Finite Group Databasebeta

← #369431

Subgroups — #369431

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic, complemented
4C2×#1989C211yesmaximal, characteristic, complemented
5C2×#1955C211yesmaximal, characteristic
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10731C212yesmaximal, complemented
10#10731C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19#2562—24no—
20#2562C418yescomplemented
21C2×#400—21no—
22C2×#394—21no—
23#2354—24no—
24#2354—24no—
25C2×#390—21no—
26C2×#388—22no—
27C2×#388—22no—
28C2×#384C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1992—24no—
36#1989—24no—
37#1955—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C23×D4D411yescharacteristic
49C24×C4C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125C2×C414yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C414yes—
64C2×C4×D4D414yes—
65C2×#117—24no—
66C2×#117—24no—
67C2×#117—24no—
68C2×#117—24no—
69C2×#117—42no—
70C2×#117—42no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D412yes—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103#400—28no—
104#394—28no—
105#390—28no—
106#388—216no—
107#388—216no—
108#384—22no—
109#384D414yes—
110#380—24no—
111#380—24no—
112#380—24no—
113#380—24no—
114#380C2×C418yes—
115#380C2×C418yes—
116C4×C4⋊C4—28no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C414yes—
120C4×C4⋊C4D414yes—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126#340—24no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#340—42no—
136C25C2×D412yes—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4C22⋊C414yes—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4C22×C414yes—
161C23×C4C2×D412yes—
162C23×C4C2×D414yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8#4114yes—
183#125—28no—
184#125—28no—
185#125—44no—
186#125—44no—
187C4×D4—28no—
188C4×D4—28no—
189C4×D4—28no—
190C4×D4—28no—
191#117—48no—
192#117—48no—
193#117—48no—
194#117—48no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—81no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4C22⋊C414yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4#4114yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—82no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4—82no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4C22⋊C418yes—
256C2×C42—28no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—81no—
263C2×C42—81no—
264C2×C42C22×C412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270C4⋊C8—44no—
271C22⋊C8—48no—
272C22⋊C8—48no—
273#97—44no—
274#95—28no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—24no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24C2×C22⋊C414yes—
294C24C22×D411yescharacteristic
295C2×D4—28no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—44no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4C2×C22⋊C414yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4#12012yes—
380C22×C4#12012yes—
381C22×C4#12312yes—
382C22×C4#12312yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+412yes—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—28no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—48no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—84no—
436C4⋊C4#12414yes—
437C22⋊C4—216no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C42—48no—
443C42—416no—
444C42—84no—
445C42C4×D414yes—
446C42C4×D414yes—
447C42#12114yes—
448C42#12114yes—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23C22×C22⋊C411yescharacteristic
485C23C2×#12011yescharacteristic
486C23C2×#12311yescharacteristic
487C23#54612yes—
488C23#54712yes—
489C23#57512yes—
490C23#57512yes—
491C23#57611yescharacteristic
492C23#57711yescharacteristic
493C23#57711yescharacteristic
494C23#57811yescharacteristic
495D4—416no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4#38614yes—
558C2×C4#38614yes—
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C4×#4112yes—
562C2×C4#52012yes—
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12411yescharacteristic
566C2×C4#52912yes—
567C2×C4#53211yescharacteristic
568C2×C4#53511yescharacteristic
569C2×C4#53611yescharacteristic
570C2×C4#54212yes—
571C2×C4#54612yes—
572C2×C4#54612yes—
573C2×C4#54712yes—
574C2×C4#54812yes—
575C2×C4#55612yes—
576C2×C4#55912yes—
577C8—44no—
578C8—81no—
579C8—81no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22C2×#38611yescharacteristic
596C22#233912yes—
597C22#243211yescharacteristic
598C22#243511yescharacteristic
599C22#244712yes—
600C22#289612yes—
601C22#289711yescharacteristic
602C22#289811yescharacteristic
603C22#300511yescharacteristic
604C22#300511yescharacteristic
605C22#301211yescharacteristic
606C22#301311yescharacteristic
607C22#309312yes—
608C22#309811yescharacteristic
609C22#309911yescharacteristic
610C22#330712yes—
611C22#330912yes—
612C22#331512yes—
613C22#331712yes—
614C22#334712yes—
615C22#334812yes—
616C22#334912yes—
617C22#335012yes—
618C4—24no—
619C4—24no—
620C4—44no—
621C4—44no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—84no—
627C4C4×#12411yescharacteristic
628C4#232611yescharacteristic
629C4#236712yes—
630C4#242712yes—
631C4#254411yescharacteristic
632C4#254711yescharacteristic
633C2—84no—
634C2#1402711yescharacteristic
635C2#2091011yescharacteristic
636C2#2091511yescharacteristic
637C2#2249612yes—
638C2#2249812yes—
639C2#2571011yescharacteristic
640C2#2571111yescharacteristic
641C2#2589211yescharacteristic
642C2#2589311yescharacteristic
643C2#2630912yes—
644C2#2631112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1992C2semidirect
4. C2×#1989C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #10849C2semidirect
9. #10731C2semidirect
10. #10731C2semidirect
11. #10703C2semidirect
20. #2562C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#23542
C2×#19922
C2×#19892
C2×#19552
#112002
#111992
#108492
#107312
#107032