Finite Group Databasebeta

← #34080608

Subgroups — #34080608

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×#18012764C211yesmaximal, characteristic, complemented
2C2×SD16×D151C211yesmaximal, characteristic, complemented
3#23752869C211yesmaximal, characteristic, complemented
4#23752846C211yesmaximal, characteristic, complemented
5#23752600C211yesmaximal, characteristic, complemented
6#23752456C211yesmaximal, characteristic, complemented
7#23752447C211yesmaximal, characteristic, complemented
8#23752426C211yesmaximal, characteristic, complemented
9#23752410C211yesmaximal, characteristic, complemented
10#23752365C211yesmaximal, characteristic, complemented
11#23752352C211yesmaximal, characteristic, complemented
12#23752344C211yesmaximal, characteristic, complemented
13#23752315C211yesmaximal, characteristic, complemented
14#23752286C211yesmaximal, characteristic, complemented
15#449×C151C211yesmaximal, characteristic, complemented
16C2×Q8×D151C2211yescharacteristic, complemented
17C22×D604C2211yescharacteristic
18C2×D4×D151—21no—
19C2×D4×D151C2211yescharacteristic
20#18012764—24no—
21C2×#14558993C2211yescharacteristic
22C2×#14558993C2211yescharacteristic, complemented
23C2×#14558992C2211yescharacteristic, complemented
24C2×#14558991C2211yescharacteristic
25C2×#14558991C2211yescharacteristic, complemented
26SD16×D151—24no—
27C2×#14558989C2211yescharacteristic
28C2×#14558989C2211yescharacteristic, complemented
29C2×#14558988C2211yescharacteristic, complemented
30C2×C8×D151C2211yescharacteristic, complemented
31#18012735C2211yescharacteristic
32#18012726—21no—
33#18012726C2211yescharacteristic, complemented
34#18012718—21no—
35#18012718C2211yescharacteristic, complemented
36#18012714C2211yescharacteristic
37#18012712—21no—
38#18012710—21no—
39#18012710C2211yescharacteristic, complemented
40#18012709—21no—
41#18012707—21no—
42Q8×Dic151C2211yescharacteristic, complemented
43#18012701C2211yescharacteristic
44C4×C22⋊D151—21no—
45C4×C22⋊D151C2211yescharacteristic
46#18012672C2211yescharacteristic, complemented
47Dic151⋊C8C2211yescharacteristic, complemented
48#18012662C2211yescharacteristic, complemented
49#18012661C2211yescharacteristic
50#18012661C2211yescharacteristic, complemented
51#18012660C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53#18012649C2211yescharacteristic, complemented
54C2×#41×C151C2211yescharacteristic, complemented
55C2×SD16×C151C2211yescharacteristic, complemented
56C2×D8×C151C2211yescharacteristic, complemented
57#121×C151C2211yescharacteristic
58#103×C151C2211yescharacteristic, complemented
59#102×C151C2211yescharacteristic, complemented
60C22⋊C8×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63C2×Dic302—21no—
64C2×Dic302C2311yescharacteristic
65Q8×D151—22no—
66Q8×D151—41no—
67C2×D604—22no—
68C2×D604—21no—
69C2×D604—21no—
70C2×D604—41no—
71C2×D604C2311yescharacteristic
72C2×D604C2311yescharacteristic
73C2×C22⋊D151—21no—
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76C2×C22⋊D151—21no—
77C2×C22⋊D151D411yescharacteristic
78D4×D151—22no—
79D4×D151—41no—
80D4×D151—44no—
81C2×C4×D151—21no—
82C2×C4×D151—21no—
83C2×C4×D151C2311yescharacteristic
84#14558993—22no—
85#14558993—41no—
86#14558992—22no—
87#14558991—22no—
88#14558991D414yescomplemented
89#14558989—22no—
90#14558989—22no—
91#14558988—22no—
92C2×C151⋊C8C2311yescharacteristic
93C2×C151⋊C8C2311yescharacteristic
94C8×D151—22no—
95C4⋊Dic151—21no—
96C4⋊Dic151C2311yescharacteristic
97Dic151⋊C4—21no—
98Dic151⋊C4D411yescharacteristic
99C22⋊Dic151—21no—
100C22⋊Dic151—21no—
101C22⋊Dic151—21no—
102C22⋊Dic151—21no—
103C22⋊Dic151D411yescharacteristic
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4D411yescharacteristic
107C4×Dic151—21no—
108C4×Dic151C2311yescharacteristic
109#41×C151—22no—
110#41×C151—41no—
111C2×Q8×C151C2311yescharacteristic
112C2×D4×C151—21no—
113C2×D4×C151—21no—
114C2×D4×C151C2311yescharacteristic
115C2×D4×C151C2311yescharacteristic
116C22×C4×C151—21no—
117C22×C4×C151C2311yescharacteristic
118SD16×C151—22no—
119D8×C151—41no—
120C2×C8×C151C2311yescharacteristic
121C2×C8×C151C2311yescharacteristic
122C4⋊C4×C151C2311yescharacteristic
123C22⋊C4×C151—21no—
124C22×D151—21no—
125C22×D151—21no—
126C22×D151—42no—
127C22×D151—42no—
128C22×D151—42no—
129C22×D151—42no—
130C22×D151—81no—
131C22×D151—81no—
132C22×D151C2×D411yescharacteristic
133Dic302—21no—
134Dic302—21no—
135Dic302—42no—
136D604—21no—
137D604—21no—
138D604—21no—
139D604—42no—
140D604—42no—
141D604C2×D412yes—
142C22⋊D151—22no—
143C22⋊D151—42no—
144C22⋊D151—42no—
145C22⋊D151—42no—
146C22⋊D151—81no—
147C2×Dic151—21no—
148C2×Dic151—21no—
149C2×Dic151C2×D411yescharacteristic
150C2×Dic151C2×D411yescharacteristic
151C2×Dic151C2×D411yescharacteristic
152C4×D151—21no—
153C4×D151—21no—
154C4×D151—42no—
155C4×D151—42no—
156C151⋊C8—21no—
157C151⋊C8C2×D412yescomplemented
158C23×C151—21no—
159C23×C151—21no—
160C23×C151C2×D411yescharacteristic
161Q8×C151—21no—
162Q8×C151C2×D412yescomplemented
163D4×C151—21no—
164D4×C151—21no—
165D4×C151—21no—
166D4×C151—21no—
167D4×C151—42no—
168D4×C151—42no—
169C2×C4×C151—21no—
170C2×C4×C151—21no—
171C2×C4×C151—21no—
172C2×C4×C151—21no—
173C2×C4×C151—42no—
174C2×C4×C151C2×D411yescharacteristic
175C2×C4×C151C2411yescharacteristic
176C8×C151—21no—
177C8×C151—21no—
178C22×C151—21no—
179C22×C151—21no—
180C22×C151—21no—
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—22no—
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—22no—
201C4×C151—41no—
202C4×C151C22×D411yescharacteristic
203C4×C151H(5,𝔽2)11yescharacteristic
204Dic151—21no—
205Dic151—21no—
206Dic151—21no—
207Dic151—41no—
208Dic151—41no—
209C2×C151—41no—
210C2×C151—81no—
211C2×C151—81no—
212C2×C151D4211yescharacteristic
213C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C151#57711yescharacteristic
215D151—22no—
216D151—42no—
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#450—1511no—
230#449D15111yescharacteristic, complemented
231#448—1511no—
232#447—1511no—
233#440—1511no—
234#435—1511no—
235C2×#41C2×D15111yescharacteristic, complemented
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×D8C2×D15111yescharacteristic, complemented
251C2×M16—1511no—
252C22×C8—1511no—
253#127—1511no—
254#124—3021no—
255#124—1511no—
256C22⋊Q8—3021no—
257C22⋊Q8—1511no—
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#107—1511no—
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⋊C8C2×D15111yescharacteristic, complemented
279#97—1511no—
280C24—3021no—
281C24—3021no—
282#41—22no—
283#41—41no—
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—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×D4C22×D15111yescharacteristic
306C2×D4C22×D15111yescharacteristic
307C22×C4—21no—
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—41no—
319D8—3022no—
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⋊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—3021no—
346C23—3021no—
347C23—6042no—
348C23—6042no—
349C23—6042no—
350C23—6042no—
351C23—1511no—
352C23—12081no—
353C23—12081no—
354C23D4×D15111yescharacteristic
355Q8—21no—
356Q8—3021no—
357Q8—3021no—
358Q8—6042no—
359Q8D4×D15112yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—3022no—
367D4—3021no—
368D4—3021no—
369D4—3021no—
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—42no—
382C2×C4—3021no—
383C2×C4—3021no—
384C2×C4—3021no—
385C2×C4—3021no—
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—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—3021no—
407C22—3021no—
408C22—3022no—
409C22—6041no—
410C22—6041no—
411C22—6041no—
412C22—6041no—
413C22—6041no—
414C22—1512no—
415C22—12084no—
416C22—12084no—
417C22C2×D4×D15111yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—3021no—
422C4—3021no—
423C4—3021no—
424C4—6041no—
425C4—6041no—
426C4C2×D4×D15111yescharacteristic
427C4#1801278411yescharacteristic
428C2—41no—
429C2—81no—
430C2—81no—
431C2—3022no—
432C2—6042no—
433C2—12081no—
434C2#2375284411yescharacteristic
435C2AGL(1,ℤ/8)×D15111yescharacteristic
436C2#2375301911yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#449×C151C2
socleC22×C151C22×D4
hypercenter#449D151
hypocenterC151#3265
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#18012764C2semidirect
2. C2×SD16×D151C2semidirect
3. #23752869C2semidirect
4. #23752846C2semidirect
5. #23752600C2semidirect
6. #23752456C2semidirect
7. #23752447C2semidirect
8. #23752426C2semidirect
9. #23752410C2semidirect
10. #23752365C2semidirect
11. #23752352C2semidirect
12. #23752344C2semidirect
13. #23752315C2semidirect
14. #23752286C2semidirect
15. #449×C151C2semidirect
16. C2×Q8×D151C22semidirect
22. C2×#14558993C22semidirect
23. C2×#14558992C22semidirect
25. C2×#14558991C22semidirect
28. C2×#14558989C22semidirect
29. C2×#14558988C22semidirect
30. C2×C8×D151C22semidirect
33. #18012726C22semidirect
35. #18012718C22semidirect
39. #18012710C22semidirect
42. Q8×Dic151C22semidirect
46. #18012672C22semidirect
47. Dic151⋊C8C22semidirect
48. #18012662C22semidirect
50. #18012661C22semidirect
51. #18012660C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. #18012649C22semidirect
54. C2×#41×C151C22semidirect
55. C2×SD16×C151C22semidirect
56. C2×D8×C151C22semidirect
58. #103×C151C22semidirect
59. #102×C151C22semidirect
60. C22⋊C8×C151C22semidirect
88. #14558991D4semidirect
157. C151⋊C8C2×D4semidirect
162. Q8×C151C2×D4semidirect
198. C2×D151AGL(1,ℤ/8)semidirect
218. C151#3265semidirect
230. #449D151semidirect
235. C2×#41C2×D151semidirect
247. C2×SD16C2×D151semidirect
250. C2×D8C2×D151semidirect
272. #103C2×D151semidirect
276. #102C2×D151semidirect
278. C22⋊C8C2×D151semidirect
359. Q8D4×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127642
C2×SD16×D1512
#237528692
#237528462
#237526002
#237524562
#237524472
#237524262
#237524102
#237523652
#237523522
#237523442
#237523152
#237522862
#449×C1512
#3265151