Finite Group Databasebeta

← #49132874

Subgroups — #49132874

Order 36224.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583637C211yesmaximal, characteristic, complemented
2C2×#22583635C211yesmaximal, characteristic, complemented
3#31827134C211yesmaximal, characteristic, complemented
4#31827099C211yesmaximal, characteristic, complemented
5#31826840C211yesmaximal, characteristic, complemented
6#31826717C211yesmaximal, characteristic, complemented
7#31826716C211yesmaximal, characteristic, complemented
8#31826663C211yesmaximal, characteristic, complemented
9#31826659C211yesmaximal, characteristic, complemented
10#31826613C211yesmaximal, characteristic, complemented
11#31826601C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826550C211yesmaximal, characteristic, complemented
14#31826540C211yesmaximal, characteristic, complemented
15#449×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C22×D1132C2211yescharacteristic
18C2×D4×D283—21no—
19C2×D4×D283C2211yescharacteristic
20#22583637—24no—
21#22583635—24no—
22C2×#17985379C2211yescharacteristic
23C2×#17985379C2211yescharacteristic, complemented
24C2×D2264C2211yescharacteristic
25C2×D2264C2211yescharacteristic, complemented
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic, complemented
28C2×#17985377C2211yescharacteristic, complemented
29C2×#17985376C2211yescharacteristic, complemented
30C2×#17985375C2211yescharacteristic, complemented
31#22583606—21no—
32#22583606C2211yescharacteristic
33#22583606C2211yescharacteristic, complemented
34#22583585C2211yescharacteristic
35#22583584—21no—
36#22583584C2211yescharacteristic, complemented
37#22583583—21no—
38#22583581—21no—
39#22583581C2211yescharacteristic, complemented
40#22583580—21no—
41#22583578—21no—
42Q8×Dic283C2211yescharacteristic, complemented
43#22583572C2211yescharacteristic
44C4×C22⋊D283—21no—
45C4×C22⋊D283C2211yescharacteristic
46#22583541C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583532C2211yescharacteristic
49#22583532C2211yescharacteristic, complemented
50#22583531C2211yescharacteristic, complemented
51#22583530C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53C8×Dic283C2211yescharacteristic, complemented
54C2×#41×C283C2211yescharacteristic, complemented
55C2×SD16×C283C2211yescharacteristic, complemented
56C2×D8×C283C2211yescharacteristic, complemented
57#121×C283C2211yescharacteristic
58#103×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic, complemented
60C22⋊C8×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63#17985393—22no—
64#17985393—41no—
65C2×D1132—22no—
66C2×D1132—21no—
67C2×D1132—21no—
68C2×D1132—21no—
69C2×D1132—41no—
70C2×D1132C2311yescharacteristic
71C2×D1132C2311yescharacteristic
72C2×D1132C2311yescharacteristic
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77C2×C22⋊D283—21no—
78C2×C22⋊D283—21no—
79C2×C22⋊D283—21no—
80C2×C22⋊D283D411yescharacteristic
81D4×D283—22no—
82D4×D283—41no—
83D4×D283—44no—
84C2×C4×D283—21no—
85C2×C4×D283—21no—
86C2×C4×D283C2311yescharacteristic
87#17985379—22no—
88#17985379D414yescomplemented
89D2264—22no—
90D2264—41no—
91#17985377—22no—
92#17985377—22no—
93#17985377—22no—
94#17985376—22no—
95#17985375—22no—
96C2×C283⋊C8C2311yescharacteristic
97C2×C283⋊C8C2311yescharacteristic
98C4⋊Dic283—21no—
99C4⋊Dic283C2311yescharacteristic
100Dic283⋊C4D411yescharacteristic
101C22⋊Dic283—21no—
102C22⋊Dic283D411yescharacteristic
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4D411yescharacteristic
107C4×Dic283—21no—
108C4×Dic283C2311yescharacteristic
109#41×C283—22no—
110#41×C283—41no—
111C2×Q8×C283C2311yescharacteristic
112C2×D4×C283—21no—
113C2×D4×C283—21no—
114C2×D4×C283C2311yescharacteristic
115C2×D4×C283C2311yescharacteristic
116C22×C4×C283—21no—
117C22×C4×C283C2311yescharacteristic
118SD16×C283—22no—
119D8×C283—41no—
120C2×C8×C283C2311yescharacteristic
121C2×C8×C283C2311yescharacteristic
122C4⋊C4×C283C2311yescharacteristic
123C22⋊C4×C283—21no—
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—21no—
127C22×D283—42no—
128C22×D283—42no—
129C22×D283—42no—
130C22×D283—42no—
131C22×D283—81no—
132C22×D283—81no—
133C22×D283C2×D411yescharacteristic
134D1132—21no—
135D1132—21no—
136D1132—21no—
137D1132—21no—
138D1132—21no—
139D1132—42no—
140D1132—42no—
141D1132—42no—
142D1132C2×D412yes—
143C22⋊D283—22no—
144C22⋊D283—42no—
145C22⋊D283—42no—
146C22⋊D283—42no—
147C22⋊D283—81no—
148C22⋊D283—81no—
149C22⋊D283—81no—
150C22⋊D283—81no—
151C2×Dic283—21no—
152C2×Dic283C2×D411yescharacteristic
153C2×Dic283C2×D411yescharacteristic
154C2×Dic283C2×D411yescharacteristic
155C4×D283—21no—
156C4×D283—21no—
157C4×D283—42no—
158C4×D283—42no—
159C283⋊C8—21no—
160C283⋊C8C2×D412yescomplemented
161C23×C283—21no—
162C23×C283—21no—
163C23×C283C2×D411yescharacteristic
164Q8×C283—21no—
165Q8×C283C2×D412yescomplemented
166D4×C283—21no—
167D4×C283—21no—
168D4×C283—21no—
169D4×C283—21no—
170D4×C283—42no—
171D4×C283—42no—
172C2×C4×C283—21no—
173C2×C4×C283—21no—
174C2×C4×C283—21no—
175C2×C4×C283—21no—
176C2×C4×C283—42no—
177C2×C4×C283C2×D411yescharacteristic
178C2×C4×C283C2411yescharacteristic
179C8×C283—21no—
180C8×C283—21no—
181C22×C283—21no—
182C22×C283—21no—
183C22×C283—21no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283—41no—
188C22×C283—41no—
189C22×C283—41no—
190C22×C283C22×D411yescharacteristic
191C2×D283—21no—
192C2×D283—22no—
193C2×D283—21no—
194C2×D283—21no—
195C2×D283—41no—
196C2×D283—41no—
197C2×D283—41no—
198C2×D283—41no—
199C2×D283—41no—
200C2×D283—41no—
201C2×D283—41no—
202C2×D283—41no—
203C2×D283—84no—
204C2×D283—84no—
205C4×C283—21no—
206C4×C283—22no—
207C4×C283—41no—
208C4×C283C22×D411yescharacteristic
209C4×C283H(5,𝔽2)11yescharacteristic
210Dic283—21no—
211Dic283—21no—
212Dic283—41no—
213Dic283AGL(1,ℤ/8)12yescomplemented
214C2×C283—41no—
215C2×C283—81no—
216C2×C283—81no—
217C2×C283D4211yescharacteristic
218C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C283#57611yescharacteristic
220D283—41no—
221D283—42no—
222D283—81no—
223D283—81no—
224C283#327411yescharacteristic, complemented, Sylow
225#3274—2831nomaximal, Sylow
226C2×AGL(1,ℤ/8)—2831no—
227C2×AGL(1,ℤ/8)—2831no—
228#550—2831no—
229D42—2831no—
230#493—2831no—
231#469—2831no—
232#469—2831no—
233#460—2831no—
234#459—2831no—
235#449—2831no—
236#449D28311yescharacteristic, complemented
237#447—2831no—
238#447—2831no—
239C4×SD16—2831no—
240#436—2831no—
241C2×#41—2831no—
242C2×#41C2×D28311yescharacteristic, complemented
243C22×D4—2831no—
244C22×D4—2831no—
245C22×D4—5661no—
246AGL(1,ℤ/8)—5664no—
247AGL(1,ℤ/8)—5664no—
248C2×SD16—2831no—
249C2×SD16—2831no—
250C2×SD16C2×D28311yescharacteristic, complemented
251C2×D8—2831no—
252C2×D8—2831no—
253C2×D8—2831no—
254C2×D8—2831no—
255C2×D8—2831no—
256C2×D8C2×D28311yescharacteristic, complemented
257C2×M16—2831no—
258C2×M16—2831no—
259#127—2831no—
260#127—2831no—
261#127—5661no—
262#121—2831no—
263#121—2831no—
264#121—2831no—
265#121—5661no—
266#121—5661no—
267#121—5661no—
268#121C2×D28311yescharacteristic
269#120—5661no—
270#120—5661no—
271C4×Q8—2831no—
272C4×D4—2831no—
273C4×D4—2831no—
274C4×D4—5661no—
275#106—2831no—
276C4⋊C8—2831no—
277#103C2×D28311yescharacteristic, complemented
278#102—2831no—
279#102—2831no—
280#102—2831no—
281#102—2831no—
282#102C2×D28311yescharacteristic, complemented
283C22⋊C8—2831no—
284C22⋊C8C2×D28311yescharacteristic, complemented
285C4×C8—2831no—
286C24—5661no—
287C24—5661no—
288#41—22no—
289#41—41no—
290#41—5662no—
291#41—11321no—
292C2×Q8C22×D28311yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2831no—
296C2×D4—2831no—
297C2×D4—2831no—
298C2×D4—2831no—
299C2×D4—5662no—
300C2×D4—5661no—
301C2×D4—5661no—
302C2×D4—5661no—
303C2×D4—5661no—
304C2×D4—5661no—
305C2×D4—5661no—
306C2×D4—5661no—
307C2×D4—5662no—
308C2×D4—5661no—
309C2×D4—5661no—
310C2×D4—5661no—
311C2×D4—11321no—
312C2×D4—11321no—
313C2×D4—11324no—
314C2×D4C22×D28311yescharacteristic
315C2×D4C22×D28311yescharacteristic
316C22×C4—21no—
317C22×C4—2831no—
318C22×C4—5661no—
319C22×C4—5661no—
320C22×C4C22×D28311yescharacteristic
321SD16—22no—
322SD16—2834no—
323SD16—5662no—
324D8—41no—
325D8—5662no—
326D8—5662no—
327D8—5662no—
328D8—5662no—
329D8—11321no—
330M16—5662no—
331M16—5662no—
332C2×C8—2831no—
333C2×C8—2831no—
334C2×C8C22×D28311yescharacteristic
335C2×C8C22×D28311yescharacteristic
336C4⋊C4—2831no—
337C4⋊C4—2831no—
338C4⋊C4—5661no—
339C4⋊C4C22×D28311yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—2831no—
342C22⋊C4—2831no—
343C22⋊C4—5661no—
344C22⋊C4—5661no—
345C22⋊C4—5661no—
346C22⋊C4—5661no—
347C42—2831no—
348C42—5661no—
349C23—21no—
350C23—21no—
351C23—2831no—
352C23—5661no—
353C23—5661no—
354C23—5661no—
355C23—11322no—
356C23—11322no—
357C23—11322no—
358C23—11322no—
359C23—22641no—
360C23—22641no—
361C23D4×D28311yescharacteristic
362Q8—21no—
363Q8D4×D28312yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—42no—
370D4—2832no—
371D4—5662no—
372D4—5661no—
373D4—5661no—
374D4—5661no—
375D4—5661no—
376D4—5661no—
377D4—11322no—
378D4—11322no—
379D4—11322no—
380D4—11322no—
381D4—11322no—
382D4—11322no—
383D4—22641no—
384D4—22641no—
385D4—22641no—
386D4—22641no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—2831no—
393C2×C4—2831no—
394C2×C4—2831no—
395C2×C4—5661no—
396C2×C4—5661no—
397C2×C4—5661no—
398C2×C4—11322no—
399C2×C4—11322no—
400C2×C4D4×D28311yescharacteristic
401C2×C4C23×D28311yescharacteristic
402C8—21no—
403C8—21no—
404C8—2832no—
405C8—5661no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—5661no—
416C22—5661no—
417C22—5661no—
418C22—5662no—
419C22—11321no—
420C22—11321no—
421C22—11321no—
422C22—11321no—
423C22—11321no—
424C22—11321no—
425C22—11321no—
426C22—11321no—
427C22—22644no—
428C22—22644no—
429C22C2×D4×D28311yescharacteristic
430C4—21no—
431C4—22no—
432C4—41no—
433C4—2832no—
434C4—5661no—
435C4—5661no—
436C4—11321no—
437C4C2×D4×D28311yescharacteristic
438C4#2258365511yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—11321no—
443C2—11322no—
444C2—22641no—
445C2—22641no—
446C2#3182709711yescharacteristic
447C2AGL(1,ℤ/8)×D28311yescharacteristic
448C2#3182726711yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#449×C283C2
socleC22×C283C22×D4
hypercenter#449D283
hypocenterC283#3274
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×#22583637C2semidirect
2. C2×#22583635C2semidirect
3. #31827134C2semidirect
4. #31827099C2semidirect
5. #31826840C2semidirect
6. #31826717C2semidirect
7. #31826716C2semidirect
8. #31826663C2semidirect
9. #31826659C2semidirect
10. #31826613C2semidirect
11. #31826601C2semidirect
12. #31826597C2semidirect
13. #31826550C2semidirect
14. #31826540C2semidirect
15. #449×C283C2semidirect
16. C2×#17985393C22semidirect
23. C2×#17985379C22semidirect
25. C2×D2264C22semidirect
27. C2×#17985377C22semidirect
28. C2×#17985377C22semidirect
29. C2×#17985376C22semidirect
30. C2×#17985375C22semidirect
33. #22583606C22semidirect
36. #22583584C22semidirect
39. #22583581C22semidirect
42. Q8×Dic283C22semidirect
46. #22583541C22semidirect
47. Dic283⋊C8C22semidirect
49. #22583532C22semidirect
50. #22583531C22semidirect
51. #22583530C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. C8×Dic283C22semidirect
54. C2×#41×C283C22semidirect
55. C2×SD16×C283C22semidirect
56. C2×D8×C283C22semidirect
58. #103×C283C22semidirect
59. #102×C283C22semidirect
60. C22⋊C8×C283C22semidirect
88. #17985379D4semidirect
160. C283⋊C8C2×D4semidirect
165. Q8×C283C2×D4semidirect
213. Dic283AGL(1,ℤ/8)semidirect
224. C283#3274semidirect
236. #449D283semidirect
242. C2×#41C2×D283semidirect
250. C2×SD16C2×D283semidirect
256. C2×D8C2×D283semidirect
277. #103C2×D283semidirect
282. #102C2×D283semidirect
284. C22⋊C8C2×D283semidirect
363. Q8D4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C2×#225836352
#318271342
#318270992
#318268402
#318267172
#318267162
#318266632
#318266592
#318266132
#318266012
#318265972
#318265502
#318265402
#449×C2832
#3274283