Finite Group Databasebeta

← #214120

Subgroups — #214120

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#9074C214yesmaximal, complemented
5#8939C214yesmaximal, complemented
6#8707C212yesmaximal, complemented
7#8310C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431C2212yescomplemented
13C22×C4⋊C8C2212yescomplemented
14C22×C22⋊C8—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121C412yes—
20C4×#121C412yes—
21C4×#121C2212yes—
22C4×#121C2212yes—
23C2×#390C411yescharacteristic
24C2×#389C411yescharacteristic
25C2×#388—21no—
26C2×#386—21no—
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—21no—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yescomplemented
37#1779—21no—
38C4×C22⋊C8C2212yes—
39C4×C22⋊C8C2212yescomplemented
40C2×#336—21no—
41C2×D4⋊C8C2218yescomplemented
42#1279C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C23×C8—22no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—24no—
53C2×#121C818yescomplemented
54C2×#121C2×C412yes—
55C2×#121C2×C412yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—44no—
63C2×C4×D4—48no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2312yes—
66C22×C4⋊C4—21no—
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×C42—21no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#431—44no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—44no—
93C2×C4⋊C8C2314yes—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—44no—
100C2×C4×C8C2314yes—
101#390—21no—
102#390—21no—
103#390C2×C412yes—
104#390C2×C412yes—
105#389—21no—
106#389—21no—
107#389C2×C412yes—
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2312yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136D4⋊C8—48no—
137C25—21no—
138C25C22⋊C411yescharacteristic
139C22×D4—28no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4C2×C814yescomplemented
149C22×D4C22×C412yes—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
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—42no—
167C23×C4—42no—
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C2×C814yes—
178C23×C4M1614yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8—44no—
193#121—28no—
194#121—28no—
195#121—24no—
196#121—24no—
197#121C2×C818yes—
198#121C2×C818yescomplemented
199C4×D4—28no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—48no—
204C4×D4—48no—
205C4×D4—48no—
206C4×D4—48no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C2×C814yescomplemented
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C22⋊C4—28no—
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—22no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—44no—
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—28no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274C4×C8—28no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—41no—
288C24—41no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24C22⋊C814yes—
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C24C2×C22⋊C411yescharacteristic
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—48no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—48no—
309C2×D4—44no—
310C2×D4—432no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4C22×C814yes—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—41no—
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—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—416no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4C22⋊C814yes—
390C22×C4C22⋊C814yes—
391C22×C4C22⋊C814yes—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C22×C812yes—
402C22×C4C22×C812yes—
403C22×C4C22×C812yes—
404C22×C4C2×M1612yes—
405C22×C4C2×M1612yes—
406C22×C4C2×M1612yes—
407C22×C4AGL(1,ℤ/8)12yes—
408C22×C4AGL(1,ℤ/8)14yes—
409C22×C4#13712yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—88no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4C22×C814yes—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—44no—
455C42—44no—
456C42H(5,𝔽2)14yes—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—28no—
472C23—28no—
473C23—24no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—48no—
480C23—48no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—48no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23C2×C22⋊C812yes—
493C23C2×C22⋊C812yes—
494C23C2×C22⋊C812yes—
495C23#41712yes—
496C23#41812yes—
497C23#42112yes—
498C23#42112yes—
499C23C22×C22⋊C411yescharacteristic
500D4—28no—
501D4—48no—
502D4—416no—
503D4—416no—
504D4—44no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—88no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4C2×C22⋊C812yes—
569C2×C4C2×C22⋊C812yes—
570C2×C4C2×C22⋊C812yes—
571C2×C4#41712yes—
572C2×C4#41812yes—
573C2×C4#51812yes—
574C2×C4#53412yes—
575C2×C4#53412yes—
576C2×C4C23×C811yescharacteristic
577C2×C4C22×M1611yescharacteristic
578C2×C4#56812yes—
579C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×#13711yescharacteristic
582C8—84no—
583C8—84no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—416no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—816no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22#147714yes—
613C22#147814yes—
614C22#230811yescharacteristic
615C22C22×C22⋊C811yescharacteristic
616C22#287911yescharacteristic
617C22C2×#41811yescharacteristic
618C22C2×#41711yescharacteristic
619C22C2×#42111yescharacteristic
620C22C2×#42111yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—44no—
632C4—82no—
633C4—82no—
634C4#297212yes—
635C4#297312yes—
636C4#319212yes—
637C4#319312yes—
638C2—24no—
639C2—24no—
640C2—42no—
641C2—84no—
642C2C2×#147811yescharacteristic
643C2C2×#147711yescharacteristic
644C2#1749211yescharacteristic
645C2#1749511yescharacteristic
646C2#2173811yescharacteristic
647C2#2176411yescharacteristic
648C2#2177411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. C2×#1797C2semidirect
4. #9074C2semidirect
5. #8939C2semidirect
6. #8707C2semidirect
7. #8310C2semidirect
12. C2×#431C22semidirect
13. C22×C4⋊C8C22semidirect
36. #1797C22semidirect
39. C4×C22⋊C8C22semidirect
41. C2×D4⋊C8C22semidirect
42. #1279C22semidirect
53. C2×#121C8semidirect
148. C22×D4C2×C8semidirect
198. #121C2×C8semidirect
216. C2×C4⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#207142
C2×#17972
#90742
#89392
#87072
#83102