Finite Group Databasebeta

← #508485

Subgroups — #508485

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32931C211yesmaximal, characteristic
2#20658C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic, complemented
4#13494C211yesmaximal, characteristic, complemented
5#13493C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic, complemented
8#13383C211yesmaximal, characteristic, complemented
9#13382C211yesmaximal, characteristic, complemented
10#13382C211yesmaximal, characteristic, complemented
11#12994C211yesmaximal, characteristic, complemented
12#12994C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic, complemented
15#12679C211yesmaximal, characteristic
16C24×C8C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2848—21no—
23#2280—21no—
24C2×#367C2211yescharacteristic, complemented
25C2×#367C2211yescharacteristic, complemented
26C2×#366C2211yescharacteristic, complemented
27C2×#366C2211yescharacteristic, complemented
28C2×#358C2211yescharacteristic, complemented
29C2×#358C2211yescharacteristic, complemented
30C2×#358C2211yescharacteristic, complemented
31C2×#358C2211yescharacteristic, complemented
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38#1892—21no—
39#1892—21no—
40#1892—21no—
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1807—21no—
49#1807—21no—
50#1735—21no—
51#1380C2211yescharacteristic, complemented
52#1380C2211yescharacteristic, complemented
53#1380C2211yescharacteristic, complemented
54#1380C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59C24×C4D411yescharacteristic
60C22×Q16C2311yescharacteristic
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—41no—
64C23×C8—41no—
65C23×C8—41no—
66C23×C8—41no—
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×#117—21no—
77C2×#117—21no—
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#467—22no—
83#467—22no—
84#467—41no—
85#467—41no—
86#467—41no—
87#467—41no—
88#427—41no—
89#427—41no—
90#427—41no—
91#427—41no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#106—21no—
100C2×#106—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104#385—41no—
105#385—41no—
106#385—41no—
107#381—41no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#379—41no—
116#379—41no—
117C4×C22⋊C4—41no—
118C4×C22⋊C4—41no—
119#367—41no—
120#367—41no—
121#366—41no—
122#366—41no—
123#358—41no—
124#358—41no—
125#358—41no—
126#358—41no—
127C22⋊C16—22no—
128C22⋊C16—22no—
129C22⋊C16—41no—
130C22⋊C16—41no—
131C22⋊C16—41no—
132C22⋊C16—41no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C25D811yescharacteristic
141C22×Q8—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4D811yescharacteristic
149C23×C4D811yescharacteristic
150C23×C4D811yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—41no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8—41no—
183C22×C8—41no—
184C22×C8—41no—
185C22×C8—44no—
186C22×C8—44no—
187C22×C8—44no—
188C22×C8—44no—
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8#4112yes—
193C22×C8#4112yes—
194C22×C8C2411yescharacteristic
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201#117—42no—
202#117—42no—
203#117—42no—
204#117—42no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#106—42no—
256#106—42no—
257#103—42no—
258#103—42no—
259#103—42no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24C2×D811yescharacteristic
275C24C2×D811yescharacteristic
276C24C2×D811yescharacteristic
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4#12312yes—
321C22×C4C2×D811yescharacteristic
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4#13712yes—
325C22×C4C22×D411yescharacteristic
326Q16—41no—
327Q16—41no—
328Q16—41no—
329Q16—41no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—44no—
343C2×C8—44no—
344C2×C8—44no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8#12312yes—
354C2×C8C2×#4111yescharacteristic
355C2×C8C2×#4111yescharacteristic
356C2×C8H(5,𝔽2)11yescharacteristic
357C2×C8H(5,𝔽2)11yescharacteristic
358C2×C8H(5,𝔽2)11yescharacteristic
359C2×C8H(5,𝔽2)11yescharacteristic
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C42—82no—
390C42—82no—
391C16—41no—
392C16—41no—
393C16—41no—
394C16—41no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23#48012yes—
421C23#50812yes—
422C23#51012yes—
423C23C22×D811yescharacteristic
424Q8—81no—
425Q8—81no—
426Q8—81no—
427Q8—81no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4#48012yes—
476C2×C4C2×#12311yescharacteristic
477C2×C4#53411yescharacteristic
478C2×C4#53411yescharacteristic
479C2×C4C2×#13711yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57511yescharacteristic
482C8—22no—
483C8—41no—
484C8—41no—
485C8#53811yescharacteristic
486C8#53811yescharacteristic
487C8#53811yescharacteristic
488C8#53811yescharacteristic
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—416no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22C2×#48011yescharacteristic
512C22#318611yescharacteristic
513C22#318611yescharacteristic
514C22C2×#50811yescharacteristic
515C22C2×#51011yescharacteristic
516C22#341411yescharacteristic
517C22#341411yescharacteristic
518C4—22no—
519C4—41no—
520C4—41no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4—161no—
525C4—161no—
526C4—161no—
527C4—161no—
528C4—161no—
529C4#262211yescharacteristic
530C4#309311yescharacteristic
531C4#319611yescharacteristic
532C4#319611yescharacteristic
533C2—42no—
534C2—42no—
535C2—42no—
536C2#2989011yescharacteristic
537C2#3364511yescharacteristic
538C2#3364911yescharacteristic
539C2#3375611yescharacteristic
540C2#3375711yescharacteristic
541C2#3376211yescharacteristic
542C2#3376311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13494C2semidirect
4. #13494C2semidirect
5. #13493C2semidirect
6. #13492C2semidirect
7. #13383C2semidirect
8. #13383C2semidirect
9. #13382C2semidirect
10. #13382C2semidirect
11. #12994C2semidirect
12. #12994C2semidirect
13. #12921C2semidirect
14. #12921C2semidirect
24. C2×#367C22semidirect
25. C2×#367C22semidirect
26. C2×#366C22semidirect
27. C2×#366C22semidirect
28. C2×#358C22semidirect
29. C2×#358C22semidirect
30. C2×#358C22semidirect
31. C2×#358C22semidirect
51. #1380C22semidirect
52. #1380C22semidirect
53. #1380C22semidirect
54. #1380C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect

Maximal subgroups

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

subgroupindex
#329312
#206582
#134942
#134932
#134922
#133832
#133822
#129942
#129212
#126792