Finite Group Databasebeta

← #508484

Subgroups — #508484

Order 512.

741 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32930C211yesmaximal, characteristic
2#20658C211yesmaximal, characteristic, complemented
3#13494C211yesmaximal, characteristic, complemented
4#13494C211yesmaximal, characteristic, complemented
5#13493C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13381C211yesmaximal, characteristic, complemented
9#13380C211yesmaximal, characteristic, complemented
10#13380C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic, complemented
12#12992C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic, complemented
15#12679C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2847—21no—
23#2280—21no—
24C2×#367C2211yescharacteristic, complemented
25C2×#367C2211yescharacteristic, complemented
26C2×#366C2211yescharacteristic, complemented
27C2×#366C2211yescharacteristic, complemented
28C2×#357C2211yescharacteristic, complemented
29C2×#357C2211yescharacteristic, complemented
30C2×#357C2211yescharacteristic, complemented
31C2×#357C2211yescharacteristic, complemented
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38#1893—21no—
39#1893—21no—
40#1893—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×D8C2311yescharacteristic
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×#121—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#120—41no—
77C2×#117—21no—
78C2×#117—21no—
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83#466—22no—
84#466—22no—
85#466—41no—
86#466—41no—
87#466—41no—
88#466—41no—
89#427—41no—
90#427—41no—
91#427—41no—
92#427—41no—
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#106—21no—
101C2×#106—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105#386—41no—
106#386—41no—
107#386—41no—
108#381—41no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#379—41no—
117C4×C22⋊C4—41no—
118C4×C22⋊C4—41no—
119#367—41no—
120#367—41no—
121#366—41no—
122#366—41no—
123#357—41no—
124#357—41no—
125#357—41no—
126#357—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×D4—21no—
142C22×D4—41no—
143C22×D4—41no—
144C22×D4—41no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4D811yescharacteristic
152C23×C4D811yescharacteristic
153C23×C4D811yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
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—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—41no—
183C22×C8—41no—
184C22×C8—41no—
185C22×C8—41no—
186C22×C8—41no—
187C22×C8—41no—
188C22×C8—44no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8#4112yes—
196C22×C8#4112yes—
197C22×C8C2411yescharacteristic
198#121—42no—
199#121—42no—
200#121—42no—
201#121—42no—
202#121—42no—
203#121—42no—
204#120—84no—
205#117—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
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×C22⋊C4—41no—
228C2×C42—41no—
229C2×C42—41no—
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—
241C2×C16—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#107—21no—
256#106—42no—
257#106—42no—
258#102—42no—
259#102—42no—
260#102—42no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24C2×D811yescharacteristic
277C24C2×D811yescharacteristic
278C24C2×D811yescharacteristic
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4#12312yes—
331C22×C4C2×D811yescharacteristic
332C22×C4C2×D811yescharacteristic
333C22×C4C2×D811yescharacteristic
334C22×C4#13712yes—
335C22×C4C22×D411yescharacteristic
336D8—41no—
337D8—41no—
338D8—41no—
339D8—41no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—44no—
353C2×C8—44no—
354C2×C8—44no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8#12312yes—
364C2×C8C2×#4111yescharacteristic
365C2×C8C2×#4111yescharacteristic
366C2×C8H(5,𝔽2)11yescharacteristic
367C2×C8H(5,𝔽2)11yescharacteristic
368C2×C8H(5,𝔽2)11yescharacteristic
369C2×C8H(5,𝔽2)11yescharacteristic
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C42—82no—
397C42—82no—
398C16—41no—
399C16—41no—
400C16—41no—
401C16—41no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23#48012yes—
435C23#50812yes—
436C23#50912yes—
437C23C22×D811yescharacteristic
438D4—81no—
439D4—81no—
440D4—81no—
441D4—81no—
442D4—164no—
443D4—164no—
444D4—164no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4#48012yes—
490C2×C4C2×#12311yescharacteristic
491C2×C4#53411yescharacteristic
492C2×C4#53411yescharacteristic
493C2×C4C2×#13711yescharacteristic
494C2×C4#57511yescharacteristic
495C2×C4#57511yescharacteristic
496C8—22no—
497C8—41no—
498C8—41no—
499C8#53811yescharacteristic
500C8#53811yescharacteristic
501C8#53811yescharacteristic
502C8#53811yescharacteristic
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—416no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—161no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22C2×#48011yescharacteristic
533C22#318611yescharacteristic
534C22#318611yescharacteristic
535C22C2×#50811yescharacteristic
536C22C2×#50911yescharacteristic
537C22#341411yescharacteristic
538C22#341411yescharacteristic
539C4—22no—
540C4—41no—
541C4—41no—
542C4—161no—
543C4—161no—
544C4—161no—
545C4—161no—
546C4—161no—
547C4—161no—
548C4—161no—
549C4#262211yescharacteristic
550C4#309311yescharacteristic
551C4#319611yescharacteristic
552C4#319611yescharacteristic
553C2—42no—
554C2—42no—
555C2—42no—
556C2—321no—
557C2#2989011yescharacteristic
558C2#3364511yescharacteristic
559C2#3364811yescharacteristic
560C2#3375411yescharacteristic
561C2#3375711yescharacteristic
562C2#3376011yescharacteristic
563C2#3376311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#32930C2
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
2. #20658C2semidirect
3. #13494C2semidirect
4. #13494C2semidirect
5. #13493C2semidirect
6. #13492C2semidirect
7. #13381C2semidirect
8. #13381C2semidirect
9. #13380C2semidirect
10. #13380C2semidirect
11. #12992C2semidirect
12. #12992C2semidirect
13. #12921C2semidirect
14. #12921C2semidirect
15. #12679C2semidirect
24. C2×#367C22semidirect
25. C2×#367C22semidirect
26. C2×#366C22semidirect
27. C2×#366C22semidirect
28. C2×#357C22semidirect
29. C2×#357C22semidirect
30. C2×#357C22semidirect
31. C2×#357C22semidirect
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
#329302
#206582
#134942
#134932
#134922
#133812
#133802
#129922
#129212
#126792