Finite Group Databasebeta

← #49130544

Subgroups — #49130544

Order 36224.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583657C211yesmaximal, characteristic, complemented
2C2×#22583634C211yesmaximal, characteristic, complemented
3C2×SD16×D283C211yesmaximal, characteristic, complemented
4#31827031C211yesmaximal, characteristic, complemented
5#31826611C211yesmaximal, characteristic, complemented
6#31826610C211yesmaximal, characteristic, complemented
7#31826607C211yesmaximal, characteristic, complemented
8#31826606C211yesmaximal, characteristic, complemented
9#31826605C211yesmaximal, characteristic, complemented
10#31826603C211yesmaximal, characteristic, complemented
11#31826602C211yesmaximal, characteristic, complemented
12#31826472C211yesmaximal, characteristic, complemented
13#31826458C211yesmaximal, characteristic, complemented
14#31826430C211yesmaximal, characteristic, complemented
15#449×C283C211yesmaximal, characteristic, complemented
16#22583657—24no—
17#22583657—24no—
18C2×#17985392—21no—
19C2×#17985392C2211yescharacteristic
20C2×#17985391—21no—
21C2×#17985391—21no—
22C2×#17985391C2211yescharacteristic
23C2×#17985391C2211yescharacteristic, complemented
24C22×Dic566—21no—
25C22×Dic566C2211yescharacteristic, complemented
26C2×Q8×D283C2211yescharacteristic
27C2×Q8×D283C2211yescharacteristic, complemented
28C2×D4×D283C2211yescharacteristic
29#22583634—24no—
30C2×#17985381C2211yescharacteristic
31C2×#17985381C2211yescharacteristic
32C2×#17985380C2211yescharacteristic
33C2×#17985380C2211yescharacteristic
34C2×#17985379C2211yescharacteristic
35SD16×D283—24no—
36C2×#17985377C2211yescharacteristic
37C2×#17985375C2211yescharacteristic
38C2×C8×D283C2211yescharacteristic
39#22583606—21no—
40#22583596—21no—
41#22583585C2211yescharacteristic
42#22583584C2211yescharacteristic
43#22583581C2211yescharacteristic
44#22583580—21no—
45#22583578—21no—
46#22583568C2211yescharacteristic
47#22583535C2211yescharacteristic, complemented
48#22583535C2211yescharacteristic, complemented
49#22583534C2211yescharacteristic, complemented
50#22583533C2211yescharacteristic, complemented
51#22583533C2211yescharacteristic, complemented
52#22583531C2211yescharacteristic, complemented
53C22⋊(C283⋊C8)C2211yescharacteristic, complemented
54(C2×D283)⋊C8C2211yescharacteristic, complemented
55(C2×D283)⋊C8C2211yescharacteristic, complemented
56C2×#41×C283C2211yescharacteristic, complemented
57C2×SD16×C283C2211yescharacteristic
58C2×D8×C283C2211yescharacteristic
59#121×C283C2211yescharacteristic
60#103×C283C2211yescharacteristic, complemented
61#102×C283C2211yescharacteristic, complemented
62C22⋊C8×C283C2211yescharacteristic, complemented
63C23×D283—21no—
64#17985392—28no—
65#17985392—22no—
66#17985392—22no—
67#17985391—28no—
68#17985391—28no—
69#17985391—22no—
70#17985391—22no—
71#17985391—22no—
72#17985391D414yescomplemented
73C2×Dic566—28no—
74C2×Dic566—22no—
75C2×Dic566—22no—
76C2×Dic566—21no—
77C2×Dic566—21no—
78C2×Dic566—21no—
79C2×Dic566D411yescharacteristic
80C2×Dic566C2311yescharacteristic
81C2×Dic566C2311yescharacteristic
82Q8×D283—22no—
83Q8×D283—22no—
84Q8×D283—22no—
85Q8×D283D414yescomplemented
86C2×D1132—21no—
87C2×D1132D411yescharacteristic
88C2×D1132C2311yescharacteristic
89C2×C22⋊D283—21no—
90C2×C22⋊D283—21no—
91C2×C22⋊D283—21no—
92C2×C22⋊D283—21no—
93C2×C22⋊D283—21no—
94C2×C22⋊D283D411yescharacteristic
95C2×C22⋊D283D411yescharacteristic
96D4×D283—22no—
97D4×D283—22no—
98C22×Dic283—21no—
99C22×Dic283—21no—
100C22×Dic283—21no—
101C2×C4×D283—21no—
102C2×C4×D283—21no—
103C2×C4×D283C2311yescharacteristic
104C2×C4×D283C2311yescharacteristic
105#17985381—22no—
106#17985381—22no—
107#17985380—22no—
108#17985380—22no—
109#17985379—22no—
110#17985377—22no—
111#17985375—22no—
112C2×C283⋊C8C2311yescharacteristic
113C2×C283⋊C8C2311yescharacteristic
114C8×D283—22no—
115C4⋊Dic283C2311yescharacteristic
116C22⋊Dic283—21no—
117(C2×D283)⋊C4—21no—
118(C2×D283)⋊C4—21no—
119(C2×D283)⋊C4—21no—
120C4×Dic283—21no—
121#41×C283—22no—
122#41×C283—22no—
123C2×Q8×C283C2311yescharacteristic
124C2×D4×C283—21no—
125C2×D4×C283—21no—
126C2×D4×C283C2311yescharacteristic
127C2×D4×C283C2311yescharacteristic
128C22×C4×C283—21no—
129C22×C4×C283C2311yescharacteristic
130SD16×C283—22no—
131D8×C283—22no—
132C2×C8×C283C2311yescharacteristic
133C2×C8×C283C2311yescharacteristic
134C4⋊C4×C283C2311yescharacteristic
135C22⋊C4×C283—21no—
136C22×D283—21no—
137C22×D283—42no—
138C22×D283—81no—
139C22×D283—81no—
140C22×D283C2×D411yescharacteristic
141C22×D283C2×D411yescharacteristic
142Dic566—24no—
143Dic566—24no—
144Dic566—21no—
145Dic566—21no—
146Dic566—24no—
147Dic566—22no—
148Dic566C2×D412yes—
149Dic566C2×D412yes—
150D1132—22no—
151D1132—21no—
152D1132—21no—
153D1132—81no—
154C22⋊D283—24no—
155C22⋊D283—24no—
156C22⋊D283—22no—
157C22⋊D283—22no—
158C22⋊D283—42no—
159C22⋊D283—81no—
160C22⋊D283—81no—
161C2×Dic283—24no—
162C2×Dic283—24no—
163C2×Dic283—21no—
164C2×Dic283—21no—
165C2×Dic283—24no—
166C2×Dic283—21no—
167C2×Dic283C2×D411yescharacteristic
168C2×Dic283C2×D411yescharacteristic
169C4×D283—24no—
170C4×D283—24no—
171C4×D283—21no—
172C4×D283—21no—
173C4×D283C2×D412yes—
174C4×D283C2×D412yes—
175C283⋊C8—21no—
176C283⋊C8—21no—
177C23×C283—21no—
178C23×C283—21no—
179C23×C283C2×D411yescharacteristic
180Q8×C283—21no—
181Q8×C283C2×D412yes—
182D4×C283—24no—
183D4×C283—21no—
184D4×C283—21no—
185D4×C283—21no—
186D4×C283—81no—
187D4×C283C2×D412yes—
188C2×C4×C283—24no—
189C2×C4×C283—21no—
190C2×C4×C283—21no—
191C2×C4×C283—21no—
192C2×C4×C283—21no—
193C2×C4×C283C2×D411yescharacteristic
194C2×C4×C283C2411yescharacteristic
195C8×C283—21no—
196C8×C283—21no—
197C22×C283—21no—
198C22×C283—22no—
199C22×C283—21no—
200C22×C283—21no—
201C22×C283—41no—
202C22×C283—41no—
203C22×C283—41no—
204C22×C283—41no—
205C22×C283—41no—
206C22×C283C22×D411yescharacteristic
207C2×D283—21no—
208C2×D283—21no—
209C2×D283—21no—
210C2×D283—21no—
211C2×D283—21no—
212C2×D283—41no—
213C2×D283—41no—
214C2×D283—41no—
215C2×D283—84no—
216C2×D283#12012yes—
217C4×C283—21no—
218C4×C283—22no—
219C4×C283—41no—
220C4×C283C22×D411yescharacteristic
221C4×C283C22×D411yescharacteristic
222Dic283—22no—
223Dic283—22no—
224Dic283—21no—
225Dic283—41no—
226Dic283#12012yes—
227C2×C283—41no—
228C2×C283—42no—
229C2×C283—81no—
230C2×C283C2×#12011yescharacteristic
231C2×C283#57711yescharacteristic
232C2×C283#57711yescharacteristic
233D283—22no—
234D283—41no—
235D283—81no—
236C283#301011yescharacteristic, complemented, Sylow
237#3010—2831nomaximal, Sylow
238C2×2−1+4—2831no—
239C2×AGL(1,ℤ/8)—2831no—
240C22×SD16—2831no—
241#535—2831no—
242#449—2831no—
243#449—2831no—
244#449—2831no—
245#449D28311yescharacteristic, complemented
246#448—2831no—
247#448—2831no—
248#448—2831no—
249#448—2831no—
250#419—2831no—
251#416—2831no—
252#408—2831no—
2532−1+4—5664no—
2542−1+4—5664no—
255C2×#41—2831no—
256C2×#41—2831no—
257C2×#41—2831no—
258C2×#41—5661no—
259C2×#41—5661no—
260C2×#41—5661no—
261C2×#41C2×D28311yescharacteristic, complemented
262C22×Q8—2831no—
263C22×Q8—2831no—
264C22×Q8—2831no—
265C22×Q8—5661no—
266C22×D4—2831no—
267AGL(1,ℤ/8)—5664no—
268C2×SD16—2831no—
269C2×SD16—2831no—
270C2×SD16—2831no—
271C2×SD16—2831no—
272C2×SD16—2831no—
273C2×SD16—5664no—
274C2×SD16C2×D28311yescharacteristic
275C2×D8—2831no—
276C2×D8C2×D28311yescharacteristic
277C2×M16—2831no—
278C22×C8—2831no—
279#127—5661no—
280#124—5661no—
281#121—2831no—
282#121—2831no—
283#121—2831no—
284#121C2×D28311yescharacteristic
285#120—5661no—
286#120—5661no—
287#117—2831no—
288#103—2831no—
289#103—2831no—
290#103—2831no—
291#103—2831no—
292#103—2831no—
293#103C2×D28311yescharacteristic, complemented
294#102—2831no—
295#102C2×D28311yescharacteristic, complemented
296C22⋊C8—2831no—
297C22⋊C8—2831no—
298C22⋊C8—2831no—
299C22⋊C8C2×D28311yescharacteristic, complemented
300C24—5661no—
301#41—22no—
302#41—22no—
303#41—2834no—
304#41—5662no—
305#41—5662no—
306#41—5668no—
307#41—5662no—
308#41—5662no—
309#41—5668no—
310#41—5668no—
311#41—5662no—
312C2×Q8—2831no—
313C2×Q8—2834no—
314C2×Q8—2831no—
315C2×Q8—2831no—
316C2×Q8—5661no—
317C2×Q8—5661no—
318C2×Q8—5662no—
319C2×Q8—5662no—
320C2×Q8—5668no—
321C2×Q8—5662no—
322C2×Q8—5662no—
323C2×Q8—5662no—
324C2×Q8—5661no—
325C2×Q8C22×D28311yescharacteristic
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—2831no—
329C2×D4—2831no—
330C2×D4—2831no—
331C2×D4—2831no—
332C2×D4—5661no—
333C2×D4—5662no—
334C2×D4—5662no—
335C2×D4—5661no—
336C2×D4—5661no—
337C2×D4—5661no—
338C2×D4—5661no—
339C2×D4—5661no—
340C2×D4C22×D28311yescharacteristic
341C2×D4C22×D28311yescharacteristic
342C22×C4—21no—
343C22×C4—2831no—
344C22×C4—2831no—
345C22×C4—5661no—
346C22×C4—5661no—
347C22×C4—5661no—
348C22×C4—5661no—
349C22×C4—5661no—
350C22×C4C22×D28311yescharacteristic
351SD16—22no—
352SD16—5662no—
353SD16—5662no—
354SD16—5662no—
355SD16—5662no—
356SD16—5662no—
357D8—22no—
358D8—5662no—
359M16—5662no—
360C2×C8—2831no—
361C2×C8—2831no—
362C2×C8—5662no—
363C2×C8C22×D28311yescharacteristic
364C2×C8C22×D28311yescharacteristic
365C4⋊C4—2831no—
366C4⋊C4C22×D28311yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—5661no—
369C22⋊C4—5661no—
370C22⋊C4—5661no—
371C22⋊C4—5661no—
372C42—5661no—
373C23—21no—
374C23—21no—
375C23—2831no—
376C23—2831no—
377C23—5661no—
378C23—11322no—
379C23—22641no—
380C23—22641no—
381C23D4×D28311yescharacteristic
382Q8—21no—
383Q8—2832no—
384Q8—2832no—
385Q8—5662no—
386Q8—5664no—
387Q8—5664no—
388Q8—5664no—
389Q8—5661no—
390Q8—5661no—
391Q8D4×D28312yes—
392D4—24no—
393D4—21no—
394D4—21no—
395D4—21no—
396D4—81no—
397D4—5662no—
398D4—5662no—
399D4—5662no—
400D4—5664no—
401D4—5664no—
402D4—5661no—
403D4—5661no—
404D4—11322no—
405D4—22641no—
406D4—22641no—
407D4—22641no—
408D4D4×D28312yes—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—2831no—
415C2×C4—2832no—
416C2×C4—2832no—
417C2×C4—2831no—
418C2×C4—5664no—
419C2×C4—5664no—
420C2×C4—5664no—
421C2×C4—5664no—
422C2×C4—5664no—
423C2×C4—5661no—
424C2×C4—5661no—
425C2×C4—5661no—
426C2×C4—5661no—
427C2×C4—5661no—
428C2×C4D4×D28311yescharacteristic
429C2×C4C23×D28311yescharacteristic
430C8—21no—
431C8—21no—
432C8—5661no—
433C8—5661no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—2832no—
444C22—5661no—
445C22—5661no—
446C22—5661no—
447C22—5661no—
448C22—5661no—
449C22—11321no—
450C22—11321no—
451C22—11321no—
452C22—22644no—
453C22C2×D4×D28311yescharacteristic
454C4—21no—
455C4—22no—
456C4—41no—
457C4—2832no—
458C4—5661no—
459C4—5662no—
460C4—5662no—
461C4—11321no—
462C4C2×D4×D28311yescharacteristic
463C4C2×D4×D28311yescharacteristic
464C2—41no—
465C2—42no—
466C2—81no—
467C2—5662no—
468C2—11321no—
469C2—22641no—
470C2#120×D28311yescharacteristic
471C2#3182727211yescharacteristic
472C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#449×C283C2
socleC22×C283C22×D4
hypercenter#449D283
hypocenterC283#3010
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22583657C2semidirect
2. C2×#22583634C2semidirect
3. C2×SD16×D283C2semidirect
4. #31827031C2semidirect
5. #31826611C2semidirect
6. #31826610C2semidirect
7. #31826607C2semidirect
8. #31826606C2semidirect
9. #31826605C2semidirect
10. #31826603C2semidirect
11. #31826602C2semidirect
12. #31826472C2semidirect
13. #31826458C2semidirect
14. #31826430C2semidirect
15. #449×C283C2semidirect
23. C2×#17985391C22semidirect
25. C22×Dic566C22semidirect
27. C2×Q8×D283C22semidirect
47. #22583535C22semidirect
48. #22583535C22semidirect
49. #22583534C22semidirect
50. #22583533C22semidirect
51. #22583533C22semidirect
52. #22583531C22semidirect
53. C22⋊(C283⋊C8)C22semidirect
54. (C2×D283)⋊C8C22semidirect
55. (C2×D283)⋊C8C22semidirect
56. C2×#41×C283C22semidirect
60. #103×C283C22semidirect
61. #102×C283C22semidirect
62. C22⋊C8×C283C22semidirect
72. #17985391D4semidirect
85. Q8×D283D4semidirect
236. C283#3010semidirect
245. #449D283semidirect
261. C2×#41C2×D283semidirect
293. #103C2×D283semidirect
295. #102C2×D283semidirect
299. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836572
C2×#225836342
C2×SD16×D2832
#318270312
#318266112
#318266102
#318266072
#318266062
#318266052
#318266032
#318266022
#318264722
#318264582
#318264302
#449×C2832
#3010283