Finite Group Databasebeta

← #196311

Subgroups — #196311

Order 512.

1655 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic, complemented
2C4×#534C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic, complemented
4C2×#1459C212yesmaximal, complemented
5#9108C214yesmaximal, complemented
6#9107C214yesmaximal, complemented
7C2×#1282C212yesmaximal, complemented
8C2×#534C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2972—24no—
12#2972—22no—
13#2972C2214yes—
14C2×C8×D4—24no—
15C2×#431—21no—
16C22×C22⋊C8—21no—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19#2325—22no—
20#2322—21no—
21#2322C2212yes—
22#2321—21no—
23#2321C2212yes—
24#2315C411yescharacteristic
25#2308C411yescharacteristic
26#2308C411yescharacteristic
27C4×#123—21no—
28C4×#123C412yes—
29C4×#121—22no—
30C4×#120—21no—
31C4×#120C412yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1459—24no—
34#1459—24no—
35#1459—22no—
36#1459C2214yes—
37C2×#324—21no—
38C2×#324C2214yescomplemented
39C2×#323—21no—
40C2×#323C2214yescomplemented
41#1282C2218yescomplemented
42C23×D4C2×C411yescharacteristic
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—22no—
50#534—24no—
51#534C818yescomplemented
52C2×#123—21no—
53C2×#123C2×C412yes—
54C2×#121—22no—
55C2×#120—21no—
56C2×#120C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C412yes—
71C2×C4×D4D412yes—
72C2×C4×D4D412yes—
73C2×C4×D4C2312yes—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C8×D4—216no—
86C8×D4—216no—
87#431—24no—
88#431—24no—
89C2×C4⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8D414yescomplemented
102C2×C22⋊C8D414yescomplemented
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×C4×C8—22no—
106#390—22no—
107#389—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—21no—
116#386—21no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386C2×C412yes—
121#386C2×C412yes—
122#385—21no—
123#385—21no—
124#385—22no—
125#385—22no—
126#385C2×C412yes—
127#385C2×C412yes—
128#381—21no—
129#381C2×C412yes—
130#380—22no—
131#379—21no—
132#379C2×C412yes—
133C4×C4⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4C2×C412yes—
139C4×C22⋊C4C2312yes—
140C2×#95—21no—
141#324—24no—
142#324—28no—
143#323—24no—
144#323—28no—
145C25C22⋊C411yescharacteristic
146C25C22×C411yescharacteristic
147C25C22×C411yescharacteristic
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—24no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—24no—
157C22×D4—22no—
158C22×D4C22⋊C412yes—
159C22×D4C2×C814yes—
160C22×D4C2×C814yescomplemented
161C22×D4M1614yes—
162C22×D4C22×C412yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4—82no—
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C412yes—
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4#4112yes—
192C23×C4#4112yes—
193C23×C4C2411yescharacteristic
194C22×C8—24no—
195C22×C8—28no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8—24no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—84no—
202C22×C8—84no—
203C22×C8—84no—
204#123—24no—
205#123—28no—
206#123C2×C818yescomplemented
207#121—216no—
208#120—28no—
209#120—24no—
210#120C2×C818yescomplemented
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—41no—
220C4×D4—42no—
221C4×D4—41no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—41no—
227C4×D4—42no—
228C4×D4—41no—
229C4×D4—84no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4C22⋊C412yes—
264C2×C22⋊C4C22⋊C412yes—
265C2×C22⋊C4C2×C814yescomplemented
266C2×C22⋊C4C22×C412yes—
267C2×C22⋊C4C22×C412yes—
268C2×C22⋊C4C22×C412yes—
269C2×C22⋊C4C2×D412yes—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—82no—
281C2×C42C22⋊C412yes—
282C2×C42C2×D412yes—
283C4⋊C8—28no—
284C22⋊C8—28no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8C2×D418yes—
293C4×C8—28no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302C24—21no—
303C24—21no—
304C24—22no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—22no—
309C24—81no—
310C24—81no—
311C24—81no—
312C24—81no—
313C24—81no—
314C24—81no—
315C24C22⋊C814yes—
316C24C2×C22⋊C412yes—
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C24C4×D412yes—
320C24C4×D412yes—
321C24C22×C812yes—
322C24C22×C812yescomplemented
323C24C2×M1612yes—
324C24#13114yes—
325C24C23×C411yescharacteristic
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—28no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—42no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4C22⋊C818yes—
349C2×D4C22×C814yes—
350C2×D4C2×M1614yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—82no—
386C22×C4—84no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—82no—
394C22×C4—81no—
395C22×C4—82no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—81no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—84no—
405C22×C4C22⋊C814yes—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C4×D412yes—
412C22×C4C4×D412yes—
413C22×C4#12011yescharacteristic
414C22×C4#12011yescharacteristic
415C22×C4#12012yes—
416C22×C4#12112yes—
417C22×C4#12112yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4#12311yescharacteristic
421C22×C4#12311yescharacteristic
422C22×C4#12312yes—
423C22×C4C22×C812yescomplemented
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2×#4112yes—
428C2×C8—28no—
429C2×C8—48no—
430C2×C8—44no—
431C2×C8—84no—
432C2×C8—84no—
433C2×C8—88no—
434C2×C8—84no—
435C2×C8—84no—
436C2×C8—84no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C22⋊C4—24no—
446C22⋊C4—22no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—28no—
453C22⋊C4—24no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—42no—
458C22⋊C4—41no—
459C22⋊C4—42no—
460C22⋊C4—41no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4C22×C814yescomplemented
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—44no—
471C42—41no—
472C42—42no—
473C42—41no—
474C42—82no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—81no—
501C23—81no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—84no—
507C23—84no—
508C23—82no—
509C23—82no—
510C23—84no—
511C23—84no—
512C23#38612yes—
513C23#38611yescharacteristic
514C23#38611yescharacteristic
515C23C2×C22⋊C814yes—
516C23C2×C22⋊C812yes—
517C23C2×C22⋊C812yes—
518C23#40812yes—
519C23#40812yes—
520C23C8×D414yes—
521C23C8×D414yes—
522C23#43514yes—
523C23#43514yes—
524C23C22×C22⋊C411yescharacteristic
525C23C2×C4×D412yes—
526C23C23×C811yescharacteristic
527C23C22×M1611yescharacteristic
528C23C2×#13112yes—
529D4—216no—
530D4—28no—
531D4—28no—
532D4—44no—
533D4—44no—
534D4—48no—
535D4—88no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—84no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4#38611yescharacteristic
593C2×C4#38612yes—
594C2×C4#38611yescharacteristic
595C2×C4C2×C22⋊C814yes—
596C2×C4C2×#12011yescharacteristic
597C2×C4C2×#12112yes—
598C2×C4C2×#12311yescharacteristic
599C8—48no—
600C8—84no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—21no—
605C22—22no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—21no—
610C22—22no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—82no—
630C22—84no—
631C22—84no—
632C22—82no—
633C22—84no—
634C22—81no—
635C22—81no—
636C22#186912yes—
637C22#186912yes—
638C22#186914yes—
639C22C2×#38611yescharacteristic
640C22C22×C22⋊C811yescharacteristic
641C22C2×#40811yescharacteristic
642C22C2×C8×D412yes—
643C22C2×#43512yes—
644C4—21no—
645C4—22no—
646C4—21no—
647C4—21no—
648C4—22no—
649C4—24no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4#191312yes—
658C2—21no—
659C2—21no—
660C2—22no—
661C2—22no—
662C2—22no—
663C2—22no—
664C2—24no—
665C2—44no—
666C2—82no—
667C2C2×#186911yescharacteristic
668C2C2×#191311yescharacteristic
669C2#2101711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC2×#534C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2972C2semidirect
3. C2×#1459C2semidirect
4. C2×#1459C2semidirect
5. #9108C2semidirect
6. #9107C2semidirect
7. C2×#1282C2semidirect
38. C2×#324C22semidirect
40. C2×#323C22semidirect
41. #1282C22semidirect
51. #534C8semidirect
101. C2×C22⋊C8D4semidirect
102. C2×C22⋊C8D4semidirect
160. C22×D4C2×C8semidirect
206. #123C2×C8semidirect
210. #120C2×C8semidirect
265. C2×C22⋊C4C2×C8semidirect
322. C24C22×C8semidirect
423. C22×C4C22×C8semidirect
465. C22⋊C4C22×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
C4×#5342
C2×#14592
#91082
#91072
C2×#12822