C2×D151—22no—
192C2×D151—21no—
193C2×D151—21no—
194C2×D151—41no—
195C2×D151—41no—
196C2×D151—84no—
197C2×D151—84no—
198C2×D151AGL(1,ℤ/8)12yescomplemented
199C4×C151—21no—
200C4×C151—21no—
201C4×C151—41no—
202C4×C151C22×D411yescharacteristic
203C4×C151H(5,𝔽2)11yescharacteristic
204Dic151—21no—
205Dic151—21no—
206Dic151—22no—
207Dic151—41no—
208Dic151—41no—
209C2×C151—81no—
210C2×C151—81no—
211C2×C151—81no—
212C2×C151D4211yescharacteristic
213C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C151#57711yescharacteristic
215D151—22no—
216D151—41no—
217D151—42no—
218C151#326511yescharacteristic, complemented, Sylow
219#3265—1511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1511no—
221C22×SD16—1511no—
222#548—1511no—
223D42—1511no—
224#496D15111yescharacteristic, complemented
225#468—1511no—
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×SD16C2×D15111yescharacteristic, complemented
247C2×SD16C2×D15111yescharacteristic, complemented
248C2×D8—1511no—
249C2×D8C2×D15111yescharacteristic, complemented
250C2×D8C2×D15111yescharacteristic, complemented
251C2×M16—1511no—
252C22×C8—1511no—
253#127C2×D15111yescharacteristic
254#124—3021no—
255#124C2×D15111yescharacteristic, complemented
256C22⋊Q8—3021no—
257C22⋊Q8—1511no—
258#121—3021no—
259#121—3021no—
260#121—1511no—
261#121—1511no—
262#121—1511no—
263#120—3021no—
264#120—3021no—
265C4×Q8—1511no—
266C4×D4—3021no—
267C4×D4—1511no—
268C4×D4—1511no—
269#107—1511no—
270C4⋊C8—1511no—
271#103—1511no—
272#103—1511no—
273#102—1511no—
274#102—1511no—
275#102—1511no—
276#102—1511no—
277C22⋊C8—1511no—
278C22⋊C8—1511no—
279#97C2×D15111yescharacteristic, complemented
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—21no—
291C2×D4—3022no—
292C2×D4—3022no—
293C2×D4—3021no—
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×D4C22×D15111yescharacteristic
305C2×D4C22×D15111yescharacteristic
306C2×D4C22×D15111yescharacteristic
307C22×C4—3021no—
308C22×C4—3021no—
309C22×C4—3021no—
310C22×C4—1511no—
311C22×C4—1511no—
312SD16—22no—
313SD16—41no—
314SD16—3022no—
315SD16—3022no—
316SD16—3022no—
317SD16—1514no—
318D8—22no—
319D8—41no—
320D8—3022no—
321M16—3022no—
322C2×C8—3022no—
323C2×C8—1511no—
324C2×C8—1511no—
325C2×C8C22×D15111yescharacteristic
326C2×C8C22×D15111yescharacteristic
327C4⋊C4—3021no—
328C4⋊C4—3021no—
329C4⋊C4—1511no—
330C4⋊C4—1511no—
331C4⋊C4—1511no—
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—
342C42C22×D15111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—3021no—
347C23—6042no—
348C23—6042no—
349C23—6042no—
350C23—6042no—
351C23—1511no—
352C23—1511no—
353C23—12081no—
354C23—12081no—
355Q8—21no—
356Q8—21no—
357Q8—3021no—
358Q8—6042no—
359Q8—1512no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—3022no—
369D4—3021no—
370D4—6042no—
371D4—6042no—
372D4—6042no—
373D4—6042no—
374D4—6042no—
375D4—6042no—
376D4—1512no—
377C2×C4—21no—
378C2×C4—3021no—
379C2×C4—3021no—
380C2×C4—3021no—
381C2×C4—3021no—
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×C4D4×D15111yescharacteristic
391C2×C4D4×D15111yescharacteristic
392C2×C4C23×D15111yescharacteristic
393C8—21no—
394C8—3021no—
395C8—3021no—
396C8D4×D15112yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—3021no—
407C22—3021no—
408C22—3021no—
409C22—3021no—
410C22—3021no—
411C22—3022no—
412C22—6041no—
413C22—6041no—
414C22—1512no—
415C22—12084no—
416C22—12084no—
417C22C2×D4×D15111yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—3021no—
422C4—3021no—
423C4—3022no—
424C4—6041no—
425C4—6041no—
426C4C2×D4×D15111yescharacteristic
427C4#1801278411yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—3022no—
432C2—6041no—
433C2—6042no—
434C2#2375284711yescharacteristic
435C2AGL(1,ℤ/8)×D15111yescharacteristic
436C2#2375302211yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#496×C151C2
socleC22×C151C22×D4
hypercenter#496D151
hypocenterC151#3265
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×#18012763C2semidirect
2. C2×SD16×D151C2semidirect
3. #23752870C2semidirect
4. #23752847C2semidirect
5. #23752455C2semidirect
6. #23752445C2semidirect
7. #23752424C2semidirect
8. #23752409C2semidirect
9. #23752361C2semidirect
10. #23752354C2semidirect
11. #23752349C2semidirect
12. #23752344C2semidirect
13. #23752314C2semidirect
14. #23752285C2semidirect
15. #496×C151C2semidirect
16. C2×#14559003C22semidirect
17. C2×Q8×D151C22semidirect
23. C2×#14558992C22semidirect
24. C2×#14558992C22semidirect
28. C2×#14558987C22semidirect
29. C2×C8×D151C22semidirect
32. #18012715C22semidirect
37. #18012710C22semidirect
40. C4×Dic302C22semidirect
44. #18012673C22semidirect
45. C4⋊(C151⋊C8)C22semidirect
46. #18012664C22semidirect
47. #18012663C22semidirect
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
59. #124×C151C22semidirect
60. #97×C151C22semidirect
87. #14558993D4semidirect
135. Dic302C2×D4semidirect
177. C8×C151C2×D4semidirect
198. C2×D151AGL(1,ℤ/8)semidirect
218. C151#3265semidirect
224. #496D151semidirect
246. C2×SD16C2×D151semidirect
247. C2×SD16C2×D151semidirect
249. C2×D8C2×D151semidirect
250. C2×D8C2×D151semidirect
255. #124C2×D151semidirect
279. #97C2×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×#180127632
C2×SD16×D1512
#237528702
#237528472
#237524552
#237524452
#237524242
#237524092
#237523612
#237523542
#237523492
#237523442
#237523142
#237522852
#496×C1512
#3265151