Finite Group Databasebeta

← #49130481

Subgroups — #49130481

Order 36224.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D283C211yesmaximal, characteristic, complemented
2C2×#22583632C211yesmaximal, characteristic, complemented
3C2×#22583630C211yesmaximal, characteristic, complemented
4#31826977C211yesmaximal, characteristic, complemented
5#31826630C211yesmaximal, characteristic, complemented
6#31826628C211yesmaximal, characteristic, complemented
7#31826622C211yesmaximal, characteristic, complemented
8#31826617C211yesmaximal, characteristic, complemented
9#31826613C211yesmaximal, characteristic, complemented
10#31826611C211yesmaximal, characteristic, complemented
11#31826605C211yesmaximal, characteristic, complemented
12#31826600C211yesmaximal, characteristic, complemented
13#31826459C211yesmaximal, characteristic, complemented
14C2×(C2×D283)⋊C8C211yesmaximal, characteristic, complemented
15#416×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985392—21no—
18C2×#17985392C2211yescharacteristic
19C2×#17985392C2211yescharacteristic, complemented
20C2×#17985391—21no—
21C2×#17985391C2211yescharacteristic
22#41×D283—24no—
23#41×D283—24no—
24C2×Q8×D283C2211yescharacteristic
25C2×D4×D283—21no—
26C2×D4×D283C2211yescharacteristic
27C22×C4×D283—21no—
28C22×C4×D283C2211yescharacteristic
29#22583632—24no—
30#22583630—24no—
31C2×Dic1132C2211yescharacteristic, complemented
32C2×#17985382C2211yescharacteristic, complemented
33C2×#17985381C2211yescharacteristic, complemented
34C2×#17985381C2211yescharacteristic, complemented
35C2×#17985380C2211yescharacteristic, complemented
36C2×#17985379C2211yescharacteristic, complemented
37C2×D2264C2211yescharacteristic, complemented
38C2×#17985377C2211yescharacteristic, complemented
39C22×C283⋊C8C2211yescharacteristic, complemented
40#22583594—21no—
41#22583590C2211yescharacteristic
42#22583590C2211yescharacteristic, complemented
43#22583585C2211yescharacteristic
44#22583585C2211yescharacteristic, complemented
45#22583580—21no—
46C4×D1132—21no—
47C4×D1132—21no—
48#22583533C2211yescharacteristic, complemented
49#22583533C2211yescharacteristic, complemented
50#22583530C2211yescharacteristic, complemented
51#22583530C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic
53(C2×D283)⋊C8C2211yescharacteristic
54(C2×D283)⋊C8C2211yescharacteristic, complemented
55(C2×D283)⋊C8C2211yescharacteristic, complemented
56C2×#41×C283C2211yescharacteristic
57C22×C8×C283C2211yescharacteristic, complemented
58#117×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#103×C283C2211yescharacteristic, complemented
61#102×C283C2211yescharacteristic, complemented
62#102×C283C2211yescharacteristic, complemented
63C23×D283—21no—
64C23×D283D411yescharacteristic
65#17985393—22no—
66#17985393—22no—
67#17985392—28no—
68#17985392—22no—
69#17985392—22no—
70#17985392—22no—
71#17985392D414yescomplemented
72#17985391—28no—
73#17985391—22no—
74#17985391—22no—
75C2×Dic566—21no—
76C2×Dic566C2311yescharacteristic
77C2×Dic566C2311yescharacteristic
78Q8×D283—22no—
79Q8×D283—22no—
80C2×D1132—21no—
81C2×D1132—21no—
82C2×D1132—21no—
83C2×D1132C2311yescharacteristic
84C2×D1132C2311yescharacteristic
85C2×C22⋊D283—21no—
86C2×C22⋊D283—21no—
87C2×C22⋊D283—21no—
88C2×C22⋊D283—21no—
89D4×D283—28no—
90D4×D283—22no—
91D4×D283—22no—
92C22×Dic283—21no—
93C22×Dic283D411yescharacteristic
94C2×C4×D283—28no—
95C2×C4×D283—21no—
96C2×C4×D283—21no—
97C2×C4×D283—21no—
98C2×C4×D283—22no—
99C2×C4×D283—22no—
100C2×C4×D283—21no—
101C2×C4×D283D411yescharacteristic
102C2×C4×D283D411yescharacteristic
103C2×C4×D283C2311yescharacteristic
104C2×C4×D283C2311yescharacteristic
105Dic1132—22no—
106#17985382—22no—
107#17985381—22no—
108#17985381—22no—
109#17985380—22no—
110#17985379—22no—
111D2264—22no—
112#17985377—22no—
113C2×C283⋊C8—22no—
114C2×C283⋊C8C2311yescharacteristic
115C2×C283⋊C8C2311yescharacteristic
116C4⋊Dic283—21no—
117C4⋊Dic283—21no—
118(C2×D283)⋊C4—21no—
119(C2×D283)⋊C4—21no—
120(C2×D283)⋊C4—21no—
121(C2×D283)⋊C4—21no—
122#41×C283—22no—
123#41×C283D414yescomplemented
124C2×Q8×C283C2311yescharacteristic
125C2×D4×C283—21no—
126C2×D4×C283C2311yescharacteristic
127C22×C4×C283—21no—
128C22×C4×C283C2311yescharacteristic
129C2×C8×C283—22no—
130C2×C8×C283C2311yescharacteristic
131C2×C8×C283C2311yescharacteristic
132C4⋊C4×C283C2311yescharacteristic
133C4⋊C4×C283C2311yescharacteristic
134C22⋊C4×C283—21no—
135C42×C283—21no—
136C22×D283—24no—
137C22×D283—22no—
138C22×D283—21no—
139C22×D283—22no—
140C22×D283—21no—
141C22×D283—41no—
142C22×D283—42no—
143C22×D283—42no—
144C22×D283C2×D411yescharacteristic
145C22×D283C2×D411yescharacteristic
146Dic566—24no—
147Dic566—21no—
148Dic566—21no—
149Dic566—21no—
150Dic566C2×D412yes—
151D1132—24no—
152D1132—21no—
153D1132—21no—
154D1132—21no—
155D1132—42no—
156D1132—42no—
157D1132C2×D412yes—
158C22⋊D283—24no—
159C22⋊D283—24no—
160C22⋊D283—24no—
161C22⋊D283—42no—
162C2×Dic283—24no—
163C2×Dic283—22no—
164C2×Dic283—21no—
165C2×Dic283—21no—
166C2×Dic283C2×D411yescharacteristic
167C2×Dic283C2×D411yescharacteristic
168C4×D283—24no—
169C4×D283—24no—
170C4×D283—22no—
171C4×D283—24no—
172C4×D283—22no—
173C4×D283—21no—
174C4×D283—21no—
175C4×D283—21no—
176C4×D283—21no—
177C4×D283—42no—
178C283⋊C8—21no—
179C283⋊C8—21no—
180C23×C283—21no—
181C23×C283C2×D411yescharacteristic
182Q8×C283—21no—
183Q8×C283C2×D412yes—
184D4×C283—24no—
185D4×C283—21no—
186D4×C283C2×D412yes—
187C2×C4×C283—21no—
188C2×C4×C283—24no—
189C2×C4×C283—21no—
190C2×C4×C283—21no—
191C2×C4×C283C2×D412yes—
192C2×C4×C283C2×D411yescharacteristic
193C2×C4×C283C2×D412yes—
194C2×C4×C283C2411yescharacteristic
195C8×C283—21no—
196C8×C283—21no—
197C22×C283—21no—
198C22×C283—21no—
199C22×C283—22no—
200C22×C283—41no—
201C22×C283—41no—
202C22×C283#12012yes—
203C22×C283C22×D411yescharacteristic
204C2×D283—21no—
205C2×D283—21no—
206C2×D283—21no—
207C2×D283—22no—
208C2×D283—21no—
209C2×D283—21no—
210C2×D283—44no—
211C2×D283—48no—
212C2×D283—41no—
213C2×D283—41no—
214C2×D283—41no—
215C2×D283—41no—
216C2×D283—41no—
217C2×D283#13512yescomplemented
218C4×C283—22no—
219C4×C283—41no—
220C4×C283—41no—
221C4×C283#12012yes—
222C4×C283C22×D411yescharacteristic
223C4×C283C22×D411yescharacteristic
224Dic283—21no—
225Dic283—22no—
226Dic283—21no—
227Dic283—41no—
228C2×C283—22no—
229C2×C283—42no—
230C2×C283C2×#12011yescharacteristic
231C2×C283C2×#13511yescharacteristic
232C2×C283#57511yescharacteristic
233D283—22no—
234D283—41no—
235D283—42no—
236D283—81no—
237C283#300211yescharacteristic, complemented, Sylow
238#3002—2831nomaximal, Sylow
239C22×#41—2831no—
240C2×#135—2831no—
241C2×#135—2831no—
242#525—2831no—
243#452—2831no—
244#452—2831no—
245#451—2831no—
246#450—2831no—
247#449—2831no—
248#449—2831no—
249#448—2831no—
250#447—2831no—
251#416—2831no—
252#416D28311yescharacteristic, complemented
253C2×C22⋊C8—2831no—
254C2×#41—2831no—
255C2×#41—2831no—
256C2×#41—2831no—
257C2×#41—2831no—
258C2×#41—5661no—
259C2×#41—5661no—
260C2×#41—5664no—
261C2×#41—5664no—
262C2×#41C2×D28311yescharacteristic
263C22×Q8—2831no—
264C22×D4—2831no—
265C22×D4—5661no—
266C23×C4—2831no—
267C23×C4—5661no—
268#135—5664no—
269#135—5664no—
270C2×Q16—2831no—
271C2×Q16—2831no—
272C2×SD16—2831no—
273C2×SD16—2831no—
274C2×SD16—2831no—
275C2×SD16—2831no—
276C2×D8—2831no—
277C2×D8—2831no—
278C22×C8—2831no—
279C22×C8C2×D28311yescharacteristic, complemented
280#123—5661no—
281C22⋊Q8—2831no—
282C22⋊Q8—2831no—
283#121—2831no—
284#121—2831no—
285#120—5661no—
286C4×D4—5661no—
287C4×D4—5661no—
288#117C2×D28311yescharacteristic, complemented
289#103—2831no—
290#103—2831no—
291#103C2×D28311yescharacteristic, complemented
292#103C2×D28311yescharacteristic, complemented
293#102—2831no—
294#102—2831no—
295#102C2×D28311yescharacteristic, complemented
296#102C2×D28311yescharacteristic, complemented
297C22⋊C8—2831no—
298C22⋊C8—2831no—
299C22⋊C8—2831no—
300C22⋊C8—2831no—
301C24—2831no—
302C24—5661no—
303#41—22no—
304#41—2834no—
305#41—5668no—
306#41—5662no—
307#41—5668no—
308#41—5662no—
309#41—5662no—
310#41—5662no—
311#41—5662no—
312#41—5662no—
313#41—5662no—
314#41D113214yescomplemented
315C2×Q8—2831no—
316C2×Q8—2831no—
317C2×Q8—5661no—
318C2×Q8—5662no—
319C2×Q8—5662no—
320C2×Q8C22×D28311yescharacteristic
321C2×D4—21no—
322C2×D4—2831no—
323C2×D4—2831no—
324C2×D4—5661no—
325C2×D4—5661no—
326C2×D4—5661no—
327C2×D4—5661no—
328C2×D4—5662no—
329C2×D4—5668no—
330C2×D4—5661no—
331C2×D4—5661no—
332C2×D4—5662no—
333C2×D4—5661no—
334C2×D4C22×D28311yescharacteristic
335C22×C4—21no—
336C22×C4—2831no—
337C22×C4—2831no—
338C22×C4—2831no—
339C22×C4—2831no—
340C22×C4—2831no—
341C22×C4—5661no—
342C22×C4—5661no—
343C22×C4—5668no—
344C22×C4—5661no—
345C22×C4—5662no—
346C22×C4—5661no—
347C22×C4—5662no—
348C22×C4—5661no—
349C22×C4C22×D28311yescharacteristic
350Q16—5662no—
351Q16—5662no—
352SD16—5662no—
353SD16—5662no—
354SD16—5662no—
355SD16—5662no—
356D8—5662no—
357D8—5662no—
358C2×C8—22no—
359C2×C8—2831no—
360C2×C8—2831no—
361C2×C8—5662no—
362C2×C8C22×D28311yescharacteristic
363C2×C8C22×D28311yescharacteristic
364C4⋊C4—5661no—
365C4⋊C4—5661no—
366C4⋊C4C22×D28311yescharacteristic
367C4⋊C4C22×D28311yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—5661no—
370C22⋊C4—5661no—
371C22⋊C4—5661no—
372C22⋊C4—5661no—
373C42—21no—
374C23—21no—
375C23—2831no—
376C23—2831no—
377C23—5662no—
378C23—5664no—
379C23—5662no—
380C23—5661no—
381C23—5661no—
382C23—11321no—
383C23—11322no—
384C23—11322no—
385C23D4×D28311yescharacteristic
386Q8—21no—
387Q8—2832no—
388Q8—5664no—
389Q8—5661no—
390Q8—5661no—
391Q8—5661no—
392Q8C2×D113212yes—
393D4—24no—
394D4—21no—
395D4—2832no—
396D4—5664no—
397D4—5664no—
398D4—5664no—
399D4—5664no—
400D4—5661no—
401D4—5661no—
402D4—5661no—
403D4—11322no—
404D4—11322no—
405D4—11322no—
406D4C2×D113212yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—2831no—
412C2×C4—2831no—
413C2×C4—5664no—
414C2×C4—5664no—
415C2×C4—5664no—
416C2×C4—5662no—
417C2×C4—5661no—
418C2×C4—5662no—
419C2×C4—5662no—
420C2×C4—5664no—
421C2×C4—5661no—
422C2×C4—5661no—
423C2×C4—5661no—
424C2×C4—5661no—
425C2×C4—5661no—
426C2×C4—11322no—
427C2×C4D4×D28312yes—
428C2×C4D4×D28311yescharacteristic
429C2×C4C2×D113212yes—
430C2×C4C23×D28311yescharacteristic
431C8—21no—
432C8—21no—
433C8—5661no—
434C8—5661no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—2832no—
441C22—5661no—
442C22—5661no—
443C22—5662no—
444C22—5661no—
445C22—5661no—
446C22—5661no—
447C22—11324no—
448C22—11328no—
449C22—11321no—
450C22—11321no—
451C22—11321no—
452C22—11321no—
453C22—11321no—
454C22#2258358012yes—
455C22C2×D4×D28311yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—5661no—
460C4—5661no—
461C4—5662no—
462C4—11321no—
463C4#2258358012yes—
464C4C2×D4×D28311yescharacteristic
465C4C22×D113211yescharacteristic
466C2—22no—
467C2—42no—
468C2—5662no—
469C2—11321no—
470C2—11322no—
471C2—22641no—
472C2C2×#2258358011yescharacteristic
473C2#135×D28311yescharacteristic
474C2#3182726211yescharacteristic
subgroupG/H
centerC2×C4D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#416×C283C2
socleC22×C283C22×D4
hypercenter#416D283
hypocenterC283#3002
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#41×D283C2semidirect
2. C2×#22583632C2semidirect
3. C2×#22583630C2semidirect
4. #31826977C2semidirect
5. #31826630C2semidirect
6. #31826628C2semidirect
7. #31826622C2semidirect
8. #31826617C2semidirect
9. #31826613C2semidirect
10. #31826611C2semidirect
11. #31826605C2semidirect
12. #31826600C2semidirect
13. #31826459C2semidirect
14. C2×(C2×D283)⋊C8C2semidirect
15. #416×C283C2semidirect
19. C2×#17985392C22semidirect
31. C2×Dic1132C22semidirect
32. C2×#17985382C22semidirect
33. C2×#17985381C22semidirect
34. C2×#17985381C22semidirect
35. C2×#17985380C22semidirect
36. C2×#17985379C22semidirect
37. C2×D2264C22semidirect
38. C2×#17985377C22semidirect
39. C22×C283⋊C8C22semidirect
42. #22583590C22semidirect
44. #22583585C22semidirect
48. #22583533C22semidirect
49. #22583533C22semidirect
50. #22583530C22semidirect
51. #22583530C22semidirect
54. (C2×D283)⋊C8C22semidirect
55. (C2×D283)⋊C8C22semidirect
57. C22×C8×C283C22semidirect
58. #117×C283C22semidirect
59. #103×C283C22semidirect
60. #103×C283C22semidirect
61. #102×C283C22semidirect
62. #102×C283C22semidirect
71. #17985392D4semidirect
123. #41×C283D4semidirect
217. C2×D283#135semidirect
237. C283#3002semidirect
252. #416D283semidirect
279. C22×C8C2×D283semidirect
288. #117C2×D283semidirect
291. #103C2×D283semidirect
292. #103C2×D283semidirect
295. #102C2×D283semidirect
296. #102C2×D283semidirect
314. #41D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2832
C2×#225836322
C2×#225836302
#318269772
#318266302
#318266282
#318266222
#318266172
#318266132
#318266112
#318266052
#318266002
#318264592
C2×(C2×D283)⋊C82
#416×C2832
#3002283