Finite Group Databasebeta

← #216209

Subgroups — #216209

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 660 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#20626C211yesmaximal, characteristic, complemented
4#9879C212yesmaximal, complemented
5#9649C212yesmaximal, complemented
6#9083C214yesmaximal, complemented
7#8557C214yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4C412yes—
10C22×C4×D4C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13C2×#423—21no—
14C2×#407C2212yes—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121—21no—
21C4×#121—21no—
22C4×#121—21no—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388C412yes—
26C2×#386C2212yes—
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1797C2212yes—
37#1779C2212yes—
38C4×C22⋊C8—21no—
39C4×C22⋊C8C2212yes—
40#1728—21no—
41C2×#323C2218yes—
42#1280C2218yes—
43C23×D4C2×C412yes—
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C2×#121—24no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—24no—
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—21no—
63C2×C4×D4—21no—
64C2×C4×D4—44no—
65C2×C4×D4—48no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2312yes—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C412yes—
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#431—44no—
90#423—44no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94#407—44no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8C2314yes—
99C2×C22⋊C8C2314yes—
100C2×C4×C8—22no—
101#390—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
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#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2312yes—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#323—48no—
137C25C22×C411yescharacteristic
138C25C22×C412yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—24no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—24no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4M1614yescomplemented
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—28no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
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×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C2×M16—44no—
189C2×M16—44no—
190C2×M16—44no—
191C22×C8—24no—
192C22×C8—24no—
193#121—28no—
194#121—28no—
195#121—28no—
196#121—28no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—44no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4M1614yescomplemented
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
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—24no—
231C2×C22⋊C4—24no—
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×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C4⋊C8—44no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C22⋊C8—24no—
270C22⋊C8—28no—
271C22⋊C8—24no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C4×C8—44no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—41no—
293C24—41no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C2×C22⋊C411yescharacteristic
299C24C2×C22⋊C411yescharacteristic
300C24C2×M1614yes—
301C24C23×C411yescharacteristic
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—44no—
311C2×D4—416no—
312C2×D4—44no—
313C2×D4—416no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—42no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
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—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
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—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
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—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—416no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—816no—
395C22×C4#10014yes—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×M1612yes—
407C22×C4C2×M1612yes—
408C22×C4C2×M1614yes—
409C22×C4#13114yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4H(5,𝔽2)12yes—
413C22×C42−1+412yes—
414M16—88no—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—44no—
456C42—44no—
457C42#9914yes—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—24no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—28no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23#41012yes—
502C23C2×#10012yes—
503C23C22×C22⋊C411yescharacteristic
504C23#52012yes—
505C23C22×M1611yescharacteristic
506C23C22×M1611yescharacteristic
507C23C2×#13112yes—
508D4—48no—
509D4—416no—
510D4—416no—
511D4—48no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
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—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
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—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—88no—
571C2×C4—88no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4#40712yes—
575C2×C4#40712yes—
576C2×C4#40812yes—
577C2×C4#40812yes—
578C2×C4C2×#9912yes—
579C2×C4C2×#9912yes—
580C2×C4C2×#9912yes—
581C2×C4C2×#10012yes—
582C2×C4C2×#10012yes—
583C2×C4#41312yes—
584C2×C4#53511yescharacteristic
585C2×C4#53611yescharacteristic
586C2×C4#53712yes—
587C8—84no—
588C8—84no—
589C22—22no—
590C22—22no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—48no—
604C22—48no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—88no—
617C22—88no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22#231111yescharacteristic
623C22C2×#40711yescharacteristic
624C22C2×#40811yescharacteristic
625C22C22×#9911yescharacteristic
626C22C2×#41011yescharacteristic
627C22C22×#10011yescharacteristic
628C22C2×#41311yescharacteristic
629C22#299012yes—
630C22#299112yes—
631C4—21no—
632C4—22no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—44no—
642C4—41no—
643C4—82no—
644C4—82no—
645C4#146514yes—
646C4#147114yes—
647C2—22no—
648C2—22no—
649C2—42no—
650C2—42no—
651C2—84no—
652C2C2×#146511yescharacteristic
653C2#1736411yescharacteristic
654C2#1737611yescharacteristic
655C2C2×#147111yescharacteristic
656C2#2184911yescharacteristic
657C2#2187411yescharacteristic
658C2#2187811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ 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. #20626C2semidirect
4. #9879C2semidirect
5. #9649C2semidirect
6. #9083C2semidirect
7. #8557C2semidirect
150. C22×D4M16semidirect
217. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#207142
#206262
#98792
#96492
#90832
#85572