Finite Group Databasebeta

← #49132804

Subgroups — #49132804

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×#22583635C211yesmaximal, characteristic, complemented
2C2×#22583633C211yesmaximal, characteristic, complemented
3#31827134C211yesmaximal, characteristic, complemented
4#31827106C211yesmaximal, characteristic, complemented
5#31826851C211yesmaximal, characteristic, complemented
6#31826713C211yesmaximal, characteristic, complemented
7#31826700C211yesmaximal, characteristic, complemented
8#31826689C211yesmaximal, characteristic, complemented
9#31826659C211yesmaximal, characteristic, complemented
10#31826613C211yesmaximal, characteristic, complemented
11#31826609C211yesmaximal, characteristic, complemented
12#31826601C211yesmaximal, characteristic, complemented
13#31826568C211yesmaximal, characteristic, complemented
14#31826539C211yesmaximal, characteristic, complemented
15#452×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985393C2211yescharacteristic, complemented
18C22×D1132C2211yescharacteristic
19#22583635—24no—
20#22583633—24no—
21C2×#17985379C2211yescharacteristic
22C2×#17985379C2211yescharacteristic
23C2×#17985379C2211yescharacteristic, complemented
24C2×D2264C2211yescharacteristic
25C2×D2264C2211yescharacteristic
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic
28C2×#17985376C2211yescharacteristic
29C2×C8×D283C2211yescharacteristic, complemented
30#22583606—21no—
31#22583606C2211yescharacteristic, complemented
32#22583599C2211yescharacteristic
33#22583595—21no—
34#22583585—21no—
35#22583585C2211yescharacteristic
36#22583584—21no—
37#22583584C2211yescharacteristic
38#22583581—21no—
39#22583581C2211yescharacteristic, complemented
40#22583580—21no—
41Q8×Dic283C2211yescharacteristic, complemented
42#22583572C2211yescharacteristic
43C4×C22⋊D283—21no—
44C4×C22⋊D283C2211yescharacteristic
45C22⋊C4×D283—21no—
46#22583543C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583534C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic
50#22583532C2211yescharacteristic
51#22583530C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583520C2211yescharacteristic, complemented
54C2×#41×C283C2211yescharacteristic
55C2×Q16×C283C2211yescharacteristic, complemented
56C2×SD16×C283C2211yescharacteristic, complemented
57C22⋊Q8×C283C2211yescharacteristic
58#103×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic, complemented
60C22⋊C8×C283C2211yescharacteristic
61C23×D283—21no—
62#17985393—22no—
63#17985393—22no—
64#17985393—41no—
65#17985393—41no—
66C2×D1132—22no—
67C2×D1132—21no—
68C2×D1132—21no—
69C2×D1132—21no—
70C2×D1132—21no—
71C2×D1132—41no—
72C2×D1132C2311yescharacteristic
73C2×D1132C2311yescharacteristic
74C2×D1132C2311yescharacteristic
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77C2×C22⋊D283—21no—
78C2×C22⋊D283—21no—
79C2×C22⋊D283D411yescharacteristic
80C2×C4×D283—21no—
81C2×C4×D283—21no—
82C2×C4×D283—21no—
83C2×C4×D283C2311yescharacteristic
84#17985379—22no—
85#17985379—22no—
86#17985379D414yescomplemented
87D2264—22no—
88D2264—41no—
89#17985377—22no—
90#17985377—22no—
91#17985376—22no—
92C2×C283⋊C8C2311yescharacteristic
93C2×C283⋊C8C2311yescharacteristic
94C8×D283—22no—
95C4⋊Dic283—21no—
96C4⋊Dic283C2311yescharacteristic
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×Q8×C283C2311yescharacteristic
112C2×D4×C283—21no—
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283—21no—
115C22×C4×C283C2311yescharacteristic
116Q16×C283—22no—
117SD16×C283—41no—
118C2×C8×C283C2311yescharacteristic
119C2×C8×C283C2311yescharacteristic
120C4⋊C4×C283—21no—
121C4⋊C4×C283C2311yescharacteristic
122C22⋊C4×C283—21no—
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—42no—
127C22×D283—42no—
128C22×D283—81no—
129C22×D283C2×D411yescharacteristic
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132—21no—
134D1132—21no—
135D1132—42no—
136D1132—42no—
137D1132—42no—
138D1132—81no—
139D1132C2×D412yes—
140C22⋊D283—41no—
141C22⋊D283—42no—
142C22⋊D283—81no—
143C22⋊D283—81no—
144C22⋊D283—81no—
145C2×Dic283—21no—
146C2×Dic283C2×D411yescharacteristic
147C2×Dic283C2×D411yescharacteristic
148C2×Dic283C2×D411yescharacteristic
149C4×D283—21no—
150C4×D283—21no—
151C4×D283—42no—
152C4×D283—42no—
153C4×D283—81no—
154C283⋊C8—21no—
155C283⋊C8C2×D412yes—
156C23×C283—21no—
157C23×C283C2×D411yescharacteristic
158Q8×C283—21no—
159Q8×C283—21no—
160Q8×C283—21no—
161Q8×C283C2×D412yes—
162D4×C283—21no—
163D4×C283—21no—
164D4×C283—42no—
165C2×C4×C283—21no—
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—21no—
177C22×C283—21no—
178C22×C283—41no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283C22×D411yescharacteristic
182C2×D283—21no—
183C2×D283—22no—
184C2×D283—21no—
185C2×D283—21no—
186C2×D283—41no—
187C2×D283—41no—
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—22no—
197C4×C283—41no—
198C4×C283—41no—
199C4×C283C22×D411yescharacteristic
200C4×C283H(5,𝔽2)11yescharacteristic
201Dic283—21no—
202Dic283—21no—
203Dic283—21no—
204Dic283—41no—
205C2×C283—41no—
206C2×C283—81no—
207C2×C283D4211yescharacteristic
208C2×C283#57511yescharacteristic
209C2×C283#57611yescharacteristic
210D283—41no—
211D283—42no—
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#452D28311yescharacteristic, complemented
226#449—2831no—
227#449—2831no—
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×Q16C2×D28311yescharacteristic, complemented
238C2×SD16—2831no—
239C2×SD16—2831no—
240C2×SD16—2831no—
241C2×SD16C2×D28311yescharacteristic, complemented
242C2×D8—2831no—
243C2×D8—2831no—
244C2×D8—2831no—
245C2×D8—2831no—
246C2×M16—2831no—
247C22×C8—2831no—
248#127—2831no—
249#127—5661no—
250#124—2831no—
251#123—5661no—
252C22⋊Q8C2×D28311yescharacteristic
253#121—2831no—
254#121—2831no—
255#121—2831no—
256#121—5661no—
257#121—5661no—
258#121—5661no—
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, complemented
273C22⋊C8—2831no—
274C22⋊C8C2×D28311yescharacteristic
275#97—2831no—
276C24—5661no—
277#41—22no—
278#41—41no—
279#41—5662no—
280#41—5662no—
281#41—11321no—
282#41—11321no—
283C2×Q8C22×D28311yescharacteristic
284C2×Q8C22×D28311yescharacteristic
285C2×D4—21no—
286C2×D4—2831no—
287C2×D4—2831no—
288C2×D4—2831no—
289C2×D4—2831no—
290C2×D4—5662no—
291C2×D4—5661no—
292C2×D4—5661no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—5661no—
298C2×D4—5661no—
299C2×D4—11321no—
300C2×D4C22×D28311yescharacteristic
301C22×C4—21no—
302C22×C4—2831no—
303C22×C4—5661no—
304C22×C4—5661no—
305C22×C4—5661no—
306C22×C4C22×D28311yescharacteristic
307Q16—22no—
308SD16—41no—
309SD16—2834no—
310SD16—5662no—
311SD16—5662no—
312D8—5662no—
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—21no—
323C4⋊C4—2831no—
324C4⋊C4—2831no—
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—2831no—
340C23—5661no—
341C23—5661no—
342C23—5661no—
343C23—11322no—
344C23—11322no—
345C23—22641no—
346C23D4×D28311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8D4×D28312yes—
351D4—21no—
352D4—21no—
353D4—42no—
354D4—2832no—
355D4—5661no—
356D4—5661no—
357D4—5661no—
358D4—5661no—
359D4—5661no—
360D4—11322no—
361D4—11322no—
362D4—11322no—
363D4—11321no—
364D4—11322no—
365D4—22641no—
366D4—22641no—
367D4—22641no—
368D4—22641no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—2831no—
376C2×C4—2831no—
377C2×C4—2831no—
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—41no—
394C22—41no—
395C22—41no—
396C22—5661no—
397C22—5661no—
398C22—5661no—
399C22—5662no—
400C22—11321no—
401C22—11321no—
402C22—11321no—
403C22—11321no—
404C22—11321no—
405C22—11321no—
406C22—11321no—
407C22—11321no—
408C22—22644no—
409C22C2×D4×D28311yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—5661no—
415C4—5661no—
416C4—5661no—
417C4—11321no—
418C4C2×D4×D28311yescharacteristic
419C4#2258365511yescharacteristic
420C2—41no—
421C2—81no—
422C2—11321no—
423C2—11322no—
424C2—22641no—
425C2—22641no—
426C2#3182709711yescharacteristic
427C2#3182726411yescharacteristic
428C2#3182726711yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#452×C283C2
socleC22×C283C22×D4
hypercenter#452D283
hypocenterC283#3268
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C2×#22583633C2semidirect
3. #31827134C2semidirect
4. #31827106C2semidirect
5. #31826851C2semidirect
6. #31826713C2semidirect
7. #31826700C2semidirect
8. #31826689C2semidirect
9. #31826659C2semidirect
10. #31826613C2semidirect
11. #31826609C2semidirect
12. #31826601C2semidirect
13. #31826568C2semidirect
14. #31826539C2semidirect
15. #452×C283C2semidirect
17. C2×#17985393C22semidirect
23. C2×#17985379C22semidirect
29. C2×C8×D283C22semidirect
31. #22583606C22semidirect
39. #22583581C22semidirect
41. Q8×Dic283C22semidirect
46. #22583543C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583534C22semidirect
51. #22583530C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583520C22semidirect
55. C2×Q16×C283C22semidirect
56. C2×SD16×C283C22semidirect
58. #103×C283C22semidirect
59. #102×C283C22semidirect
86. #17985379D4semidirect
214. C283#3268semidirect
225. #452D283semidirect
237. C2×Q16C2×D283semidirect
241. C2×SD16C2×D283semidirect
268. #103C2×D283semidirect
272. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C2×#225836332
#318271342
#318271062
#318268512
#318267132
#318267002
#318266892
#318266592
#318266132
#318266092
#318266012
#318265682
#318265392
#452×C2832
#3268283