Finite Group Databasebeta

← #49132805

Subgroups — #49132805

Order 36224.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583634C211yesmaximal, characteristic, complemented
2C2×#22583630C211yesmaximal, characteristic, complemented
3#31827133C211yesmaximal, characteristic, complemented
4#31827103C211yesmaximal, characteristic, complemented
5#31826851C211yesmaximal, characteristic, complemented
6#31826714C211yesmaximal, characteristic, complemented
7#31826699C211yesmaximal, characteristic, complemented
8#31826690C211yesmaximal, characteristic, complemented
9#31826658C211yesmaximal, characteristic, complemented
10#31826624C211yesmaximal, characteristic, complemented
11#31826612C211yesmaximal, characteristic, complemented
12#31826600C211yesmaximal, characteristic, complemented
13#31826562C211yesmaximal, characteristic, complemented
14#31826533C211yesmaximal, characteristic, complemented
15#449×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic, complemented
17C2×#17985391C2211yescharacteristic
18C2×D4×D283C2211yescharacteristic
19#22583634—24no—
20#22583630—24no—
21C2×#17985382C2211yescharacteristic, complemented
22C2×#17985381C2211yescharacteristic
23C2×#17985380C2211yescharacteristic
24C2×#17985379C2211yescharacteristic, complemented
25C2×D2264C2211yescharacteristic
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic
28C2×#17985375C2211yescharacteristic
29C22×C283⋊C8C2211yescharacteristic, complemented
30#22583606—21no—
31#22583606C2211yescharacteristic, complemented
32#22583599C2211yescharacteristic
33#22583593—21no—
34#22583586C2211yescharacteristic
35#22583585C2211yescharacteristic
36#22583584C2211yescharacteristic
37#22583583—21no—
38#22583582—21no—
39#22583582—21no—
40#22583578—21no—
41#22583575C2211yescharacteristic, complemented
42D4×Dic283C2211yescharacteristic
43C4×C22⋊D283C2211yescharacteristic
44(C22⋊D283)⋊C4—21no—
45C2×(C2×D283)⋊C4—21no—
46#22583543C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583535C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic, complemented
50#22583531C2211yescharacteristic, complemented
51#22583530C2211yescharacteristic
52(C2×D283)⋊C8C2211yescharacteristic
53#22583520C2211yescharacteristic, complemented
54C2×#41×C283C2211yescharacteristic
55C2×SD16×C283C2211yescharacteristic, complemented
56C2×D8×C283C2211yescharacteristic
57#121×C283C2211yescharacteristic, complemented
58#103×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic
60C22⋊C8×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985392—22no—
63#17985392—41no—
64#17985391—22no—
65#17985391—41no—
66C2×Dic566C2311yescharacteristic
67C2×D1132—21no—
68C2×D1132C2311yescharacteristic
69C2×D1132C2311yescharacteristic
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283D411yescharacteristic
77D4×D283—22no—
78D4×D283—41no—
79C22×Dic283—21no—
80C22×Dic283—21no—
81C2×C4×D283—21no—
82C2×C4×D283C2311yescharacteristic
83#17985382—22no—
84#17985381—22no—
85#17985380D414yescomplemented
86#17985379—22no—
87D2264—41no—
88#17985377—22no—
89#17985377—22no—
90#17985375—22no—
91C2×C283⋊C8—22no—
92C2×C283⋊C8C2311yescharacteristic
93C2×C283⋊C8C2311yescharacteristic
94C4⋊Dic283C2311yescharacteristic
95Dic283⋊C4—21no—
96Dic283⋊C4—21no—
97Dic283⋊C4D411yescharacteristic
98C22⋊Dic283—21no—
99C22⋊Dic283D411yescharacteristic
100(C2×D283)⋊C4—21no—
101(C2×D283)⋊C4—21no—
102(C2×D283)⋊C4—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4D411yescharacteristic
106C4×Dic283—21no—
107C4×Dic283C2311yescharacteristic
108#41×C283—22no—
109#41×C283—41no—
110C2×Q8×C283C2311yescharacteristic
111C2×D4×C283—21no—
112C2×D4×C283—21no—
113C2×D4×C283C2311yescharacteristic
114C2×D4×C283C2311yescharacteristic
115C22×C4×C283—21no—
116C22×C4×C283C2311yescharacteristic
117SD16×C283—41no—
118D8×C283—22no—
119C2×C8×C283C2311yescharacteristic
120C2×C8×C283C2311yescharacteristic
121C4⋊C4×C283C2311yescharacteristic
122C22⋊C4×C283—21no—
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—81no—
128C22×D283C2×D411yescharacteristic
129Dic566—21no—
130Dic566C2×D412yes—
131D1132—21no—
132D1132—21no—
133D1132—21no—
134D1132—21no—
135D1132—81no—
136C22⋊D283—41no—
137C22⋊D283—42no—
138C22⋊D283—42no—
139C22⋊D283—42no—
140C22⋊D283—42no—
141C22⋊D283—81no—
142C22⋊D283—81no—
143C2×Dic283—21no—
144C2×Dic283—21no—
145C2×Dic283—42no—
146C2×Dic283—81no—
147C2×Dic283C2×D411yescharacteristic
148C2×Dic283C2×D411yescharacteristic
149C2×Dic283C2×D411yescharacteristic
150C4×D283—21no—
151C4×D283—21no—
152C4×D283—42no—
153C283⋊C8—21no—
154C283⋊C8C2×D412yes—
155C23×C283—21no—
156C23×C283—21no—
157C23×C283C2×D411yescharacteristic
158Q8×C283—21no—
159Q8×C283—21no—
160D4×C283—21no—
161D4×C283—21no—
162D4×C283—21no—
163D4×C283—42no—
164D4×C283—81no—
165D4×C283C2×D412yes—
166C2×C4×C283—21no—
167C2×C4×C283—21no—
168C2×C4×C283—21no—
169C2×C4×C283—21no—
170C2×C4×C283—42no—
171C2×C4×C283C2×D411yescharacteristic
172C2×C4×C283C2411yescharacteristic
173C8×C283—21no—
174C8×C283—21no—
175C22×C283—21no—
176C22×C283—22no—
177C22×C283—21no—
178C22×C283—21no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283C22×D411yescharacteristic
185C2×D283—21no—
186C2×D283—21no—
187C2×D283—21no—
188C2×D283—41no—
189C2×D283—41no—
190C2×D283—41no—
191C2×D283—41no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—84no—
195C4×C283—21no—
196C4×C283—41no—
197C4×C283—41no—
198C4×C283C22×D411yescharacteristic
199C4×C283H(5,𝔽2)11yescharacteristic
200Dic283—21no—
201Dic283—22no—
202Dic283—21no—
203Dic283—21no—
204Dic283—41no—
205C2×C283—42no—
206C2×C283—41no—
207C2×C283—81no—
208C2×C283D4211yescharacteristic
209C2×C283#57511yescharacteristic
210C2×C283#57611yescharacteristic
211D283—41no—
212D283—81no—
213D283—81no—
214C283#326811yescharacteristic, complemented, Sylow
215#3268—2831nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2831no—
217C2×#135—2831no—
218#550—2831no—
219#546—2831no—
220#496—2831no—
221#468—2831no—
222#466—2831no—
223#464—2831no—
224#459—2831no—
225#452—2831no—
226#449—2831no—
227#449D28311yescharacteristic, complemented
228#447—2831no—
229#440—2831no—
230#435—2831no—
231C2×#41—2831no—
232C2×#41—2831no—
233C2×#41C2×D28311yescharacteristic
234C22×D4—2831no—
235AGL(1,ℤ/8)—5664no—
236#135—5664no—
237C2×Q16—2831no—
238C2×SD16—2831no—
239C2×SD16—2831no—
240C2×SD16—2831no—
241C2×SD16C2×D28311yescharacteristic, complemented
242C2×D8—2831no—
243C2×D8—2831no—
244C2×D8—2831no—
245C2×D8C2×D28311yescharacteristic
246C2×M16—2831no—
247C22×C8—2831no—
248#127—2831no—
249#127—5661no—
250#124—2831no—
251#123—5661no—
252C22⋊Q8—2831no—
253#121—2831no—
254#121—2831no—
255#121—5661no—
256#121—5661no—
257#121—5661no—
258#121C2×D28311yescharacteristic, complemented
259#120—5661no—
260C4×Q8—2831no—
261C4×D4—2831no—
262C4×D4—2831no—
263C4×D4—5661no—
264C2×C22⋊C4—5661no—
265#107—2831no—
266C4⋊C8—2831no—
267#103—2831no—
268#103C2×D28311yescharacteristic, complemented
269#102—2831no—
270#102—2831no—
271#102—2831no—
272#102C2×D28311yescharacteristic
273C22⋊C8—2831no—
274C22⋊C8C2×D28311yescharacteristic, complemented
275#97—2831no—
276C24—5661no—
277#41—22no—
278#41—41no—
279#41—5662no—
280#41—5662no—
281#41—11321no—
282#41—11321no—
283C2×Q8—2831no—
284C2×Q8C22×D28311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2831no—
288C2×D4—2831no—
289C2×D4—2831no—
290C2×D4—5661no—
291C2×D4—5661no—
292C2×D4—5661no—
293C2×D4—5661no—
294C2×D4—5662no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—5661no—
298C2×D4—11321no—
299C2×D4C22×D28311yescharacteristic
300C2×D4C22×D28311yescharacteristic
301C22×C4—21no—
302C22×C4—2831no—
303C22×C4—5661no—
304C22×C4—5661no—
305C22×C4—5661no—
306C22×C4C22×D28311yescharacteristic
307Q16—5662no—
308SD16—41no—
309SD16—2834no—
310SD16—5662no—
311SD16—5662no—
312D8—22no—
313D8—5662no—
314D8—5662no—
315D8—11321no—
316M16—5662no—
317C2×C8—2831no—
318C2×C8—2831no—
319C2×C8—5662no—
320C2×C8C22×D28311yescharacteristic
321C2×C8C22×D28311yescharacteristic
322C4⋊C4—2831no—
323C4⋊C4—2831no—
324C4⋊C4—5661no—
325C4⋊C4—5661no—
326C4⋊C4C22×D28311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—2831no—
329C22⋊C4—2831no—
330C22⋊C4—5661no—
331C22⋊C4—5661no—
332C22⋊C4—5661no—
333C22⋊C4—5661no—
334C22⋊C4—5661no—
335C22⋊C4—5661no—
336C42—2831no—
337C42—5661no—
338C23—21no—
339C23—21no—
340C23—2831no—
341C23—5661no—
342C23—5661no—
343C23—11322no—
344C23—11322no—
345C23—22641no—
346C23D4×D28311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—2832no—
350Q8—5661no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—81no—
356D4—5661no—
357D4—5661no—
358D4—5661no—
359D4—5661no—
360D4—11321no—
361D4—11322no—
362D4—11322no—
363D4—11322no—
364D4—11322no—
365D4—22641no—
366D4—22641no—
367D4—22641no—
368D4D4×D28312yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—2831no—
375C2×C4—2831no—
376C2×C4—2831no—
377C2×C4—5661no—
378C2×C4—5661no—
379C2×C4—5661no—
380C2×C4—5661no—
381C2×C4—11322no—
382C2×C4—11322no—
383C2×C4—22641no—
384C2×C4D4×D28311yescharacteristic
385C2×C4C23×D28311yescharacteristic
386C8—21no—
387C8—21no—
388C8—2832no—
389C8—5661no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—5661no—
400C22—5661no—
401C22—5661no—
402C22—11321no—
403C22—11321no—
404C22—11321no—
405C22—11321no—
406C22—11321no—
407C22—11321no—
408C22—22644no—
409C22C2×D4×D28311yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—5661no—
414C4—5661no—
415C4—5661no—
416C4—5662no—
417C4—11321no—
418C4C2×D4×D28311yescharacteristic
419C4#2258365511yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—11321no—
424C2—22641no—
425C2—22641no—
426C2#3182709711yescharacteristic
427C2#3182726211yescharacteristic
428C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#449×C283C2
socleC22×C283C22×D4
hypercenter#449D283
hypocenterC283#3268
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×#22583634C2semidirect
2. C2×#22583630C2semidirect
3. #31827133C2semidirect
4. #31827103C2semidirect
5. #31826851C2semidirect
6. #31826714C2semidirect
7. #31826699C2semidirect
8. #31826690C2semidirect
9. #31826658C2semidirect
10. #31826624C2semidirect
11. #31826612C2semidirect
12. #31826600C2semidirect
13. #31826562C2semidirect
14. #31826533C2semidirect
15. #449×C283C2semidirect
16. C2×#17985392C22semidirect
21. C2×#17985382C22semidirect
24. C2×#17985379C22semidirect
29. C22×C283⋊C8C22semidirect
31. #22583606C22semidirect
41. #22583575C22semidirect
46. #22583543C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583535C22semidirect
49. #22583532C22semidirect
50. #22583531C22semidirect
53. #22583520C22semidirect
55. C2×SD16×C283C22semidirect
57. #121×C283C22semidirect
58. #103×C283C22semidirect
60. C22⋊C8×C283C22semidirect
85. #17985380D4semidirect
214. C283#3268semidirect
227. #449D283semidirect
241. C2×SD16C2×D283semidirect
258. #121C2×D283semidirect
268. #103C2×D283semidirect
274. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836342
C2×#225836302
#318271332
#318271032
#318268512
#318267142
#318266992
#318266902
#318266582
#318266242
#318266122
#318266002
#318265622
#318265332
#449×C2832
#3268283