Finite Group Databasebeta

← #49130518

Subgroups — #49130518

Order 36224.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D283C211yesmaximal, characteristic
2C2×#22583642C211yesmaximal, characteristic, complemented
3C2×#22583639C211yesmaximal, characteristic, complemented
4C22⋊Q8×D283C211yesmaximal, characteristic, complemented
5#31826629C211yesmaximal, characteristic, complemented
6#31826627C211yesmaximal, characteristic, complemented
7#31826625C211yesmaximal, characteristic, complemented
8#31826622C211yesmaximal, characteristic, complemented
9#31826619C211yesmaximal, characteristic, complemented
10#31826617C211yesmaximal, characteristic, complemented
11#31826614C211yesmaximal, characteristic, complemented
12#31826465C211yesmaximal, characteristic, complemented
13#31826463C211yesmaximal, characteristic, complemented
14#31826427C211yesmaximal, characteristic, complemented
15#452×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985392—21no—
18C2×#17985392C2211yescharacteristic
19C2×#17985391—21no—
20C2×#17985391C2211yescharacteristic
21#41×D283—24no—
22#41×D283—24no—
23C2×Q8×D283C2211yescharacteristic
24C2×Q8×D283C2211yescharacteristic, complemented
25C2×D4×D283—21no—
26C2×D4×D283C2211yescharacteristic
27C22×C4×D283—21no—
28C22×C4×D283C2211yescharacteristic
29#22583642—24no—
30#22583639—24no—
31C2×Dic1132C2211yescharacteristic, complemented
32C2×#17985382C2211yescharacteristic, complemented
33C2×#17985382C2211yescharacteristic, complemented
34C2×#17985381C2211yescharacteristic, complemented
35C2×#17985380C2211yescharacteristic, complemented
36C2×#17985379C2211yescharacteristic, complemented
37C2×#17985375C2211yescharacteristic, complemented
38C2×#17985375C2211yescharacteristic, complemented
39#22583590C2211yescharacteristic
40#22583589C2211yescharacteristic, complemented
41#22583588—21no—
42#22583587—21no—
43#22583586C2211yescharacteristic
44C4⋊C4×D283—21no—
45C4⋊C4×D283C2211yescharacteristic, complemented
46C22⋊C4×D283—21no—
47#22583535C2211yescharacteristic, complemented
48#22583534C2211yescharacteristic, complemented
49#22583533C2211yescharacteristic, complemented
50#22583532C2211yescharacteristic, complemented
51#22583531C2211yescharacteristic, complemented
52#22583530C2211yescharacteristic, complemented
53C22⋊(C283⋊C8)C2211yescharacteristic, complemented
54(C2×D283)⋊C8C2211yescharacteristic
55(C2×D283)⋊C8C2211yescharacteristic
56C2×#41×C283C2211yescharacteristic
57C2×Q16×C283C2211yescharacteristic, complemented
58C2×SD16×C283C2211yescharacteristic, complemented
59C22⋊Q8×C283C2211yescharacteristic, complemented
60#103×C283C2211yescharacteristic, complemented
61#102×C283C2211yescharacteristic, complemented
62C22⋊C8×C283C2211yescharacteristic, complemented
63C23×D283—21no—
64C23×D283D411yescharacteristic
65#17985393—22no—
66#17985393D414yes—
67#17985392—28no—
68#17985392—22no—
69#17985392—22no—
70#17985391—28no—
71#17985391—22no—
72#17985391D414yes—
73C2×Dic566—21no—
74C2×Dic566—21no—
75C2×Dic566C2311yescharacteristic
76C2×Dic566C2311yescharacteristic
77Q8×D283—22no—
78Q8×D283—22no—
79Q8×D283—22no—
80Q8×D283—22no—
81C2×D1132—21no—
82C2×D1132C2311yescharacteristic
83C2×C22⋊D283—21no—
84C2×C22⋊D283—21no—
85C2×C22⋊D283—21no—
86D4×D283—28no—
87D4×D283—22no—
88D4×D283—22no—
89C22×Dic283—21no—
90C22×Dic283D411yescharacteristic
91C2×C4×D283—28no—
92C2×C4×D283—21no—
93C2×C4×D283—21no—
94C2×C4×D283—22no—
95C2×C4×D283—22no—
96C2×C4×D283—21no—
97C2×C4×D283—21no—
98C2×C4×D283—21no—
99C2×C4×D283D411yescharacteristic
100C2×C4×D283D411yescharacteristic
101C2×C4×D283C2311yescharacteristic
102C2×C4×D283C2311yescharacteristic
103Dic1132—22no—
104#17985382—22no—
105#17985382—22no—
106#17985381—22no—
107#17985380—22no—
108#17985379—22no—
109#17985375—22no—
110#17985375—22no—
111C2×C283⋊C8C2311yescharacteristic
112C2×C283⋊C8C2311yescharacteristic
113C4⋊Dic283—21no—
114C4⋊Dic283C2311yescharacteristic
115Dic283⋊C4—21no—
116Dic283⋊C4—21no—
117Dic283⋊C4—21no—
118C22⋊Dic283—21no—
119(C2×D283)⋊C4—21no—
120(C2×D283)⋊C4—21no—
121#41×C283—22no—
122#41×C283—22no—
123C2×Q8×C283C2311yescharacteristic
124C2×Q8×C283C2311yescharacteristic
125C2×D4×C283—21no—
126C2×D4×C283C2311yescharacteristic
127C22×C4×C283—21no—
128C22×C4×C283C2311yescharacteristic
129Q16×C283—22no—
130SD16×C283—22no—
131C2×C8×C283C2311yescharacteristic
132C2×C8×C283C2311yescharacteristic
133C4⋊C4×C283—21no—
134C4⋊C4×C283C2311yescharacteristic
135C22⋊C4×C283—21no—
136C22×D283—24no—
137C22×D283—21no—
138C22×D283—22no—
139C22×D283—22no—
140C22×D283—41no—
141C22×D283—42no—
142C22×D283—42no—
143C22×D283C2×D411yescharacteristic
144C22×D283C2×D411yescharacteristic
145Dic566—24no—
146Dic566—21no—
147Dic566—21no—
148Dic566—21no—
149Dic566—42no—
150Dic566C2×D412yes—
151D1132—24no—
152D1132—21no—
153D1132C2×D412yes—
154C22⋊D283—24no—
155C22⋊D283—24no—
156C22⋊D283—24no—
157C2×Dic283—21no—
158C2×Dic283—24no—
159C2×Dic283—22no—
160C2×Dic283—21no—
161C2×Dic283—21no—
162C2×Dic283C2×D411yescharacteristic
163C2×Dic283C2×D411yescharacteristic
164C4×D283—24no—
165C4×D283—24no—
166C4×D283—22no—
167C4×D283—21no—
168C4×D283—21no—
169C4×D283—22no—
170C4×D283—24no—
171C4×D283—42no—
172C4×D283—42no—
173C4×D283C2×D412yes—
174C4×D283C2×D412yes—
175C283⋊C8—21no—
176C283⋊C8—21no—
177C23×C283—21no—
178C23×C283C2×D411yescharacteristic
179Q8×C283—21no—
180Q8×C283—21no—
181Q8×C283—21no—
182Q8×C283C2×D412yes—
183D4×C283—24no—
184D4×C283—21no—
185D4×C283C2×D412yes—
186C2×C4×C283—24no—
187C2×C4×C283—21no—
188C2×C4×C283—21no—
189C2×C4×C283—21no—
190C2×C4×C283—21no—
191C2×C4×C283—21no—
192C2×C4×C283C2×D411yescharacteristic
193C2×C4×C283C2411yescharacteristic
194C8×C283—21no—
195C8×C283—21no—
196C22×C283—21no—
197C22×C283—22no—
198C22×C283—21no—
199C22×C283—21no—
200C22×C283—41no—
201C22×C283—41no—
202C22×C283C22×D411yescharacteristic
203C2×D283—21no—
204C2×D283—21no—
205C2×D283—21no—
206C2×D283—21no—
207C2×D283—22no—
208C2×D283—44no—
209C2×D283—48no—
210C2×D283—41no—
211C2×D283—41no—
212C2×D283#12012yes—
213C2×D283#13712yescomplemented
214C4×C283—21no—
215C4×C283—22no—
216C4×C283—41no—
217C4×C283—41no—
218C4×C283C22×D411yescharacteristic
219C4×C283C22×D411yescharacteristic
220Dic283—22no—
221Dic283—21no—
222Dic283—41no—
223Dic283—41no—
224Dic283#12012yes—
225C2×C283—41no—
226C2×C283—42no—
227C2×C283C2×#12011yescharacteristic
228C2×C283C2×#13711yescharacteristic
229C2×C283#57511yescharacteristic
230D283—22no—
231D283—22no—
232D283—42no—
233C283#300711yescharacteristic, complemented, Sylow
234#3007—2831nomaximal, Sylow
235C22×#41—2831no—
236C2×#137—2831no—
237C2×#137—2831no—
238C2×C22⋊Q8—2831no—
239#452—2831no—
240#452—2831no—
241#452—2831no—
242#452D28311yescharacteristic, complemented
243#451—2831no—
244#451—2831no—
245#450—2831no—
246#450—2831no—
247#417—2831no—
248#417—2831no—
249#407—2831no—
250C2×#41—2831no—
251C2×#41—2831no—
252C2×#41—2831no—
253C2×#41—5661no—
254C2×#41—5661no—
255C2×#41—5664no—
256C2×#41—5664no—
257C2×#41C2×D28311yescharacteristic
258C22×Q8—2831no—
259C22×Q8—2831no—
260C22×D4—2831no—
261C22×D4—5661no—
262C23×C4—2831no—
263C23×C4—5661no—
264#137—5664no—
265#137—5664no—
266C2×Q16—2831no—
267C2×Q16—2831no—
268C2×Q16—2831no—
269C2×Q16C2×D28311yescharacteristic, complemented
270C2×SD16—2831no—
271C2×SD16—2831no—
272C2×SD16—2831no—
273C2×SD16C2×D28311yescharacteristic, complemented
274C2×M16—2831no—
275C2×M16—2831no—
276C22⋊Q8—2831no—
277C22⋊Q8—2831no—
278C22⋊Q8—2831no—
279C22⋊Q8—5661no—
280C22⋊Q8—5661no—
281C22⋊Q8C2×D28311yescharacteristic, complemented
282C2×C4⋊C4—2831no—
283C2×C4⋊C4—5661no—
284C2×C22⋊C4—5661no—
285#103—2831no—
286#103—2831no—
287#103—2831no—
288#103C2×D28311yescharacteristic, complemented
289#102—2831no—
290#102—2831no—
291#102—2831no—
292#102C2×D28311yescharacteristic, complemented
293C22⋊C8—2831no—
294C22⋊C8—2831no—
295C22⋊C8—2831no—
296C22⋊C8C2×D28311yescharacteristic, complemented
297C24—2831no—
298C24—5661no—
299#41—22no—
300#41—22no—
301#41—2834no—
302#41—2834no—
303#41—5662no—
304#41—5662no—
305#41—5668no—
306#41—5662no—
307#41—5662no—
308#41—5668no—
309C2×Q8—2831no—
310C2×Q8—2831no—
311C2×Q8—5661no—
312C2×Q8—5661no—
313C2×Q8—5662no—
314C2×Q8—5662no—
315C2×Q8—5662no—
316C2×Q8—5662no—
317C2×Q8C22×D28311yescharacteristic
318C2×Q8C22×D28311yescharacteristic
319C2×D4—21no—
320C2×D4—2831no—
321C2×D4—5661no—
322C2×D4—5661no—
323C2×D4—5662no—
324C2×D4—5668no—
325C2×D4—5661no—
326C2×D4—5661no—
327C2×D4—5662no—
328C2×D4C22×D28311yescharacteristic
329C22×C4—21no—
330C22×C4—2831no—
331C22×C4—2831no—
332C22×C4—2831no—
333C22×C4—2831no—
334C22×C4—2831no—
335C22×C4—5662no—
336C22×C4—5662no—
337C22×C4—5661no—
338C22×C4—5661no—
339C22×C4—5668no—
340C22×C4—5661no—
341C22×C4—5661no—
342C22×C4—5661no—
343C22×C4—5661no—
344C22×C4C22×D28311yescharacteristic
345Q16—22no—
346Q16—5662no—
347Q16—5662no—
348Q16—5662no—
349SD16—22no—
350SD16—5662no—
351SD16—5662no—
352SD16—5662no—
353M16—5662no—
354M16—5662no—
355C2×C8—2831no—
356C2×C8—2831no—
357C2×C8C22×D28311yescharacteristic
358C2×C8C22×D28311yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—2831no—
361C4⋊C4—5661no—
362C4⋊C4—5661no—
363C4⋊C4—5661no—
364C4⋊C4—5661no—
365C4⋊C4C22×D28311yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—5661no—
368C22⋊C4—5661no—
369C22⋊C4—5661no—
370C23—21no—
371C23—2831no—
372C23—2831no—
373C23—5662no—
374C23—5662no—
375C23—5664no—
376C23—5661no—
377C23—11321no—
378C23—11322no—
379C23—11322no—
380C23D4×D28311yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—2832no—
385Q8—5664no—
386Q8—5661no—
387Q8—5661no—
388Q8—5661no—
389Q8—11322no—
390Q8D4×D28312yes—
391D4—24no—
392D4—21no—
393D4—2832no—
394D4—5664no—
395D4—5664no—
396D4—5664no—
397D4—5664no—
398D4—5661no—
399D4D4×D28312yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—2831no—
407C2×C4—2832no—
408C2×C4—2832no—
409C2×C4—2831no—
410C2×C4—5662no—
411C2×C4—5664no—
412C2×C4—5662no—
413C2×C4—5664no—
414C2×C4—5664no—
415C2×C4—5662no—
416C2×C4—5664no—
417C2×C4—5661no—
418C2×C4—5661no—
419C2×C4—5661no—
420C2×C4—5661no—
421C2×C4—5661no—
422C2×C4—11322no—
423C2×C4—11322no—
424C2×C4D4×D28311yescharacteristic
425C2×C4C23×D28311yescharacteristic
426C8—21no—
427C8—21no—
428C8—5661no—
429C8—5661no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—2832no—
437C22—2832no—
438C22—5661no—
439C22—5661no—
440C22—5661no—
441C22—5661no—
442C22—5662no—
443C22—11324no—
444C22—11328no—
445C22—11321no—
446C22—11321no—
447C22C2×D4×D28311yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—2832no—
453C4—5661no—
454C4—5662no—
455C4—11321no—
456C4—11321no—
457C4C2×D4×D28311yescharacteristic
458C4C2×D4×D28311yescharacteristic
459C2—41no—
460C2—42no—
461C2—5662no—
462C2—5662no—
463C2—11322no—
464C2#120×D28311yescharacteristic
465C2#137×D28311yescharacteristic
466C2#3182726211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#452×C283C2
socleC22×C283C22×D4
prime radicalC2×#41×D283C2
hypercenter#452D283
hypocenterC283#3007
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
2. C2×#22583642C2semidirect
3. C2×#22583639C2semidirect
4. C22⋊Q8×D283C2semidirect
5. #31826629C2semidirect
6. #31826627C2semidirect
7. #31826625C2semidirect
8. #31826622C2semidirect
9. #31826619C2semidirect
10. #31826617C2semidirect
11. #31826614C2semidirect
12. #31826465C2semidirect
13. #31826463C2semidirect
14. #31826427C2semidirect
15. #452×C283C2semidirect
24. C2×Q8×D283C22semidirect
31. C2×Dic1132C22semidirect
32. C2×#17985382C22semidirect
33. C2×#17985382C22semidirect
34. C2×#17985381C22semidirect
35. C2×#17985380C22semidirect
36. C2×#17985379C22semidirect
37. C2×#17985375C22semidirect
38. C2×#17985375C22semidirect
40. #22583589C22semidirect
45. C4⋊C4×D283C22semidirect
47. #22583535C22semidirect
48. #22583534C22semidirect
49. #22583533C22semidirect
50. #22583532C22semidirect
51. #22583531C22semidirect
52. #22583530C22semidirect
53. C22⋊(C283⋊C8)C22semidirect
57. C2×Q16×C283C22semidirect
58. C2×SD16×C283C22semidirect
59. C22⋊Q8×C283C22semidirect
60. #103×C283C22semidirect
61. #102×C283C22semidirect
62. C22⋊C8×C283C22semidirect
213. C2×D283#137semidirect
233. C283#3007semidirect
242. #452D283semidirect
269. C2×Q16C2×D283semidirect
273. C2×SD16C2×D283semidirect
281. C22⋊Q8C2×D283semidirect
288. #103C2×D283semidirect
292. #102C2×D283semidirect
296. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2832
C2×#225836422
C2×#225836392
C22⋊Q8×D2832
#318266292
#318266272
#318266252
#318266222
#318266192
#318266172
#318266142
#318264652
#318264632
#318264272
#452×C2832
#3007283