Finite Group Databasebeta

← #49130875

Subgroups — #49130875

Order 36224.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583637C211yesmaximal, characteristic, complemented
2C2×M16×D283C211yesmaximal, characteristic, complemented
3#31826978C211yesmaximal, characteristic, complemented
4#121×D283C211yesmaximal, characteristic, complemented
5#31826718C211yesmaximal, characteristic, complemented
6#31826716C211yesmaximal, characteristic, complemented
7#31826713C211yesmaximal, characteristic, complemented
8#31826702C211yesmaximal, characteristic, complemented
9#31826700C211yesmaximal, characteristic, complemented
10#31826697C211yesmaximal, characteristic, complemented
11#31826691C211yesmaximal, characteristic, complemented
12#31826506C211yesmaximal, characteristic, complemented
13#31826468C211yesmaximal, characteristic, complemented
14#31826465C211yesmaximal, characteristic, complemented
15#468×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×D4×D283—21no—
18C2×D4×D283C2211yescharacteristic
19C22×C4×D283C2211yescharacteristic
20#22583637—24no—
21C2×#17985379C2211yescharacteristic, complemented
22C2×D2264C2211yescharacteristic, complemented
23C2×#17985377C2211yescharacteristic, complemented
24C2×#17985376C2211yescharacteristic, complemented
25C2×#17985375C2211yescharacteristic
26C2×#17985375C2211yescharacteristic
27M16×D283—24no—
28C2×C8×D283C2211yescharacteristic, complemented
29C2×C8×D283C2211yescharacteristic, complemented
30#22583595—21no—
31#22583586C2211yescharacteristic, complemented
32#22583585C2211yescharacteristic
33#22583585C2211yescharacteristic, complemented
34#22583584C2211yescharacteristic
35#22583584C2211yescharacteristic, complemented
36#22583583—21no—
37#22583582—21no—
38#22583581C2211yescharacteristic
39#22583580—21no—
40#22583572—21no—
41(C22⋊D283)⋊C4—21no—
42#22583568C2211yescharacteristic
43C4⋊C4×D283C2211yescharacteristic
44C22⋊C4×D283—21no—
45#22583543C2211yescharacteristic, complemented
46#22583542C2211yescharacteristic, complemented
47#22583541C2211yescharacteristic, complemented
48#22583534C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic, complemented
50#22583532C2211yescharacteristic, complemented
51#22583531C2211yescharacteristic, complemented
52#22583530C2211yescharacteristic, complemented
53#22583530C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic, complemented
55C2×M16×C283C2211yescharacteristic, complemented
56C22⋊Q8×C283C2211yescharacteristic, complemented
57#121×C283C2211yescharacteristic
58#107×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#102×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283D411yescharacteristic
63#17985393—22no—
64#17985393—22no—
65C2×D1132—21no—
66C2×D1132—21no—
67C2×D1132—21no—
68C2×D1132C2311yescharacteristic
69C2×D1132C2311yescharacteristic
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74D4×D283—22no—
75D4×D283—22no—
76D4×D283—44no—
77C22×Dic283D411yescharacteristic
78C2×C4×D283—22no—
79C2×C4×D283—22no—
80C2×C4×D283—21no—
81C2×C4×D283—21no—
82C2×C4×D283D411yescharacteristic
83C2×C4×D283D411yescharacteristic
84C2×C4×D283C2311yescharacteristic
85C2×C4×D283C2311yescharacteristic
86#17985379—22no—
87D2264—22no—
88#17985377—22no—
89#17985376—22no—
90#17985375—22no—
91#17985375D414yescomplemented
92C2×C283⋊C8C2311yescharacteristic
93C2×C283⋊C8C2311yescharacteristic
94C8×D283—22no—
95C8×D283—22no—
96C4⋊Dic283C2311yescharacteristic
97C4⋊Dic283C2311yescharacteristic
98Dic283⋊C4—21no—
99Dic283⋊C4—21no—
100C22⋊Dic283—21no—
101(C2×D283)⋊C4—21no—
102(C2×D283)⋊C4—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107C4×Dic283—21no—
108C2×Q8×C283C2311yescharacteristic
109C2×D4×C283—21no—
110C2×D4×C283C2311yescharacteristic
111C22×C4×C283C2311yescharacteristic
112SD16×C283—22no—
113M16×C283—22no—
114C2×C8×C283C2311yescharacteristic
115C2×C8×C283C2311yescharacteristic
116C4⋊C4×C283—21no—
117C4⋊C4×C283C2311yescharacteristic
118C4⋊C4×C283C2311yescharacteristic
119C22⋊C4×C283—21no—
120C22⋊C4×C283—21no—
121C22×D283—22no—
122C22×D283—22no—
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—41no—
126C22×D283—42no—
127C22×D283—42no—
128C22×D283—42no—
129C22×D283C2×D411yescharacteristic
130C22×D283C2×D411yescharacteristic
131D1132—21no—
132D1132—21no—
133D1132—21no—
134D1132—21no—
135D1132—42no—
136D1132—42no—
137D1132—42no—
138C22⋊D283—42no—
139C22⋊D283—42no—
140C22⋊D283—42no—
141C22⋊D283—42no—
142C2×Dic283—22no—
143C2×Dic283—21no—
144C2×Dic283—21no—
145C2×Dic283C2×D411yescharacteristic
146C2×Dic283C2×D411yescharacteristic
147C4×D283—22no—
148C4×D283—21no—
149C4×D283—21no—
150C4×D283—22no—
151C4×D283—42no—
152C4×D283—42no—
153C4×D283C2×D412yes—
154C4×D283#4112yes—
155C283⋊C8—21no—
156C283⋊C8C2×D412yescomplemented
157C23×C283—21no—
158C23×C283C2×D411yescharacteristic
159Q8×C283—21no—
160Q8×C283—21no—
161D4×C283—21no—
162D4×C283—21no—
163D4×C283—42no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—21no—
167C2×C4×C283—21no—
168C2×C4×C283—21no—
169C2×C4×C283C2×D411yescharacteristic
170C2×C4×C283C2411yescharacteristic
171C8×C283—21no—
172C8×C283C2×D412yescomplemented
173C22×C283—21no—
174C22×C283—21no—
175C22×C283—21no—
176C22×C283—41no—
177C22×C283—41no—
178C22×C283—41no—
179C22×C283C22×D411yescharacteristic
180C2×D283—21no—
181C2×D283—21no—
182C2×D283—21no—
183C2×D283—21no—
184C2×D283—44no—
185C2×D283—41no—
186C2×D283—41no—
187C2×D283—41no—
188C2×D283—41no—
189C2×D283—41no—
190C2×D283—41no—
191C2×D283—84no—
192C2×D283#12112yes—
193C2×D283AGL(1,ℤ/8)12yescomplemented
194C4×C283—21no—
195C4×C283—41no—
196C4×C283—41no—
197C4×C283—41no—
198C4×C283C22×D411yescharacteristic
199C4×C283C2×#4111yescharacteristic
200Dic283—21no—
201Dic283—41no—
202Dic283—41no—
203Dic283#12112yes—
204C2×C283—41no—
205C2×C283—81no—
206C2×C283C2×#12111yescharacteristic
207C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C283#57511yescharacteristic
209D283—22no—
210D283—22no—
211D283—81no—
212D283—81no—
213C283#305511yescharacteristic, complemented, Sylow
214#3055—2831nomaximal, Sylow
215C2×AGL(1,ℤ/8)—2831no—
216C22×M16—2831no—
217#525—2831no—
218C2×#121—2831no—
219#469—2831no—
220#469—2831no—
221#468—2831no—
222#468D28311yescharacteristic, complemented
223#466—2831no—
224#466—2831no—
225#465—2831no—
226#465—2831no—
227#428—2831no—
228#418—2831no—
229#417—2831no—
230C2×#41—2831no—
231C22×D4—2831no—
232C22×D4—5661no—
233C23×C4—2831no—
234AGL(1,ℤ/8)—5664no—
235C2×SD16—2831no—
236C2×SD16C2×D28311yescharacteristic, complemented
237C2×D8—2831no—
238C2×D8—2831no—
239C2×M16—2831no—
240C2×M16—2831no—
241C2×M16—2831no—
242C2×M16—5664no—
243C2×M16C2×D28311yescharacteristic, complemented
244C22×C8—2831no—
245C22×C8—2831no—
246#123—5661no—
247C22⋊Q8—2831no—
248C22⋊Q8C2×D28311yescharacteristic, complemented
249#121—2831no—
250#121—2831no—
251#121—2831no—
252#121—2831no—
253#121—2831no—
254#121—5661no—
255#121—5661no—
256#121C2×D28311yescharacteristic
257#120—5661no—
258C4×D4—5661no—
259C4×D4—5661no—
260#117—2831no—
261C2×C4⋊C4—2831no—
262C2×C22⋊C4—5661no—
263#107—2831no—
264#107C2×D28311yescharacteristic, complemented
265#106—2831no—
266#106—2831no—
267#103—2831no—
268#103C2×D28311yescharacteristic, complemented
269#102—2831no—
270#102—2831no—
271#102—2831no—
272#102—2831no—
273#102—2831no—
274#102C2×D28311yescharacteristic, complemented
275C24—2831no—
276C24—5661no—
277#41—5662no—
278#41—5662no—
279C2×Q8C22×D28311yescharacteristic
280C2×D4—21no—
281C2×D4—2831no—
282C2×D4—2831no—
283C2×D4—5661no—
284C2×D4—5661no—
285C2×D4—5661no—
286C2×D4—5662no—
287C2×D4—5662no—
288C2×D4—5661no—
289C2×D4—5661no—
290C2×D4—5661no—
291C2×D4—5661no—
292C2×D4—11324no—
293C2×D4C22×D28311yescharacteristic
294C22×C4—2831no—
295C22×C4—2831no—
296C22×C4—2831no—
297C22×C4—2831no—
298C22×C4—2831no—
299C22×C4—5662no—
300C22×C4—5662no—
301C22×C4—5661no—
302C22×C4—5661no—
303C22×C4C22×D28311yescharacteristic
304SD16—22no—
305SD16—5662no—
306D8—5662no—
307D8—5662no—
308M16—22no—
309M16—2834no—
310M16—5662no—
311M16—5662no—
312C2×C8—2831no—
313C2×C8—2831no—
314C2×C8—5662no—
315C2×C8—5662no—
316C2×C8C22×D28311yescharacteristic
317C2×C8C22×D28311yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—2831no—
320C4⋊C4—2831no—
321C4⋊C4—5661no—
322C4⋊C4—5661no—
323C4⋊C4C22×D28311yescharacteristic
324C4⋊C4C22×D28311yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—5661no—
328C22⋊C4—5661no—
329C22⋊C4—5661no—
330C22⋊C4—5661no—
331C22⋊C4—5661no—
332C22⋊C4—5661no—
333C22⋊C4—5661no—
334C42—5661no—
335C23—21no—
336C23—2831no—
337C23—2831no—
338C23—5662no—
339C23—5662no—
340C23—5661no—
341C23—5661no—
342C23—11321no—
343C23—11322no—
344C23—11322no—
345C23—11322no—
346C23D4×D28311yescharacteristic
347Q8—21no—
348Q8—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—5661no—
353D4—5661no—
354D4—5661no—
355D4—5661no—
356D4—11322no—
357D4—11322no—
358D4—11322no—
359D4—11322no—
360D4—11322no—
361D4—11322no—
362D4—11322no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—2831no—
369C2×C4—2832no—
370C2×C4—2832no—
371C2×C4—2831no—
372C2×C4—5662no—
373C2×C4—5662no—
374C2×C4—5662no—
375C2×C4—5661no—
376C2×C4—5661no—
377C2×C4—5661no—
378C2×C4—5661no—
379C2×C4—11322no—
380C2×C4—11322no—
381C2×C4D4×D28311yescharacteristic
382C2×C4C23×D28311yescharacteristic
383C8—21no—
384C8—2832no—
385C8—5661no—
386C8D4×D28312yescomplemented
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—2832no—
394C22—2832no—
395C22—5661no—
396C22—5661no—
397C22—5661no—
398C22—5661no—
399C22—11324no—
400C22—11321no—
401C22—11321no—
402C22—11321no—
403C22—11321no—
404C22—11321no—
405C22—11321no—
406C22—22644no—
407C22C2×D4×D28311yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—2832no—
413C4—5661no—
414C4—11321no—
415C4—11321no—
416C4C2×D4×D28311yescharacteristic
417C4#41×D28311yescharacteristic
418C2—41no—
419C2—81no—
420C2—5662no—
421C2—5662no—
422C2—22641no—
423C2—22641no—
424C2#121×D28311yescharacteristic
425C2AGL(1,ℤ/8)×D28311yescharacteristic
426C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#468×C283C2
socleC22×C283C22×D4
hypercenter#468D283
hypocenterC283#3055
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. C2×M16×D283C2semidirect
3. #31826978C2semidirect
4. #121×D283C2semidirect
5. #31826718C2semidirect
6. #31826716C2semidirect
7. #31826713C2semidirect
8. #31826702C2semidirect
9. #31826700C2semidirect
10. #31826697C2semidirect
11. #31826691C2semidirect
12. #31826506C2semidirect
13. #31826468C2semidirect
14. #31826465C2semidirect
15. #468×C283C2semidirect
21. C2×#17985379C22semidirect
22. C2×D2264C22semidirect
23. C2×#17985377C22semidirect
24. C2×#17985376C22semidirect
28. C2×C8×D283C22semidirect
29. C2×C8×D283C22semidirect
31. #22583586C22semidirect
33. #22583585C22semidirect
35. #22583584C22semidirect
45. #22583543C22semidirect
46. #22583542C22semidirect
47. #22583541C22semidirect
48. #22583534C22semidirect
49. #22583532C22semidirect
50. #22583532C22semidirect
51. #22583531C22semidirect
52. #22583530C22semidirect
53. #22583530C22semidirect
54. C2×SD16×C283C22semidirect
55. C2×M16×C283C22semidirect
56. C22⋊Q8×C283C22semidirect
58. #107×C283C22semidirect
59. #103×C283C22semidirect
60. #102×C283C22semidirect
91. #17985375D4semidirect
156. C283⋊C8C2×D4semidirect
172. C8×C283C2×D4semidirect
193. C2×D283AGL(1,ℤ/8)semidirect
213. C283#3055semidirect
222. #468D283semidirect
236. C2×SD16C2×D283semidirect
243. C2×M16C2×D283semidirect
248. C22⋊Q8C2×D283semidirect
264. #107C2×D283semidirect
268. #103C2×D283semidirect
274. #102C2×D283semidirect
386. C8D4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C2×M16×D2832
#318269782
#121×D2832
#318267182
#318267162
#318267132
#318267022
#318267002
#318266972
#318266912
#318265062
#318264682
#318264652
#468×C2832
#3055283