Finite Group Databasebeta

← #49133032

Subgroups — #49133032

Order 36224.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827280C211yesmaximal, characteristic, complemented
2#31827273C211yesmaximal, characteristic, complemented
3#31827272C211yesmaximal, characteristic, complemented
4#31826858C211yesmaximal, characteristic, complemented
5#31826738C211yesmaximal, characteristic, complemented
6#31826737C211yesmaximal, characteristic, complemented
7#31826730C211yesmaximal, characteristic, complemented
8#31826728C211yesmaximal, characteristic, complemented
9#31826648C211yesmaximal, characteristic, complemented
10#31826646C211yesmaximal, characteristic, complemented
11#31826641C211yesmaximal, characteristic, complemented
12#31826638C211yesmaximal, characteristic, complemented
13#31826587C211yesmaximal, characteristic, complemented
14#31826585C211yesmaximal, characteristic, complemented
15#578×C283C211yesmaximal, characteristic, complemented
16#22583657C2211yescharacteristic
17#22583656C2211yescharacteristic
18#22583642—21no—
19#22583641—21no—
20#22583641—21no—
21#22583641C2211yescharacteristic
22#22583641C2211yescharacteristic
23#22583640—21no—
24#22583640C2211yescharacteristic
25#22583640C2211yescharacteristic, complemented
26#22583635—21no—
27#22583635C2211yescharacteristic
28#22583635C2211yescharacteristic
29#22583634—21no—
30#22583631—21no—
31#22583630—21no—
32#22583630—21no—
33#22583630C2211yescharacteristic
34C2×#17985382—21no—
35C2×#17985382C2211yescharacteristic, complemented
36C2×#17985381C2211yescharacteristic
37C2×#17985379—21no—
38C2×#17985379C2211yescharacteristic
39SD16×D283—21no—
40#22583617C2211yescharacteristic
41#22583617C2211yescharacteristic, complemented
42#22583616C2211yescharacteristic
43#22583608C2211yescharacteristic, complemented
44#22583599C2211yescharacteristic
45#22583546C2211yescharacteristic, complemented
46#22583545C2211yescharacteristic, complemented
47#22583537C2211yescharacteristic
48#22583537C2211yescharacteristic, complemented
49#22583537C2211yescharacteristic, complemented
50#22583536C2211yescharacteristic
51#22583529C2211yescharacteristic, complemented
52#22583529C2211yescharacteristic, complemented
53#22583528C2211yescharacteristic, complemented
54#22583526C2211yescharacteristic
55#22583520C2211yescharacteristic, complemented
562−1+4×C283C2211yescharacteristic
572−1+4×C283C2211yescharacteristic, complemented
58#137×C283—21no—
59#137×C283—21no—
60#137×C283C2211yescharacteristic
61#137×C283C2211yescharacteristic, complemented
62#135×C283—21no—
63#135×C283C2211yescharacteristic
64C2×Q16×C283—21no—
65C2×Q16×C283C2211yescharacteristic, complemented
66#131×C283C2211yescharacteristic, complemented
67#17985393—21no—
68#17985392—21no—
69#17985392—21no—
70#17985392C2311yescharacteristic
71#17985391—21no—
72#17985391—41no—
73C2×Dic566—21no—
74C2×Dic566C2311yescharacteristic
75Q8×D283—21no—
76C2×D1132—21no—
77C2×D1132C2311yescharacteristic
78D4×D283—21no—
79D4×D283—41no—
80#17985382—21no—
81#17985382—21no—
82#17985382—21no—
83#17985382—21no—
84#17985382—21no—
85#17985382—21no—
86#17985382D411yescharacteristic
87#17985381—21no—
88#17985381—21no—
89#17985380—21no—
90#17985380—21no—
91#17985380D411yescharacteristic
92#17985379—21no—
93#17985379—21no—
94#17985379—21no—
95#17985379—21no—
96#17985379—21no—
97#17985379—21no—
98#17985379D411yescharacteristic
99#17985377—21no—
100#17985377—21no—
101#17985377D411yescharacteristic
102#17985376—21no—
103#17985376—21no—
104#17985376C2311yescharacteristic
105#17985376C2311yescharacteristic
106#17985376C2311yescharacteristic
107#17985375—21no—
108C2×C283⋊C8—21no—
109C2×C283⋊C8—21no—
110C2×C283⋊C8C2311yescharacteristic
111C8×D283—21no—
112Dic283⋊C4—41no—
113(C2×D283)⋊C4—41no—
114C4×Dic283C2311yescharacteristic
115#41×C283—21no—
116#41×C283—21no—
117#41×C283—21no—
118#41×C283—21no—
119#41×C283—41no—
120#41×C283—41no—
121#41×C283C2311yescharacteristic
122#41×C283C2311yescharacteristic
123#41×C283C2311yescharacteristic
124C2×Q8×C283—21no—
125C2×Q8×C283—21no—
126C2×Q8×C283—21no—
127C2×Q8×C283—21no—
128C2×Q8×C283C2311yescharacteristic
129C2×Q8×C283C2311yescharacteristic
130Q16×C283—21no—
131Q16×C283—21no—
132Q16×C283—21no—
133Q16×C283—21no—
134Q16×C283D411yescharacteristic
135SD16×C283—21no—
136SD16×C283—21no—
137SD16×C283D411yescharacteristic
138SD16×C283D411yescharacteristic
139D8×C283D411yescharacteristic
140M16×C283—21no—
141M16×C283C2311yescharacteristic
142C2×C8×C283—21no—
143C2×C8×C283C2311yescharacteristic
144C22×D283—21no—
145C22×D283—41no—
146Dic566—21no—
147Dic566—21no—
148Dic566—41no—
149Dic566C2×D411yescharacteristic
150D1132—21no—
151D1132—21no—
152D1132—41no—
153D1132C2×D411yescharacteristic
154C22⋊D283—21no—
155C22⋊D283—41no—
156C22⋊D283—41no—
157C2×Dic283—21no—
158C2×Dic283—21no—
159C2×Dic283—41no—
160C4×D283—21no—
161C4×D283—41no—
162C4×D283—41no—
163C283⋊C8—21no—
164C283⋊C8—21no—
165C283⋊C8—21no—
166C283⋊C8C2×D411yescharacteristic
167C283⋊C8C2×D411yescharacteristic
168Q8×C283—21no—
169Q8×C283—21no—
170Q8×C283—21no—
171Q8×C283—21no—
172Q8×C283—41no—
173Q8×C283—41no—
174Q8×C283C2×D411yescharacteristic
175Q8×C283C2×D411yescharacteristic
176Q8×C283C2×D411yescharacteristic
177D4×C283—21no—
178D4×C283—21no—
179D4×C283—21no—
180D4×C283—41no—
181D4×C283—41no—
182D4×C283C2×D411yescharacteristic
183D4×C283C2×D411yescharacteristic
184D4×C283C2×D411yescharacteristic
185C2×C4×C283—21no—
186C2×C4×C283—21no—
187C2×C4×C283—21no—
188C2×C4×C283—21no—
189C2×C4×C283—21no—
190C2×C4×C283—41no—
191C2×C4×C283—41no—
192C2×C4×C283—41no—
193C2×C4×C283—41no—
194C2×C4×C283C2411yescharacteristic
195C8×C283—21no—
196C8×C283C2×D411yescharacteristic
197C8×C283C2×D411yescharacteristic
198C22×C283—21no—
199C22×C283—21no—
200C22×C283—21no—
201C22×C283H(5,𝔽2)11yescharacteristic
202C2×D283—21no—
203C2×D283—41no—
204C2×D283—81no—
205C2×D283—161no—
206C4×C283—21no—
207C4×C283—21no—
208C4×C283—21no—
209C4×C283—41no—
210C4×C283—41no—
211C4×C283C22×D411yescharacteristic
212C4×C283C22×D411yescharacteristic
213Dic283—21no—
214Dic283—41no—
215Dic283—41no—
216C2×C283—21no—
217C2×C283—41no—
218C2×C283—41no—
219C2×C283—41no—
220C2×C283D4211yescharacteristic
221D283—41no—
222D283—81no—
223C283#328911yescharacteristic, complemented, Sylow
224#3289—2831nomaximal, Sylow
225#578—2831no—
226#578D28311yescharacteristic, complemented
227#577—2831no—
228#577—2831no—
229#497—2831no—
230#473—2831no—
231#473—2831no—
232#471—2831no—
233#471—2831no—
234#456—2831no—
235#456—2831no—
236#455—2831no—
237#454—2831no—
238#444—2831no—
239#444—2831no—
2402−1+4—2831no—
2412−1+4C2×D28311yescharacteristic
2422−1+4C2×D28311yescharacteristic, complemented
243H(5,𝔽2)—2831no—
244#137—21no—
245#137—21no—
246#137—2831no—
247#137—2831no—
248#137—2831no—
249#137—2831no—
250#137—5661no—
251#137—5661no—
252#137—5661no—
253#137—5661no—
254#137C2×D28311yescharacteristic
255#137C2×D28311yescharacteristic, complemented
256AGL(1,ℤ/8)—2831no—
257AGL(1,ℤ/8)—2831no—
258AGL(1,ℤ/8)—5661no—
259AGL(1,ℤ/8)—5661no—
260#135—21no—
261#135—2831no—
262#135—5661no—
263#135—5661no—
264#135—5661no—
265#135C2×D28311yescharacteristic
266C2×Q16—21no—
267C2×Q16—2831no—
268C2×Q16—5661no—
269C2×Q16C2×D28311yescharacteristic, complemented
270C2×SD16—2831no—
271C2×SD16—2831no—
272C2×SD16—5661no—
273C2×SD16—5661no—
274#131—2831no—
275#131—2831no—
276#131—2831no—
277#131C2×D28311yescharacteristic, complemented
278C4⋊Q8—2831no—
279#124—2831no—
280#108—2831no—
281#108—2831no—
282#104—2831no—
283#104—2831no—
284#104—2831no—
285#104—2831no—
286#101—2831no—
287#101—2831no—
288#101—2831no—
289#100—2831no—
290#97—2831no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—41no—
296#41—41no—
297#41—2831no—
298#41—5661no—
299#41—5661no—
300#41—5661no—
301#41—5661no—
302#41—11321no—
303#41C22×D28311yescharacteristic
304#41C22×D28311yescharacteristic
305#41C22×D28311yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—2831no—
311C2×Q8—5661no—
312C2×Q8—5661no—
313C2×Q8C22×D28311yescharacteristic
314C2×Q8C22×D28311yescharacteristic
315C2×D4—2831no—
316C2×D4—5661no—
317C2×D4—5661no—
318C2×D4—11321no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—2831no—
324Q16—5661no—
325Q16—5661no—
326Q16—5661no—
327Q16—5661no—
328Q16—5661no—
329Q16—5661no—
330Q16C22⋊D28311yescharacteristic
331SD16—21no—
332SD16—21no—
333SD16—2831no—
334SD16—2831no—
335SD16—5661no—
336SD16—5661no—
337SD16—5661no—
338SD16—5661no—
339SD16—5661no—
340SD16—5661no—
341SD16—5661no—
342SD16—5661no—
343SD16—5661no—
344SD16—5661no—
345SD16C22⋊D28311yescharacteristic
346SD16C22⋊D28311yescharacteristic
347D8—2831no—
348D8—5661no—
349D8—5661no—
350D8C22⋊D28311yescharacteristic
351M16—21no—
352M16—2831no—
353M16—2831no—
354M16—2831no—
355M16—5661no—
356M16—5661no—
357M16—5661no—
358M16C22×D28311yescharacteristic
359C2×C8—21no—
360C2×C8—2831no—
361C2×C8—5661no—
362C2×C8—5661no—
363C2×C8—5661no—
364C2×C8C22×D28311yescharacteristic
365C4⋊C4—11321no—
366C22⋊C4—11321no—
367C42—2831no—
368C23—5661no—
369C23—11321no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—41no—
376Q8—2831no—
377Q8—5661no—
378Q8—5661no—
379Q8—11321no—
380Q8D4×D28311yescharacteristic
381Q8C2×C22⋊D28311yescharacteristic
382Q8C2×C22⋊D28311yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388D4—2831no—
389D4—5661no—
390D4—5661no—
391D4—5661no—
392D4—11321no—
393D4—11321no—
394D4—11321no—
395D4D4×D28311yescharacteristic
396D4C2×C22⋊D28311yescharacteristic
397D4C2×C22⋊D28311yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—5661no—
408C2×C4—5661no—
409C2×C4—5661no—
410C2×C4—11321no—
411C2×C4—11321no—
412C2×C4—11321no—
413C2×C4C23×D28311yescharacteristic
414C8—21no—
415C8—2831no—
416C8—2831no—
417C8—5661no—
418C8—5661no—
419C8—5661no—
420C8C2×C22⋊D28311yescharacteristic
421C8C2×C22⋊D28311yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—5661no—
426C22—11321no—
427C22—45281no—
428C22—22641no—
429C22#2258365511yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—5661no—
436C4—11321no—
437C4—11321no—
438C4C2×D4×D28311yescharacteristic
439C4C22×C22⋊D28311yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—11321no—
445C2—22641no—
446C2D4×C22⋊D28311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#578×C283C2
socleC2×C283D42
hypercenter#578D283
hypocenterC283#3289
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #31827280C2semidirect
2. #31827273C2semidirect
3. #31827272C2semidirect
4. #31826858C2semidirect
5. #31826738C2semidirect
6. #31826737C2semidirect
7. #31826730C2semidirect
8. #31826728C2semidirect
9. #31826648C2semidirect
10. #31826646C2semidirect
11. #31826641C2semidirect
12. #31826638C2semidirect
13. #31826587C2semidirect
14. #31826585C2semidirect
15. #578×C283C2semidirect
25. #22583640C22semidirect
35. C2×#17985382C22semidirect
41. #22583617C22semidirect
43. #22583608C22semidirect
45. #22583546C22semidirect
46. #22583545C22semidirect
48. #22583537C22semidirect
49. #22583537C22semidirect
51. #22583529C22semidirect
52. #22583529C22semidirect
53. #22583528C22semidirect
55. #22583520C22semidirect
57. 2−1+4×C283C22semidirect
61. #137×C283C22semidirect
65. C2×Q16×C283C22semidirect
66. #131×C283C22semidirect
223. C283#3289semidirect
226. #578D283semidirect
242. 2−1+4C2×D283semidirect
255. #137C2×D283semidirect
269. C2×Q16C2×D283semidirect
277. #131C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272802
#318272732
#318272722
#318268582
#318267382
#318267372
#318267302
#318267282
#318266482
#318266462
#318266412
#318266382
#318265872
#318265852
#578×C2832
#3289283