Finite Group Databasebeta

← #219733

Subgroups — #219733

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2367C211yesmaximal, characteristic
2#22552C212yesmaximal, complemented
3C2×#1461C211yesmaximal, characteristic, complemented
4C2×#1459C211yesmaximal, characteristic
5#9651C212yesmaximal, complemented
6#9081C212yesmaximal, complemented
7#9080C212yesmaximal, complemented
8#8558C211yesmaximal, characteristic, complemented
9#8303C211yesmaximal, characteristic, complemented
10#8301C211yesmaximal, characteristic
11C4×#324C211yesmaximal, characteristic, complemented
12C22×#124C411yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#431C2212yes—
16C22×C22⋊C8C2212yes—
17#2367—22no—
18#2367—22no—
19#2367—22no—
20#2367—22no—
21C2×#391C411yescharacteristic
22C2×#390C411yescharacteristic
23C2×#388C411yescharacteristic
24C2×#388C411yescharacteristic
25C2×#388C2211yescharacteristic
26C2×#388C2211yescharacteristic
27C2×#385C411yescharacteristic
28C2×#385C2211yescharacteristic
29C2×C4×C4⋊C4C411yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C411yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1923—22no—
34#1923—22no—
35#1850—22no—
36#1817C2212yes—
37#1797C2212yes—
38#1779C2212yes—
39C8×C22⋊C4—22no—
40C4×C22⋊C8C2212yes—
41#1461—24no—
42#1459—24no—
43C2×#324C2211yescharacteristic
44C2×#324C2211yescharacteristic
45C2×#324C2211yescharacteristic
46C2×#324C2212yes—
47C2×#324C2212yes—
48C2×#324C2211yescharacteristic
49C2×#323C2212yes—
50C2×#323C2212yes—
51#1280C2212yes—
52#1280C2212yes—
53C23×Q8C2×C411yescharacteristic
54C23×D4C2×C411yescharacteristic
55C24×C4C2×C411yescharacteristic
56C24×C4C2311yescharacteristic
57C23×C8—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—24no—
61C2×#124C818yescomplemented
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—42no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—42no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C411yescharacteristic
85C22×C22⋊C4C2×C411yescharacteristic
86C22×C22⋊C4C2×C411yescharacteristic
87C22×C22⋊C4C2311yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C411yescharacteristic
90C22×C42C2×C411yescharacteristic
91C22×C42C2×C411yescharacteristic
92C22×C42C2×C411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#431—24no—
96#431—42no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2312yes—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110#391—22no—
111#391—22no—
112#390—22no—
113#390—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#388—22no—
121#388—22no—
122#385—22no—
123#385—22no—
124#385—22no—
125#385—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4D412yes—
137C4×C22⋊C4D412yes—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4D412yes—
140C2×#95C2×C411yescharacteristic
141C2×#95C2×C411yescharacteristic
142C2×#95C2×C411yescharacteristic
143C2×#95C2×C411yescharacteristic
144C2×#95C2×C411yescharacteristic
145C2×#95C2×C411yescharacteristic
146C2×#95C2×C411yescharacteristic
147C2×#95C2311yescharacteristic
148#336—22no—
149#336—22no—
150#336—22no—
151#336—22no—
152#324—22no—
153#324—22no—
154#324—22no—
155#324—24no—
156#324—42no—
157#324D414yes—
158#323—24no—
159#323—42no—
160C25C22×C411yescharacteristic
161C25C22×C411yescharacteristic
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8C2×C814yescomplemented
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4C2×C814yes—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—41no—
184C23×C4—42no—
185C23×C4—81no—
186C23×C4—81no—
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2411yescharacteristic
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—44no—
207C22×C8—44no—
208C22×C8—82no—
209#124—216no—
210#124—216no—
211C4×Q8—44no—
212C4×Q8—44no—
213C4×Q8—44no—
214C4×Q8—44no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—44no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4#4112yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—81no—
260C2×C22⋊C4—81no—
261C2×C22⋊C4C22⋊C412yes—
262C2×C22⋊C4C22⋊C412yes—
263C2×C22⋊C4C22⋊C412yes—
264C2×C22⋊C4C22⋊C412yes—
265C2×C22⋊C4C2×C814yes—
266C2×C22⋊C4C2×C814yes—
267C2×C22⋊C4C2×C814yescomplemented
268C2×C22⋊C4C2×C814yescomplemented
269C2×C22⋊C4M1614yes—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—41no—
288C2×C42—41no—
289C2×C42—41no—
290C2×C42—81no—
291C2×C42—81no—
292C2×C42C2×C814yescomplemented
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C4⋊C8—44no—
298C4⋊C8—82no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—44no—
307C22⋊C8—44no—
308C22⋊C8—82no—
309C22⋊C8C2×D414yes—
310C4×C8—44no—
311C4×C8—82no—
312#95—21no—
313#95—21no—
314#95—22no—
315#95—21no—
316#95—21no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95C22⋊C411yescharacteristic
321#95C22⋊C411yescharacteristic
322#95C22⋊C411yescharacteristic
323#95C22⋊C411yescharacteristic
324#95C22⋊C412yes—
325#95C22⋊C412yes—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—41no—
337C24—41no—
338C24—41no—
339C24—41no—
340C24—41no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24C2×C22⋊C412yes—
345C24C22×C812yes—
346C24C22×C812yes—
347C24C2×M1612yes—
348C24C23×C411yescharacteristic
349C2×Q8—28no—
350C2×Q8—28no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C412yes—
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C4×D412yes—
444C22×C4C4×D412yes—
445C22×C4C22×C812yes—
446C22×C4C22×C812yes—
447C22×C4C22×C812yes—
448C22×C4C22×C812yes—
449C22×C4C22×C812yes—
450C22×C4C2×M1612yes—
451C22×C4C2×M1612yes—
452C22×C4#13112yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4H(5,𝔽2)12yes—
457C2×C8—24no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—88no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4C22⋊C818yes—
494C22⋊C4C22⋊C818yes—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—24no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—81no—
510C42—81no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—81no—
545C23—81no—
546C23—81no—
547C23—81no—
548C23—81no—
549C23—81no—
550C23—81no—
551C23—81no—
552C23—81no—
553C23—81no—
554C23—81no—
555C23—81no—
556C23—81no—
557C23—81no—
558C23—81no—
559C23—81no—
560C23C2×C22⋊C814yes—
561C23C22×C22⋊C411yescharacteristic
562C23C2×C4×D411yescharacteristic
563C23#51811yescharacteristic
564C23C23×C811yescharacteristic
565C23C22×M1611yescharacteristic
566C23C2×#13111yescharacteristic
567C23#56811yescharacteristic
568Q8—416no—
569D4—416no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—84no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—84no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4C2×C22⋊C812yes—
651C2×C4C2×C22⋊C812yes—
652C2×C4C2×C22⋊C814yes—
653C2×C4#41012yes—
654C2×C4#41312yes—
655C2×C4C8×D414yes—
656C2×C4D4212yes—
657C2×C4#54612yes—
658C8—44no—
659C8—82no—
660C8—82no—
661C8—82no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22D4×C22⋊C411yescharacteristic
697C22C22×C22⋊C811yescharacteristic
698C22#288011yescharacteristic
699C22C2×#41011yescharacteristic
700C22#288311yescharacteristic
701C22C2×#41311yescharacteristic
702C22#288911yescharacteristic
703C22C2×C8×D411yescharacteristic
704C22D4×M1611yescharacteristic
705C22#297211yescharacteristic
706C22#297411yescharacteristic
707C4—24no—
708C4—24no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—42no—
716C4—42no—
717C4—41no—
718C4—42no—
719C4—41no—
720C4—42no—
721C4—81no—
722C4—81no—
723C4—81no—
724C2—21no—
725C2—21no—
726C2—21no—
727C2—21no—
728C2—44no—
729C2—82no—
730C2#1734911yescharacteristic
731C2#1735111yescharacteristic
732C2#1742011yescharacteristic
733C2#1742911yescharacteristic
734C2D4×C22⋊C811yescharacteristic
735C2#2240511yescharacteristic
736C2#2241011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22552C2semidirect
3. C2×#1461C2semidirect
5. #9651C2semidirect
6. #9081C2semidirect
7. #9080C2semidirect
8. #8558C2semidirect
9. #8303C2semidirect
11. C4×#324C2semidirect
61. C2×#124C8semidirect
168. C22×Q8C2×C8semidirect
267. C2×C22⋊C4C2×C8semidirect
268. C2×C22⋊C4C2×C8semidirect
292. C2×C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23672
#225522
C2×#14612
C2×#14592
#96512
#90812
#90802
#85582
#83032
#83012
C4×#3242