Finite Group Databasebeta

← #34080603

Subgroups — #34080603

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×#18012765C211yesmaximal, characteristic, complemented
2C2×SD16×D151C211yesmaximal, characteristic, complemented
3#23752872C211yesmaximal, characteristic, complemented
4D4×C22⋊D151C211yesmaximal, characteristic, complemented
5#23752598C211yesmaximal, characteristic, complemented
6#23752460C211yesmaximal, characteristic, complemented
7#23752447C211yesmaximal, characteristic, complemented
8#23752429C211yesmaximal, characteristic, complemented
9#23752411C211yesmaximal, characteristic, complemented
10#23752359C211yesmaximal, characteristic, complemented
11#23752349C211yesmaximal, characteristic, complemented
12#23752347C211yesmaximal, characteristic, complemented
13#23752309C211yesmaximal, characteristic, complemented
14#23752286C211yesmaximal, characteristic, complemented
15#450×C151C211yesmaximal, characteristic, complemented
16C2×#14559004C2211yescharacteristic, complemented
17C2×Q8×D151C2211yescharacteristic, complemented
18C22×C22⋊D151—21no—
19C2×D4×D151C2211yescharacteristic
20#18012765—24no—
21C2×#14558993C2211yescharacteristic, complemented
22C2×#14558992C2211yescharacteristic
23C2×#14558992C2211yescharacteristic, complemented
24C2×#14558991C2211yescharacteristic, complemented
25SD16×D151—24no—
26C2×D1208C2211yescharacteristic, complemented
27C2×#14558989C2211yescharacteristic
28C2×#14558989C2211yescharacteristic, complemented
29C2×#14558988C2211yescharacteristic, complemented
30C2×C8×D151C2211yescharacteristic, complemented
31#18012735C2211yescharacteristic
32#18012728C2211yescharacteristic, complemented
33#18012725—21no—
34#18012717—21no—
35#18012714C2211yescharacteristic, complemented
36#18012713C2211yescharacteristic
37#18012712—21no—
38#18012711—21no—
39#18012710C2211yescharacteristic
40#18012708—21no—
41#18012707—21no—
42#18012704C2211yescharacteristic, complemented
43D4×Dic151C2211yescharacteristic
44#18012701—21no—
45C4×C22⋊D151C2211yescharacteristic
46#18012672C2211yescharacteristic, complemented
47Dic151⋊C8C2211yescharacteristic, complemented
48#18012664C2211yescharacteristic, complemented
49#18012663C2211yescharacteristic, complemented
50#18012661C2211yescharacteristic, complemented
51#18012659C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53#18012649C2211yescharacteristic, complemented
54C22×D4×C151C2211yescharacteristic
55C2×SD16×C151C2211yescharacteristic
56C2×SD16×C151C2211yescharacteristic, complemented
57C22⋊Q8×C151C2211yescharacteristic, complemented
58#102×C151C2211yescharacteristic
59#102×C151C2211yescharacteristic, complemented
60C22⋊C8×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62#14559004—22no—
63#14559004—41no—
64C2×Dic302—21no—
65C2×Dic302C2311yescharacteristic
66Q8×D151—22no—
67Q8×D151—41no—
68C2×D604—21no—
69C2×D604C2311yescharacteristic
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—
77C2×C22⋊D151—44no—
78C2×C22⋊D151D411yescharacteristic
79D4×D151—22no—
80D4×D151—41no—
81C22×Dic151—21no—
82C2×C4×D151—21no—
83C2×C4×D151—21no—
84C2×C4×D151C2311yescharacteristic
85#14558993—22no—
86#14558992—22no—
87#14558992D414yescomplemented
88#14558991—22no—
89D1208—41no—
90#14558989—22no—
91#14558989—22no—
92#14558988—22no—
93C2×C151⋊C8C2311yescharacteristic
94C2×C151⋊C8C2311yescharacteristic
95C8×D151—22no—
96C4⋊Dic151C2311yescharacteristic
97Dic151⋊C4—21no—
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)⋊C4D411yescharacteristic
107C4×Dic151—21no—
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—42no—
127C22×D151—42no—
128C22×D151—81no—
129C22×D151C2×D411yescharacteristic
130Dic302—21no—
131Dic302—42no—
132Dic302C2×D412yescomplemented
133D604—21no—
134D604—21no—
135D604—21no—
136D604—21no—
137D604—42no—
138C22⋊D151—22no—
139C22⋊D151—42no—
140C22⋊D151—42no—
141C22⋊D151—42no—
142C22⋊D151—42no—
143C22⋊D151—42no—
144C22⋊D151—81no—
145C2×Dic151—21no—
146C2×Dic151—21no—
147C2×Dic151—42no—
148C2×Dic151C2×D411yescharacteristic
149C2×Dic151C2×D411yescharacteristic
150C2×Dic151C2×D411yescharacteristic
151C4×D151—21no—
152C4×D151—21no—
153C4×D151—42no—
154C4×D151—42no—
155C151⋊C8—21no—
156C151⋊C8C2×D412yescomplemented
157C23×C151—21no—
158C23×C151—21no—
159C23×C151—42no—
160C23×C151—42no—
161C23×C151—81no—
162C23×C151C2×D411yescharacteristic
163Q8×C151—21no—
164Q8×C151—21no—
165D4×C151—21no—
166D4×C151—21no—
167D4×C151—21no—
168D4×C151—42no—
169D4×C151C2×D412yes—
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×C151—21no—
178C22×C151—21no—
179C22×C151—22no—
180C22×C151—21no—
181C22×C151—21no—
182C22×C151—41no—
183C22×C151—41no—
184C22×C151—41no—
185C22×C151—41no—
186C22×C151—41no—
187C22×C151—84no—
188C22×C151C22×D411yescharacteristic
189C2×D151—21no—
190C2×D151—21no—
191C2×D151—41no—
192C2×D151—41no—
193C2×D151—41no—
194C2×D151—41no—
195C2×D151—41no—
196C2×D151—41no—
197C2×D151—84no—
198C2×D151AGL(1,ℤ/8)12yescomplemented
199C4×C151—21no—
200C4×C151—41no—
201C4×C151—41no—
202C4×C151C22×D411yescharacteristic
203C4×C151H(5,𝔽2)11yescharacteristic
204Dic151—21no—
205Dic151—22no—
206Dic151—21no—
207Dic151—21no—
208Dic151—41no—
209C2×C151—42no—
210C2×C151—41no—
211C2×C151—81no—
212C2×C151D4211yescharacteristic
213C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C151#57711yescharacteristic
215D151—22no—
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#468—1511no—
226#466—1511no—
227#463—1511no—
228#460—1511no—
229#450D15111yescharacteristic, complemented
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×D4C2×D15111yescharacteristic
240AGL(1,ℤ/8)—3024no—
241C2×SD16—3024no—
242C2×SD16—1511no—
243C2×SD16—1511no—
244C2×SD16—1511no—
245C2×SD16—1511no—
246C2×SD16C2×D15111yescharacteristic
247C2×SD16C2×D15111yescharacteristic, complemented
248C2×D8—1511no—
249C2×D8—1511no—
250C2×D8—1511no—
251C2×M16—1511no—
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#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#102C2×D15111yescharacteristic
276#102C2×D15111yescharacteristic, complemented
277C22⋊C8—1511no—
278C22⋊C8C2×D15111yescharacteristic, complemented
279#97—1511no—
280C24—21no—
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—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—3021no—
293C2×D4—3022no—
294C2×D4—3021no—
295C2×D4—3021no—
296C2×D4—3021no—
297C2×D4—3021no—
298C2×D4—3021no—
299C2×D4—3021no—
300C2×D4—6041no—
301C2×D4—6044no—
302C2×D4—1511no—
303C2×D4—1511no—
304C2×D4—1511no—
305C2×D4C22×D15111yescharacteristic
306C2×D4C22×D15111yescharacteristic
307C22×C4—3021no—
308C22×C4—3021no—
309C22×C4—3021no—
310C22×C4—1511no—
311C22×C4C22×D15111yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—3022no—
315SD16—3022no—
316SD16—3022no—
317SD16—1514no—
318D8—3022no—
319D8—3022no—
320D8—6041no—
321M16—3022no—
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⋊C4—1511no—
331C4⋊C4C22×D15111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3021no—
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—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—3021no—
349C23—3021no—
350C23—6042no—
351C23—6042no—
352C23—1511no—
353C23—12081no—
354C23D4×D15111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3021no—
358Q8—6042no—
359Q8—1512no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—3022no—
365D4—3021no—
366D4—3021no—
367D4—3021no—
368D4—3021no—
369D4—6042no—
370D4—6042no—
371D4—6042no—
372D4—6042no—
373D4—6042no—
374D4—6042no—
375D4—12081no—
376D4D4×D15112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
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×C4—1511no—
391C2×C4D4×D15111yescharacteristic
392C2×C4C23×D15111yescharacteristic
393C8—21no—
394C8—21no—
395C8—3021no—
396C8—1512no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—3021no—
408C22—3021no—
409C22—6041no—
410C22—6041no—
411C22—6041no—
412C22—6041no—
413C22—6041no—
414C22—6041no—
415C22—1512no—
416C22—12084no—
417C22C2×D4×D15111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—3021no—
422C4—3021no—
423C4—3021no—
424C4—3022no—
425C4—6041no—
426C4C2×D4×D15111yescharacteristic
427C4#1801278411yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—3022no—
432C2—12081no—
433C2—12081no—
434C2#2375284411yescharacteristic
435C2AGL(1,ℤ/8)×D15111yescharacteristic
436C2#2375301811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#450×C151C2
socleC22×C151C22×D4
hypercenter#450D151
hypocenterC151#3265
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×#18012765C2semidirect
2. C2×SD16×D151C2semidirect
3. #23752872C2semidirect
4. D4×C22⋊D151C2semidirect
5. #23752598C2semidirect
6. #23752460C2semidirect
7. #23752447C2semidirect
8. #23752429C2semidirect
9. #23752411C2semidirect
10. #23752359C2semidirect
11. #23752349C2semidirect
12. #23752347C2semidirect
13. #23752309C2semidirect
14. #23752286C2semidirect
15. #450×C151C2semidirect
16. C2×#14559004C22semidirect
17. C2×Q8×D151C22semidirect
21. C2×#14558993C22semidirect
23. C2×#14558992C22semidirect
24. C2×#14558991C22semidirect
26. C2×D1208C22semidirect
28. C2×#14558989C22semidirect
29. C2×#14558988C22semidirect
30. C2×C8×D151C22semidirect
32. #18012728C22semidirect
35. #18012714C22semidirect
42. #18012704C22semidirect
46. #18012672C22semidirect
47. Dic151⋊C8C22semidirect
48. #18012664C22semidirect
49. #18012663C22semidirect
50. #18012661C22semidirect
51. #18012659C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. #18012649C22semidirect
56. C2×SD16×C151C22semidirect
57. C22⋊Q8×C151C22semidirect
59. #102×C151C22semidirect
60. C22⋊C8×C151C22semidirect
87. #14558992D4semidirect
132. Dic302C2×D4semidirect
156. C151⋊C8C2×D4semidirect
198. C2×D151AGL(1,ℤ/8)semidirect
218. C151#3265semidirect
229. #450D151semidirect
247. C2×SD16C2×D151semidirect
257. C22⋊Q8C2×D151semidirect
276. #102C2×D151semidirect
278. C22⋊C8C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127652
C2×SD16×D1512
#237528722
D4×C22⋊D1512
#237525982
#237524602
#237524472
#237524292
#237524112
#237523592
#237523492
#237523472
#237523092
#237522862
#450×C1512
#3265151