Finite Group Databasebeta

← #223076

Subgroups — #223076

Order 512.

1315 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36154C211yesmaximal, characteristic
2D4×C22⋊C8C211yesmaximal, characteristic
3#13920C211yesmaximal, characteristic
4#9908C211yesmaximal, characteristic, complemented
5#9884C211yesmaximal, characteristic, complemented
6#9151C212yesmaximal, complemented
7#9111C212yesmaximal, complemented
8#8713C212yesmaximal, complemented
9#8572C211yesmaximal, characteristic, complemented
10C4×#328C211yesmaximal, characteristic, complemented
11#8023C212yesmaximal, complemented
12C2×#554C411yescharacteristic, complemented
13C22×C4×D4C2211yescharacteristic
14C2×C8×D4—21no—
15C22×C22⋊C8C2212yes—
16#2857C2211yescharacteristic
17#2848C2212yes—
18#2563C411yescharacteristic, complemented
19#2538C411yescharacteristic, complemented
20#2487C412yescomplemented
21#2454—21no—
22#2450—22no—
23#2442—21no—
24#2437—22no—
25#2417C412yescomplemented
26#2349—21no—
27D4×C4⋊C4C411yescharacteristic, complemented
28D4×C22⋊C4—21no—
29C4×C22⋊Q8C2211yescharacteristic
30C2×#390—21no—
31C2×#389—21no—
32C2×#380C2212yes—
33C2×#380C2212yes—
34#1954—21no—
35#1869—22no—
36#1797C2211yescharacteristic
37#1779C2212yes—
38C4×C22⋊C8C2211yescharacteristic
39C2×#328C2211yescharacteristic
40C2×#328C2211yescharacteristic
41C2×#328C2212yes—
42#1439—22no—
43C4⋊#95C2211yescharacteristic
44C22⋊#95C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2211yescharacteristic
51#1294C2212yes—
52#1294C2212yes—
53C23×D4D411yescharacteristic
54C24×C4C2312yes—
55C23×C8—22no—
56#554—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#125—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2311yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—44no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C412yes—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D412yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4D411yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D412yes—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C8×D4—44no—
86C2×C4⋊C8—21no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2311yescharacteristic
95C2×C4×C8—21no—
96#400—22no—
97#399—22no—
98#398—22no—
99#396—22no—
100#396—22no—
101#393—22no—
102#391—21no—
103#391C2×C411yescharacteristic
104#390—21no—
105#390—21no—
106#390—44no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—44no—
111#389C2×C412yes—
112#389C2×C411yescharacteristic
113#387—21no—
114#386—22no—
115#386—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—21no—
120#385—22no—
121#385—22no—
122#384C2×C411yescharacteristic
123#382—21no—
124#380—21no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380—24no—
129#380—42no—
130#380C2×C412yes—
131#380C2×C412yes—
132#380C2×C412yes—
133#380C2×C412yes—
134#380C2312yes—
135#380C2311yescharacteristic
136#379—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C22⋊C4—21no—
140C2×#95—22no—
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147#336—21no—
148#336—21no—
149#328—22no—
150#328—22no—
151#328—42no—
152C25C2×D412yes—
153C22×Q8—21no—
154C22×Q8C22×C411yescharacteristic
155C22×D4—42no—
156C22×D4—42no—
157C22×D4C22⋊C411yescharacteristic
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22⋊C412yes—
160C22×D4SD1614yes—
161C22×D4Q1614yes—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—44no—
172C23×C4—82no—
173C23×C4C22×C412yes—
174C23×C4C22×C412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2411yescharacteristic
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—44no—
185C22×C8—44no—
186C22×C8—82no—
187C4⋊Q8—42no—
188#125—44no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—48no—
194C22⋊Q8—44no—
195C22⋊Q8—81no—
196C22⋊Q8—81no—
197C22⋊Q8—82no—
198C4×Q8—42no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—44no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4—82no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4C22⋊C412yes—
254C2×C22⋊C4C22⋊C412yes—
255C2×C22⋊C4C22⋊C412yes—
256C2×C22⋊C4C22⋊C412yes—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C4⋊C8—42no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—21no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C22⋊C8—82no—
276C22⋊C8#4112yes—
277C4×C8—42no—
278#95—22no—
279#95—22no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—21no—
288#95—22no—
289#95—21no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95—21no—
294#95—42no—
295#95—44no—
296#95—81no—
297#95#4112yes—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24C2×C22⋊C412yes—
307C24C2×C22⋊C412yes—
308C24C2×SD1614yes—
309C24C2×Q1614yes—
310C24C22×D411yescharacteristic
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—81no—
317C2×Q8—81no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4#10314yes—
327C2×D4#10314yes—
328C2×D4#10318yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
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—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×SD1612yes—
401C22×C4C2×Q1612yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C42−1+411yescharacteristic
408C2×C8—22no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—41no—
417C2×C8—88no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—81no—
438C4⋊C4—84no—
439C4⋊C4—88no—
440C4⋊C4#9914yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—42no—
451C22⋊C4—41no—
452C22⋊C4—42no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C42—22no—
460C42—22no—
461C42—41no—
462C42—41no—
463C42—42no—
464C42—42no—
465C42—84no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23C2×#10314yes—
499C23C2×#10314yes—
500C23C22×C22⋊C411yescharacteristic
501C23#54611yescharacteristic
502C23#54711yescharacteristic
503C23C22×SD1611yescharacteristic
504C23C22×Q1611yescharacteristic
505C23#57611yescharacteristic
506C23#57711yescharacteristic
507Q8—81no—
508Q8—81no—
509Q8—162no—
510D4—28no—
511D4—28no—
512D4—48no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—164no—
588C2×C4—164no—
589C2×C4C2×#9914yes—
590C2×C4C2×#9912yes—
591C2×C4C2×#10312yes—
592C2×C4C2×#10312yes—
593C2×C4#42112yes—
594C2×C4C4×#4111yescharacteristic
595C2×C4#52011yescharacteristic
596C2×C4#54611yescharacteristic
597C2×C4#54711yescharacteristic
598C8—42no—
599C8—81no—
600C8—82no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22#233911yescharacteristic
632C22C22×#9911yescharacteristic
633C22#288311yescharacteristic
634C22C22×#10311yescharacteristic
635C22#289711yescharacteristic
636C22C2×#42111yescharacteristic
637C22#290111yescharacteristic
638C22#329511yescharacteristic
639C22#329911yescharacteristic
640C22#333511yescharacteristic
641C22#333811yescharacteristic
642C4—22no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—24no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—162no—
660C4#152014yes—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—44no—
666C2—44no—
667C2C2×#152011yescharacteristic
668C2#1759911yescharacteristic
669C2#1761111yescharacteristic
670C2#1761411yescharacteristic
671C2#2247611yescharacteristic
672C2#2248711yescharacteristic
673C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9908C2semidirect
5. #9884C2semidirect
6. #9151C2semidirect
7. #9111C2semidirect
8. #8713C2semidirect
9. #8572C2semidirect
10. C4×#328C2semidirect
11. #8023C2semidirect
12. C2×#554C4semidirect
18. #2563C4semidirect
19. #2538C4semidirect
20. #2487C4semidirect
25. #2417C4semidirect
27. D4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#361542
D4×C22⋊C82
#139202
#99082
#98842
#91512
#91112
#87132
#85722
C4×#3282
#80232