Finite Group Databasebeta

← #191063

Subgroups — #191063

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3#14682C211yesmaximal, characteristic, complemented
4#8712C212yesmaximal, complemented
5#8571C214yesmaximal, complemented
6#8328C212yesmaximal
7#8220C214yesmaximal, complemented
8C22×#121C2211yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic, complemented
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13C4×#121—21no—
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121C412yescomplemented
18C4×#121C412yescomplemented
19C4×#121C2212yes—
20C4×#121C2212yes—
21C2×#390C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#388—21no—
24C2×#386—21no—
25C2×#382—21no—
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic, complemented
29C2×C4×C22⋊C4—21no—
30#1817—21no—
31#1797—21no—
32#1797C2212yescomplemented
33#1779C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8C2212yescomplemented
36C2×#327C2218yes—
37C4⋊#95C2212yescomplemented
38#1434—21no—
39#1434C2212yescomplemented
40C22⋊#95—21no—
41C22⋊#95C2212yes—
42#1294C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C2×#121—21no—
50C2×#121—21no—
51C2×#121—28no—
52C2×#121C2×C412yes—
53C2×#121C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—44no—
61C2×C4×D4—44no—
62C2×C4×D4—44no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C412yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#431—44no—
91C2×C4⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2314yes—
95C2×C4×C8—22no—
96#390—21no—
97#390—21no—
98#390C2×C412yes—
99#390C2×C412yes—
100#389—21no—
101#389—21no—
102#389C2×C412yes—
103#389C2×C412yes—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#382—44no—
113#380—21no—
114#380—21no—
115#380—44no—
116#380C2×C412yes—
117#380C2312yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2314yes—
134#336—22no—
135#336—22no—
136#327—216no—
137C25—21no—
138C25C2×D411yescharacteristic
139C22×D4—28no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—21no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—44no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4C22×C412yes—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4D814yes—
176C23×C4SD1614yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8—24no—
188#121—28no—
189#121—28no—
190#121—24no—
191#121—24no—
192#121—24no—
193#121#4118yes—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—44no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C4⋊C8—44no—
270C22⋊C8—28no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C4×C8—44no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—28no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24C2×C22⋊C411yescharacteristic
300C24C2×C22⋊C411yescharacteristic
301C24C2×D812yes—
302C24C2×SD1612yes—
303C24C22×D411yescharacteristic
304C2×D4—24no—
305C2×D4—28no—
306C2×D4—28no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—416no—
311C2×D4—42no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—42no—
320C2×D4#11714yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—416no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—416no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—816no—
398C22×C4#10214yes—
399C22×C4#10214yes—
400C22×C4#10214yes—
401C22×C4#10214yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×D812yes—
413C22×C4C2×D812yes—
414C22×C4C2×SD1612yes—
415C22×C4C2×SD1612yes—
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4114yes—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—88no—
431C4⋊C4#11714yes—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—88no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—44no—
460C42—44no—
461C42#9914yes—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—24no—
471C23—24no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—48no—
488C23—48no—
489C23—48no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23#41012yes—
501C23C2×#10212yes—
502C23C2×#10212yes—
503C23#42112yes—
504C23C22×C22⋊C411yescharacteristic
505C23#52512yes—
506C23C22×D811yescharacteristic
507C23C22×SD1611yescharacteristic
508C23#57512yes—
509D4—44no—
510D4—44no—
511D4—48no—
512D4—416no—
513D4—48no—
514D4—48no—
515C2×C4—28no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—88no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—88no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4C2×#9912yes—
583C2×C4C2×#9912yes—
584C2×C4C2×#9912yes—
585C2×C4C2×#10212yes—
586C2×C4C2×#10212yes—
587C2×C4C2×#10212yes—
588C2×C4C2×#10212yes—
589C2×C4#42112yes—
590C2×C4C2×#11712yes—
591C2×C4#52512yes—
592C8—84no—
593C22—22no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—48no—
611C22—48no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—816no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22#151714yes—
625C22#228011yescharacteristic
626C22C22×#9911yescharacteristic
627C22C2×#41011yescharacteristic
628C22C22×#10211yescharacteristic
629C22#289611yescharacteristic
630C22C2×#42111yescharacteristic
631C22C2×#42111yescharacteristic
632C22#310012yes—
633C22#310112yes—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—164no—
649C4#151914yes—
650C2—22no—
651C2—24no—
652C2—22no—
653C2—42no—
654C2—84no—
655C2C2×#151911yescharacteristic
656C2C2×#151711yescharacteristic
657C2#1758011yescharacteristic
658C2#1758211yescharacteristic
659C2#2072211yescharacteristic
660C2#2073211yescharacteristic
661C2#2074811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. #14682C2semidirect
4. #8712C2semidirect
5. #8571C2semidirect
7. #8220C2semidirect
10. C22×C4×D4C4semidirect
17. C4×#121C4semidirect
18. C4×#121C4semidirect
21. C2×#390C4semidirect
22. C2×#389C4semidirect
28. C2×C4×C4⋊C4C4semidirect
32. #1797C22semidirect
35. C4×C22⋊C8C22semidirect
37. C4⋊#95C22semidirect
39. #1434C22semidirect
42. #1294C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#207142
#146822
#87122
#85712
#83282
#82202