Finite Group Databasebeta

← #49130535

Subgroups — #49130535

Order 36224.

752 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583656C211yesmaximal, characteristic
2C2×#22583642C211yesmaximal, characteristic, complemented
3C2×SD16×D283C211yesmaximal, characteristic, complemented
4#31827038C211yesmaximal, characteristic, complemented
5#31826630C211yesmaximal, characteristic, complemented
6#31826627C211yesmaximal, characteristic, complemented
7#31826626C211yesmaximal, characteristic, complemented
8#31826618C211yesmaximal, characteristic, complemented
9#31826617C211yesmaximal, characteristic, complemented
10#31826615C211yesmaximal, characteristic, complemented
11#31826614C211yesmaximal, characteristic, complemented
12#31826468C211yesmaximal, characteristic, complemented
13#31826460C211yesmaximal, characteristic, complemented
14#31826430C211yesmaximal, characteristic, complemented
15#452×C283C211yesmaximal, characteristic, complemented
16#22583656—24no—
17#22583656—24no—
18C2×#17985393C2211yescharacteristic
19C2×#17985393C2211yescharacteristic
20C2×#17985392—21no—
21C2×#17985392C2211yescharacteristic
22C2×Q8×D283C2211yescharacteristic, complemented
23C22×D1132—21no—
24C22×D1132C2211yescharacteristic
25C2×D4×D283—21no—
26C2×D4×D283—21no—
27C2×D4×D283C2211yescharacteristic
28C2×D4×D283C2211yescharacteristic
29#22583642—24no—
30C2×#17985382C2211yescharacteristic, complemented
31C2×#17985381C2211yescharacteristic, complemented
32C2×#17985381C2211yescharacteristic, complemented
33C2×#17985380C2211yescharacteristic, complemented
34C2×#17985379C2211yescharacteristic, complemented
35C2×#17985379C2211yescharacteristic, complemented
36SD16×D283—24no—
37C2×#17985375C2211yescharacteristic, complemented
38C2×C8×D283C2211yescharacteristic, complemented
39#22583609—21no—
40#22583596—21no—
41#22583595—21no—
42#22583592—21no—
43#22583590C2211yescharacteristic, complemented
44#22583589C2211yescharacteristic, complemented
45#22583586C2211yescharacteristic, complemented
46#22583568C2211yescharacteristic, complemented
47#22583534C2211yescharacteristic, complemented
48#22583532C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic, complemented
50#22583531C2211yescharacteristic, complemented
51#22583530C2211yescharacteristic, complemented
52#22583530C2211yescharacteristic, complemented
53C22⋊(C283⋊C8)C2211yescharacteristic, complemented
54(C2×D283)⋊C8C2211yescharacteristic, complemented
55(C2×D283)⋊C8C2211yescharacteristic, complemented
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×D283—21no—
65C23×D283—21no—
66#17985393—22no—
67#17985393—22no—
68#17985393—22no—
69#17985393D414yes—
70#17985392—28no—
71#17985392—22no—
72#17985392—22no—
73C2×Dic566—21no—
74C2×Dic566D411yescharacteristic
75C2×Dic566C2311yescharacteristic
76Q8×D283—22no—
77Q8×D283—22no—
78C2×D1132—28no—
79C2×D1132—21no—
80C2×D1132—21no—
81C2×D1132—22no—
82C2×D1132—22no—
83C2×D1132—21no—
84C2×D1132D411yescharacteristic
85C2×D1132C2311yescharacteristic
86C2×D1132C2311yescharacteristic
87C2×C22⋊D283—21no—
88C2×C22⋊D283—21no—
89C2×C22⋊D283D411yescharacteristic
90C2×C22⋊D283D411yescharacteristic
91D4×D283—28no—
92D4×D283—28no—
93D4×D283—22no—
94D4×D283—22no—
95D4×D283—22no—
96D4×D283D414yes—
97C2×C4×D283—21no—
98C2×C4×D283—21no—
99C2×C4×D283—21no—
100C2×C4×D283C2311yescharacteristic
101C2×C4×D283C2311yescharacteristic
102#17985382—22no—
103#17985381—22no—
104#17985381—22no—
105#17985380—22no—
106#17985379—22no—
107#17985379—22no—
108#17985375—22no—
109C2×C283⋊C8C2311yescharacteristic
110C2×C283⋊C8C2311yescharacteristic
111C8×D283—22no—
112C4⋊Dic283—21no—
113C4⋊Dic283C2311yescharacteristic
114Dic283⋊C4—21no—
115Dic283⋊C4—21no—
116C22⋊Dic283—21no—
117(C2×D283)⋊C4—21no—
118(C2×D283)⋊C4—21no—
119(C2×D283)⋊C4—21no—
120C4×Dic283—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—21no—
137C22×D283—21no—
138C22×D283—24no—
139C22×D283—24no—
140C22×D283—24no—
141C22×D283—81no—
142C22×D283—81no—
143C22×D283—81no—
144C22×D283—81no—
145C22×D283—81no—
146C22×D283—81no—
147C22×D283C2×D411yescharacteristic
148C22×D283C2×D411yescharacteristic
149Dic566—22no—
150Dic566—21no—
151Dic566—21no—
152Dic566—42no—
153D1132—24no—
154D1132—24no—
155D1132—24no—
156D1132—22no—
157D1132—21no—
158D1132—21no—
159D1132C2×D412yes—
160D1132C2×D412yes—
161C22⋊D283—24no—
162C22⋊D283—22no—
163C22⋊D283—22no—
164C22⋊D283—24no—
165C2×Dic283—21no—
166C2×Dic283—21no—
167C2×Dic283C2×D411yescharacteristic
168C2×Dic283C2×D411yescharacteristic
169C4×D283—24no—
170C4×D283—21no—
171C4×D283—21no—
172C4×D283—24no—
173C4×D283—42no—
174C4×D283C2×D412yes—
175C4×D283C2×D412yes—
176C283⋊C8—21no—
177C283⋊C8—21no—
178C23×C283—21no—
179C23×C283C2×D411yescharacteristic
180Q8×C283—21no—
181Q8×C283—21no—
182Q8×C283—21no—
183Q8×C283C2×D412yes—
184D4×C283—24no—
185D4×C283—21no—
186D4×C283C2×D412yes—
187C2×C4×C283—24no—
188C2×C4×C283—21no—
189C2×C4×C283—21no—
190C2×C4×C283—21no—
191C2×C4×C283—21no—
192C2×C4×C283—21no—
193C2×C4×C283C2×D411yescharacteristic
194C2×C4×C283C2411yescharacteristic
195C8×C283—21no—
196C8×C283—21no—
197C22×C283—21no—
198C22×C283—22no—
199C22×C283—21no—
200C22×C283—21no—
201C22×C283—41no—
202C22×C283—41no—
203C22×C283C22×D411yescharacteristic
204C2×D283—21no—
205C2×D283—21no—
206C2×D283—21no—
207C2×D283—22no—
208C2×D283—22no—
209C2×D283—21no—
210C2×D283—21no—
211C2×D283—41no—
212C2×D283—41no—
213C2×D283—41no—
214C2×D283—41no—
215C2×D283—84no—
216C2×D283—84no—
217C2×D283—84no—
218C2×D283#12012yes—
219C4×C283—21no—
220C4×C283—22no—
221C4×C283—41no—
222C4×C283—41no—
223C4×C283C22×D411yescharacteristic
224C4×C283C22×D411yescharacteristic
225Dic283—21no—
226Dic283—41no—
227Dic283—41no—
228Dic283#12012yes—
229C2×C283—41no—
230C2×C283—42no—
231C2×C283C2×#12011yescharacteristic
232C2×C283#57711yescharacteristic
233C2×C283#57711yescharacteristic
234D283—22no—
235D283—41no—
236D283—42no—
237D283—42no—
238C283#300911yescharacteristic, complemented, Sylow
239#3009—2831nomaximal, Sylow
240C2×H(5,𝔽2)—2831no—
241C2×#137—2831no—
242C22×SD16—2831no—
243#536—2831no—
244#452—2831no—
245#452—2831no—
246#452—2831no—
247#452D28311yescharacteristic, complemented
248#450—2831no—
249#450—2831no—
250#450—2831no—
251#450—2831no—
252#418—2831no—
253#416—2831no—
254#408—2831no—
255H(5,𝔽2)—5664no—
256H(5,𝔽2)—5664no—
257C2×#41—2831no—
258C2×#41—2831no—
259C2×#41—2831no—
260C2×#41—5661no—
261C2×#41C2×D28311yescharacteristic
262C22×Q8—2831no—
263C22×D4—2831no—
264C22×D4—2831no—
265C22×D4—2831no—
266C22×D4—5661no—
267C22×D4—5661no—
268C22×D4—5661no—
269#137—5664no—
270C2×Q16—2831no—
271C2×Q16C2×D28311yescharacteristic, complemented
272C2×SD16—2831no—
273C2×SD16—2831no—
274C2×SD16—2831no—
275C2×SD16—2831no—
276C2×SD16—2831no—
277C2×SD16—5664no—
278C2×SD16C2×D28311yescharacteristic, complemented
279C2×M16—2831no—
280C22×C8—2831no—
281C4⋊Q8—5661no—
282#124—5661no—
283#123—5661no—
284#123—5661no—
285C22⋊Q8—2831no—
286C22⋊Q8—2831no—
287C22⋊Q8—2831no—
288C22⋊Q8C2×D28311yescharacteristic, complemented
289#117—2831no—
290#103—2831no—
291#103C2×D28311yescharacteristic, complemented
292#102—2831no—
293#102—2831no—
294#102—2831no—
295#102—2831no—
296#102—2831no—
297#102C2×D28311yescharacteristic, complemented
298C22⋊C8—2831no—
299C22⋊C8—2831no—
300C22⋊C8—2831no—
301C22⋊C8C2×D28311yescharacteristic, complemented
302C24—5661no—
303C24—5661no—
304C24—5661no—
305#41—22no—
306#41—22no—
307#41—2834no—
308#41—5662no—
309#41—5662no—
310#41—5668no—
311#41—5662no—
312#41—5662no—
313#41—5662no—
314C2×Q8—2831no—
315C2×Q8—2831no—
316C2×Q8—5661no—
317C2×Q8—5662no—
318C2×Q8—5662no—
319C2×Q8C22×D28311yescharacteristic
320C2×Q8C22×D28311yescharacteristic
321C2×D4—21no—
322C2×D4—2834no—
323C2×D4—2831no—
324C2×D4—2831no—
325C2×D4—2831no—
326C2×D4—2831no—
327C2×D4—2831no—
328C2×D4—5662no—
329C2×D4—5662no—
330C2×D4—5661no—
331C2×D4—5661no—
332C2×D4—5668no—
333C2×D4—5662no—
334C2×D4—5661no—
335C2×D4—5661no—
336C2×D4—5662no—
337C2×D4—5668no—
338C2×D4—5668no—
339C2×D4—5662no—
340C2×D4—5661no—
341C2×D4C22×D28311yescharacteristic
342C22×C4—21no—
343C22×C4—2831no—
344C22×C4—2831no—
345C22×C4—5661no—
346C22×C4—5661no—
347C22×C4—5661no—
348C22×C4C22×D28311yescharacteristic
349Q16—22no—
350Q16—5662no—
351SD16—22no—
352SD16—5662no—
353SD16—5662no—
354SD16—5662no—
355SD16—5662no—
356SD16—5662no—
357M16—5662no—
358C2×C8—2831no—
359C2×C8—2831no—
360C2×C8—5662no—
361C2×C8C22×D28311yescharacteristic
362C2×C8C22×D28311yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—2831no—
365C4⋊C4—5661no—
366C4⋊C4—5661no—
367C4⋊C4—5661no—
368C4⋊C4C22×D28311yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—5661no—
371C22⋊C4—5661no—
372C22⋊C4—5661no—
373C22⋊C4—5661no—
374C42—5661no—
375C23—21no—
376C23—2831no—
377C23—2831no—
378C23—5664no—
379C23—5664no—
380C23—5664no—
381C23—5661no—
382C23—5661no—
383C23—22641no—
384C23—22641no—
385C23—22641no—
386C23—22641no—
387C23—22641no—
388C23—22641no—
389C23D4×D28311yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—5662no—
394Q8—5661no—
395Q8—5661no—
396Q8—11322no—
397Q8D4×D28312yes—
398D4—24no—
399D4—21no—
400D4—2832no—
401D4—2832no—
402D4—5662no—
403D4—5664no—
404D4—5664no—
405D4—5664no—
406D4—5662no—
407D4—5664no—
408D4—5662no—
409D4—5664no—
410D4—5661no—
411D4—5661no—
412D4D4×D28312yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—2831no—
420C2×C4—2832no—
421C2×C4—2832no—
422C2×C4—2831no—
423C2×C4—5664no—
424C2×C4—5664no—
425C2×C4—5661no—
426C2×C4—5661no—
427C2×C4—5661no—
428C2×C4—5661no—
429C2×C4—11322no—
430C2×C4D4×D28311yescharacteristic
431C2×C4C23×D28311yescharacteristic
432C8—21no—
433C8—21no—
434C8—5661no—
435C8—5661no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—2832no—
443C22—5661no—
444C22—5661no—
445C22—5662no—
446C22—5662no—
447C22—5661no—
448C22—5661no—
449C22—5661no—
450C22—11321no—
451C22—11321no—
452C22—11321no—
453C22—11321no—
454C22—22644no—
455C22—22644no—
456C22—22644no—
457C22C2×D4×D28311yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—2832no—
463C4—5661no—
464C4—11321no—
465C4—11321no—
466C4C2×D4×D28311yescharacteristic
467C4C2×D4×D28311yescharacteristic
468C2—41no—
469C2—42no—
470C2—5662no—
471C2—11321no—
472C2—11322no—
473C2—11322no—
474C2#120×D28311yescharacteristic
475C2#3182727211yescharacteristic
476C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#452×C283C2
socleC22×C283C22×D4
prime radicalC2×#22583656C2
hypercenter#452D283
hypocenterC283#3009
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×SD16×D283C2semidirect
4. #31827038C2semidirect
5. #31826630C2semidirect
6. #31826627C2semidirect
7. #31826626C2semidirect
8. #31826618C2semidirect
9. #31826617C2semidirect
10. #31826615C2semidirect
11. #31826614C2semidirect
12. #31826468C2semidirect
13. #31826460C2semidirect
14. #31826430C2semidirect
15. #452×C283C2semidirect
22. C2×Q8×D283C22semidirect
30. C2×#17985382C22semidirect
31. C2×#17985381C22semidirect
32. C2×#17985381C22semidirect
33. C2×#17985380C22semidirect
34. C2×#17985379C22semidirect
35. C2×#17985379C22semidirect
37. C2×#17985375C22semidirect
38. C2×C8×D283C22semidirect
43. #22583590C22semidirect
44. #22583589C22semidirect
45. #22583586C22semidirect
46. #22583568C22semidirect
47. #22583534C22semidirect
48. #22583532C22semidirect
49. #22583532C22semidirect
50. #22583531C22semidirect
51. #22583530C22semidirect
52. #22583530C22semidirect
53. C22⋊(C283⋊C8)C22semidirect
54. (C2×D283)⋊C8C22semidirect
55. (C2×D283)⋊C8C22semidirect
57. C2×Q16×C283C22semidirect
58. C2×SD16×C283C22semidirect
59. C22⋊Q8×C283C22semidirect
60. #103×C283C22semidirect
61. #102×C283C22semidirect
62. C22⋊C8×C283C22semidirect
238. C283#3009semidirect
247. #452D283semidirect
271. C2×Q16C2×D283semidirect
278. C2×SD16C2×D283semidirect
288. C22⋊Q8C2×D283semidirect
291. #103C2×D283semidirect
297. #102C2×D283semidirect
301. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836562
C2×#225836422
C2×SD16×D2832
#318270382
#318266302
#318266272
#318266262
#318266182
#318266172
#318266152
#318266142
#318264682
#318264602
#318264302
#452×C2832
#3009283