Finite Group Databasebeta

← #473836

Subgroups — #473836

Order 512.

1386 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14698C211yesmaximal, characteristic
2#14682C211yesmaximal, characteristic, complemented
3#11316C212yesmaximal, complemented
4#11268C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11199C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#11051C211yesmaximal, characteristic, complemented
9#11051C211yesmaximal, characteristic, complemented
10#10969C212yesmaximal
11#10712C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×C22⋊C8C2212yes—
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#386—22no—
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1437—22no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37#1434—22no—
38C22⋊#95—22no—
39C23×D4—21no—
40C23×D4C2311yescharacteristic
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—24no—
48C2×#121—42no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C42—22no—
60C22×C42—22no—
61C22×C42—42no—
62C22×C42—42no—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#102—24no—
67C2×#102—42no—
68C2×#102—42no—
69C2×#102—42no—
70C2×#102—42no—
71C2×#102—42no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—42no—
74C2×C22⋊C8—42no—
75#390—22no—
76#390—22no—
77#390—42no—
78#386—48no—
79#382—42no—
80#382—42no—
81#380—24no—
82#380—42no—
83#380—42no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92#340—24no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336—22no—
99#336—22no—
100#336—42no—
101C25—21no—
102C25C2×D411yescharacteristic
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—42no—
114C22×D4—44no—
115C22×D4—42no—
116C22×D4—44no—
117C22×D4#4112yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4#4114yes—
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158#121—48no—
159#121—48no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4#4112yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—48no—
206C2×C42—48no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42#4112yes—
210C2×C42#4112yes—
211#102—48no—
212#102—48no—
213C22⋊C8—48no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—42no—
221#95—44no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24#12412yes—
243C24AGL(1,ℤ/8)14yes—
244C24C22×D411yescharacteristic
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—48no—
250C2×D4—416no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—84no—
264C2×D4—88no—
265C2×D4—84no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4#12314yes—
337C22×C4#12412yes—
338C22×C4#13514yes—
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+411yescharacteristic
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—24no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C22⋊C4—48no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C22⋊C4—84no—
382C42—24no—
383C42—24no—
384C42—48no—
385C42—42no—
386C42—42no—
387C23—22no—
388C23—24no—
389C23—24no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—42no—
414C23—82no—
415C23—82no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—81no—
431C23—81no—
432C23#49014yes—
433C23C2×#12311yescharacteristic
434C23#52511yescharacteristic
435C23C2×#12411yescharacteristic
436C23#53411yescharacteristic
437C23#53411yescharacteristic
438C23#53511yescharacteristic
439C23#53711yescharacteristic
440C23C2×#13512yes—
441C23C2×AGL(1,ℤ/8)12yes—
442C23#57512yes—
443C23#57512yes—
444D4—44no—
445D4—44no—
446D4—816no—
447D4—82no—
448D4—82no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—432no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4#53812yes—
505C2×C4#53812yes—
506C2×C4#53812yes—
507C2×C4#53812yes—
508C2×C4#53912yes—
509C2×C4#54012yes—
510C2×C4#55112yes—
511C2×C4#55112yes—
512C2×C4#55211yescharacteristic
513C2×C4#55312yes—
514C2×C4#55311yescharacteristic
515C2×C4#55312yes—
516C2×C4#55511yescharacteristic
517C2×C4#55611yescharacteristic
518C2×C4#56012yes—
519C2×C4#56212yes—
520C8—82no—
521C8—82no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—42no—
526C22—44no—
527C22—42no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—42no—
532C22—42no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22—82no—
538C22—161no—
539C22—161no—
540C22—162no—
541C22—162no—
542C22—161no—
543C22—161no—
544C22#248511yescharacteristic
545C22#250411yescharacteristic
546C22#256111yescharacteristic
547C22#262211yescharacteristic
548C22#263511yescharacteristic
549C22#263911yescharacteristic
550C22#281311yescharacteristic
551C22#309112yes—
552C22#309412yes—
553C22#310212yes—
554C22#311412yes—
555C22#313111yescharacteristic
556C22C2×#49011yescharacteristic
557C22#314011yescharacteristic
558C22#314211yescharacteristic
559C22#318812yes—
560C22#319112yes—
561C22#319412yes—
562C22#319712yes—
563C22#321212yes—
564C22#321312yes—
565C22#324712yes—
566C22#325612yes—
567C4—44no—
568C4—44no—
569C4—42no—
570C4—82no—
571C4—161no—
572C4—162no—
573C4—162no—
574C4—161no—
575C4#275012yes—
576C4#275612yes—
577C4#277812yes—
578C4#278612yes—
579C2—44no—
580C2—162no—
581C2#1688411yescharacteristic
582C2#2763312yes—
583C2#2797611yescharacteristic
584C2#2798211yescharacteristic
585C2#2910811yescharacteristic
586C2#2911011yescharacteristic
587C2#2989612yes—
588C2#3011112yes—
589C2#3018911yescharacteristic
590C2#3019811yescharacteristic
591C2#3272712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14682C2semidirect
3. #11316C2semidirect
4. #11268C2semidirect
5. #11268C2semidirect
6. #11199C2semidirect
7. #11189C2semidirect
8. #11051C2semidirect
9. #11051C2semidirect
11. #10712C2semidirect

Maximal subgroups

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

subgroupindex
#146982
#146822
#113162
#112682
#111992
#111892
#110512
#109692
#107122