Finite Group Databasebeta

← #49132868

Subgroups — #49132868

Order 36224.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583636C211yesmaximal, characteristic, complemented
2C2×#22583635C211yesmaximal, characteristic, complemented
3#31827111C211yesmaximal, characteristic, complemented
4#31826847C211yesmaximal, characteristic, complemented
5#31826713C211yesmaximal, characteristic, complemented
6#31826709C211yesmaximal, characteristic, complemented
7#31826680C211yesmaximal, characteristic, complemented
8#31826677C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic, complemented
10#31826625C211yesmaximal, characteristic, complemented
11#31826599C211yesmaximal, characteristic, complemented
12#31826598C211yesmaximal, characteristic, complemented
13C4×#17985377C211yesmaximal, characteristic, complemented
14#31826543C211yesmaximal, characteristic, complemented
15#546×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×D1132C2211yescharacteristic, complemented
18#22583636—24no—
19#22583635—24no—
20C2×#17985382C2211yescharacteristic, complemented
21C2×#17985380C2211yescharacteristic, complemented
22C2×#17985380C2211yescharacteristic, complemented
23C2×#17985379C2211yescharacteristic, complemented
24C2×#17985379C2211yescharacteristic, complemented
25C2×#17985377C2211yescharacteristic
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic
28C2×#17985377C2211yescharacteristic
29C2×#17985376C2211yescharacteristic, complemented
30C2×#17985376C2211yescharacteristic, complemented
31#22583598C2211yescharacteristic, complemented
32#22583595—21no—
33#22583589C2211yescharacteristic, complemented
34#22583584C2211yescharacteristic
35#22583583—21no—
36#22583578—21no—
37C4×D1132C2211yescharacteristic, complemented
38C22⋊C4×D283—21no—
39#22583543C2211yescharacteristic, complemented
40C4⋊(C283⋊C8)C2211yescharacteristic, complemented
41#22583534C2211yescharacteristic, complemented
42#22583533C2211yescharacteristic, complemented
43#22583531C2211yescharacteristic, complemented
44#22583531C2211yescharacteristic, complemented
45#22583530C2211yescharacteristic, complemented
46#22583530C2211yescharacteristic, complemented
47C22⋊(C283⋊C8)C2211yescharacteristic, complemented
48C22⋊(C283⋊C8)C2211yescharacteristic, complemented
49C4×C283⋊C8C2211yescharacteristic, complemented
50C2×#41×C283C2211yescharacteristic
51C22×D4×C283C2211yescharacteristic, complemented
52#124×C283C2211yescharacteristic, complemented
53#123×C283—21no—
54C22⋊Q8×C283C2211yescharacteristic, complemented
55#121×C283—21no—
56#121×C283C2211yescharacteristic
57#120×C283—21no—
58C4×D4×C283C2211yescharacteristic
59C4×D4×C283C2211yescharacteristic, complemented
60C2×C22⋊C4×C283—21no—
61C23×D283—21no—
62#17985392—22no—
63#17985392—41no—
64C2×Dic566C2311yescharacteristic
65C2×D1132—22no—
66C2×D1132—21no—
67C2×D1132—41no—
68C2×D1132C2311yescharacteristic
69C2×D1132C2311yescharacteristic
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C4×D283—21no—
73C2×C4×D283—21no—
74#17985382—41no—
75#17985380—22no—
76#17985380—22no—
77#17985379—22no—
78#17985379—22no—
79#17985377—22no—
80#17985377—22no—
81#17985377—41no—
82#17985377D414yescomplemented
83#17985376—22no—
84#17985376—22no—
85C2×C283⋊C8C2311yescharacteristic
86C2×C283⋊C8C2311yescharacteristic
87C2×C283⋊C8C2311yescharacteristic
88C2×C283⋊C8C2311yescharacteristic
89C4⋊Dic283C2311yescharacteristic
90Dic283⋊C4—21no—
91C22⋊Dic283—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4—21no—
96C24×C283—21no—
97#41×C283—22no—
98#41×C283—41no—
99C2×Q8×C283C2311yescharacteristic
100C2×D4×C283—22no—
101C2×D4×C283—21no—
102C2×D4×C283—21no—
103C2×D4×C283—21no—
104C2×D4×C283—41no—
105C2×D4×C283D411yescharacteristic
106C2×D4×C283C2311yescharacteristic
107C2×D4×C283C2311yescharacteristic
108C22×C4×C283—21no—
109C22×C4×C283—21no—
110C22×C4×C283C2311yescharacteristic
111C22×C4×C283C2311yescharacteristic
112C4⋊C4×C283—21no—
113C4⋊C4×C283D411yescharacteristic
114C4⋊C4×C283C2311yescharacteristic
115C22⋊C4×C283—21no—
116C22⋊C4×C283—21no—
117C22⋊C4×C283—21no—
118C22⋊C4×C283—21no—
119C22⋊C4×C283—21no—
120C22⋊C4×C283D411yescharacteristic
121C22⋊C4×C283D411yescharacteristic
122C42×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—81no—
128Dic566—21no—
129Dic566—21no—
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132—42no—
134D1132C2×D412yes—
135C22⋊D283—42no—
136C22⋊D283—81no—
137C2×Dic283—21no—
138C2×Dic283—21no—
139C4×D283—42no—
140C4×D283—81no—
141C283⋊C8—21no—
142C283⋊C8—21no—
143C283⋊C8—21no—
144C283⋊C8C2×D412yes—
145C23×C283—21no—
146C23×C283—21no—
147C23×C283—42no—
148C23×C283—42no—
149C23×C283—81no—
150C23×C283C2×D411yescharacteristic
151C23×C283C2×D411yescharacteristic
152Q8×C283—21no—
153Q8×C283—21no—
154D4×C283—21no—
155D4×C283—21no—
156D4×C283—21no—
157D4×C283—41no—
158D4×C283—42no—
159D4×C283—42no—
160D4×C283—81no—
161D4×C283C2×D412yes—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—21no—
167C2×C4×C283—21no—
168C2×C4×C283—42no—
169C2×C4×C283—81no—
170C2×C4×C283C2×D411yescharacteristic
171C2×C4×C283C2×D411yescharacteristic
172C2×C4×C283C2×D411yescharacteristic
173C2×C4×C283C2×D411yescharacteristic
174C2×C4×C283C2411yescharacteristic
175C22×C283—21no—
176C22×C283—21no—
177C22×C283—22no—
178C22×C283—21no—
179C22×C283—21no—
180C22×C283—21no—
181C22×C283—21no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283—84no—
188C22×C283C22×D411yescharacteristic
189C2×D283—22no—
190C2×D283—41no—
191C2×D283—41no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—41no—
195C2×D283—84no—
196C4×C283—21no—
197C4×C283—21no—
198C4×C283—21no—
199C4×C283—21no—
200C4×C283—41no—
201C4×C283—41no—
202C4×C283C22×D411yescharacteristic
203C4×C283H(5,𝔽2)11yescharacteristic
204Dic283—41no—
205Dic283—41no—
206C2×C283—41no—
207C2×C283—41no—
208C2×C283—42no—
209C2×C283—81no—
210C2×C283D4211yescharacteristic
211C2×C283#57511yescharacteristic
212C2×C283#57611yescharacteristic
213D283—42no—
214D283—81no—
215C283#327311yescharacteristic, complemented, Sylow
216#3273—2831nomaximal, Sylow
217C2×AGL(1,ℤ/8)—2831no—
218C2×AGL(1,ℤ/8)—2831no—
219#546—2831no—
220#546D28311yescharacteristic, complemented
221#495—2831no—
222#468—2831no—
223#468—2831no—
224#463—2831no—
225#463—2831no—
226#452—2831no—
227#452—2831no—
228#447—2831no—
229#447—2831no—
230C4×D8—2831no—
231#436—2831no—
232C2×#41—2831no—
233C2×#41C2×D28311yescharacteristic
234C22×D4—2831no—
235C22×D4C2×D28311yescharacteristic, complemented
236AGL(1,ℤ/8)—5664no—
237AGL(1,ℤ/8)—5664no—
238C2×Q16—2831no—
239C2×SD16—2831no—
240C2×SD16—2831no—
241C2×SD16—2831no—
242C2×SD16—2831no—
243C2×D8—2831no—
244C2×D8—2831no—
245C2×D8—2831no—
246C2×D8—2831no—
247C2×M16—2831no—
248C2×M16—2831no—
249#124—2831no—
250#124C2×D28311yescharacteristic, complemented
251#123—21no—
252#123—5661no—
253C22⋊Q8—2831no—
254C22⋊Q8C2×D28311yescharacteristic, complemented
255#121—21no—
256#121—2831no—
257#121—5661no—
258#121C2×D28311yescharacteristic
259#120—21no—
260#120—5661no—
261C4×D4—2831no—
262C4×D4C2×D28311yescharacteristic
263C4×D4C2×D28311yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—5661no—
266#107—2831no—
267C4⋊C8—2831no—
268#103—2831no—
269#103—2831no—
270#102—2831no—
271#102—2831no—
272#102—2831no—
273#102—2831no—
274C22⋊C8—2831no—
275C22⋊C8—2831no—
276C4×C8—2831no—
277C24—21no—
278C24—5661no—
279#41—22no—
280#41—41no—
281#41—5662no—
282#41—11321no—
283C2×Q8—2831no—
284C2×Q8C22×D28311yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2831no—
291C2×D4—2831no—
292C2×D4—5662no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—11321no—
297C2×D4C22⋊D28311yescharacteristic
298C2×D4C22×D28311yescharacteristic
299C2×D4C22×D28311yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—5661no—
303C22×C4—5661no—
304C22×C4C22×D28311yescharacteristic
305C22×C4C22×D28311yescharacteristic
306Q16—11321no—
307SD16—5662no—
308SD16—5662no—
309SD16—5662no—
310SD16—5662no—
311D8—2834no—
312D8—5662no—
313D8—5662no—
314D8—11321no—
315M16—5662no—
316M16—5662no—
317C2×C8—2831no—
318C2×C8—2831no—
319C2×C8—2831no—
320C2×C8—2831no—
321C4⋊C4—21no—
322C4⋊C4—2831no—
323C4⋊C4—5661no—
324C4⋊C4C22⋊D28311yescharacteristic
325C4⋊C4C22×D28311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—5661no—
332C22⋊C4—5661no—
333C22⋊C4—5661no—
334C22⋊C4—5661no—
335C22⋊C4—5661no—
336C22⋊C4C22⋊D28311yescharacteristic
337C22⋊C4C22⋊D28311yescharacteristic
338C42C22×D28311yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—5661no—
345C23—5661no—
346C23—11322no—
347C23—11322no—
348C23—22641no—
349C23C2×C22⋊D28311yescharacteristic
350C23C2×C22⋊D28311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—5661no—
354Q8—5661no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—2832no—
363D4—5661no—
364D4—5661no—
365D4—5661no—
366D4—11322no—
367D4—11322no—
368D4—22641no—
369D4D4×D28312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—5661no—
379C2×C4—5661no—
380C2×C4—11322no—
381C2×C4—22641no—
382C2×C4C2×C22⋊D28311yescharacteristic
383C2×C4C2×C22⋊D28311yescharacteristic
384C2×C4C2×C22⋊D28311yescharacteristic
385C2×C4C2×C22⋊D28311yescharacteristic
386C2×C4C23×D28311yescharacteristic
387C8—2832no—
388C8—5661no—
389C8—5661no—
390C8—5661no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—5662no—
405C22—11321no—
406C22—11321no—
407C22—11321no—
408C22—11321no—
409C22—11321no—
410C22—22644no—
411C22C22×C22⋊D28311yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—11321no—
419C4—11321no—
420C4C2×D4×D28311yescharacteristic
421C4#2258365511yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—11322no—
427C2—22641no—
428C2D4×C22⋊D28311yescharacteristic
429C2#3182726311yescharacteristic
430C2#3182726811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#546×C283C2
socleC22×C283C22×D4
hypercenter#546D283
hypocenterC283#3273
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#22583636C2semidirect
2. C2×#22583635C2semidirect
3. #31827111C2semidirect
4. #31826847C2semidirect
5. #31826713C2semidirect
6. #31826709C2semidirect
7. #31826680C2semidirect
8. #31826677C2semidirect
9. #31826627C2semidirect
10. #31826625C2semidirect
11. #31826599C2semidirect
12. #31826598C2semidirect
13. C4×#17985377C2semidirect
14. #31826543C2semidirect
15. #546×C283C2semidirect
17. C22×D1132C22semidirect
20. C2×#17985382C22semidirect
21. C2×#17985380C22semidirect
22. C2×#17985380C22semidirect
23. C2×#17985379C22semidirect
24. C2×#17985379C22semidirect
29. C2×#17985376C22semidirect
30. C2×#17985376C22semidirect
31. #22583598C22semidirect
33. #22583589C22semidirect
37. C4×D1132C22semidirect
39. #22583543C22semidirect
40. C4⋊(C283⋊C8)C22semidirect
41. #22583534C22semidirect
42. #22583533C22semidirect
43. #22583531C22semidirect
44. #22583531C22semidirect
45. #22583530C22semidirect
46. #22583530C22semidirect
47. C22⋊(C283⋊C8)C22semidirect
48. C22⋊(C283⋊C8)C22semidirect
49. C4×C283⋊C8C22semidirect
51. C22×D4×C283C22semidirect
52. #124×C283C22semidirect
54. C22⋊Q8×C283C22semidirect
59. C4×D4×C283C22semidirect
82. #17985377D4semidirect
215. C283#3273semidirect
220. #546D283semidirect
235. C22×D4C2×D283semidirect
250. #124C2×D283semidirect
254. C22⋊Q8C2×D283semidirect
263. C4×D4C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836362
C2×#225836352
#318271112
#318268472
#318267132
#318267092
#318266802
#318266772
#318266272
#318266252
#318265992
#318265982
C4×#179853772
#318265432
#546×C2832
#3273283