Finite Group Databasebeta

← #189827

Subgroups — #189827

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34194C211yesmaximal, characteristic
2#21788C211yesmaximal, characteristic
3#13785C211yesmaximal, characteristic
4#8715C212yesmaximal, complemented
5#8715C212yesmaximal, complemented
6#8713C212yesmaximal, complemented
7#8712C212yesmaximal, complemented
8#8022C211yesmaximal, characteristic, complemented
9#8021C211yesmaximal, characteristic, complemented
10#8014C211yesmaximal, characteristic, complemented
11#8013C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2354C411yescharacteristic, complemented
16#2354C411yescharacteristic, complemented
17#2352C412yescomplemented
18#2352C412yescomplemented
19D4×C4⋊C4C411yescharacteristic, complemented
20D4×C4⋊C4C411yescharacteristic, complemented
21#2345—21no—
22#2342—22no—
23D4×C22⋊C4—21no—
24#2319—21no—
25#2318—22no—
26C2×#390—21no—
27#2285C2211yescharacteristic
28#2283C2211yescharacteristic
29#2280C2212yes—
30C2×#380C2212yes—
31#2275—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1955—22no—
35#1954—21no—
36#1954—21no—
37#1779C2212yes—
38#1779C2212yes—
39#1437—22no—
40C4⋊#95—21no—
41C22⋊#95C2212yes—
42C22⋊#95C2212yes—
43#1294C2211yescharacteristic
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C23×D4D411yescharacteristic
54C24×C4C2312yes—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×#117—21no—
68C2×#117C2×C412yes—
69C2×#117C2×C412yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#432—22no—
82#432—22no—
83#431—22no—
84#431—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2311yescharacteristic
93C2×C22⋊C8C2312yes—
94C2×#97—21no—
95C2×C4×C8—21no—
96#390—28no—
97#390—21no—
98#390—21no—
99#389—21no—
100#389—21no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#383—21no—
114#382—21no—
115#382C2×C411yescharacteristic
116#382C2×C411yescharacteristic
117#381—22no—
118#381—21no—
119#380—22no—
120#380—21no—
121#380—22no—
122#380C2×C412yes—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2×C412yes—
126#380C2311yescharacteristic
127#380C2312yes—
128#380C2311yescharacteristic
129#379—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C412yes—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143#376—21no—
144#376—22no—
145#376—22no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C25C2×D412yes—
154C22×D4—24no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—42no—
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22⋊C412yes—
160C22×D4C22⋊C411yescharacteristic
161C23×C4—22no—
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—82no—
171C23×C4—82no—
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2411yescharacteristic
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—24no—
192C4×D4—22no—
193C4×D4—24no—
194C4×D4—22no—
195C4×D4—44no—
196C4×D4—44no—
197#117—81no—
198#117—81no—
199#117—82no—
200#117—82no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C412yes—
244C2×C22⋊C4C22⋊C412yes—
245C2×C22⋊C4C22⋊C412yes—
246C2×C42—24no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—81no—
256C2×C42—81no—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C4⋊C8—42no—
266C4⋊C8—42no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—21no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—82no—
274C22⋊C8—82no—
275C22⋊C8—82no—
276C22⋊C8—82no—
277#97—42no—
278C4×C8—42no—
279#95—22no—
280#95—22no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—22no—
285#95—21no—
286#95—21no—
287#95—22no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—22no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—42no—
300#95—82no—
301#95—82no—
302C24—24no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24C2×C22⋊C412yes—
313C24C2×C22⋊C412yes—
314C24C22×D411yescharacteristic
315C2×D4—24no—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—42no—
325C2×D4#10414yes—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
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—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4#12312yes—
398C22×C4#12312yes—
399C22×C4#12312yes—
400C22×C4#12312yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—42no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—48no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—44no—
437C22⋊C4—41no—
438C22⋊C4—42no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—42no—
442C22⋊C4—41no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—48no—
451C42—42no—
452C42—42no—
453C42—82no—
454C42—82no—
455C42—82no—
456C42—84no—
457C42#11712yes—
458C42#11712yes—
459C42#11712yes—
460C42#11712yes—
461C42#12312yes—
462C42#12312yes—
463C42#12312yes—
464C42#12312yes—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
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—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23C2×#10414yes—
499C23C22×C22⋊C411yescharacteristic
500C23C2×#12311yescharacteristic
501C23C2×#12311yescharacteristic
502C23#57611yescharacteristic
503C23#57711yescharacteristic
504C23#57711yescharacteristic
505C23#57811yescharacteristic
506D4—28no—
507D4—48no—
508D4—44no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
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—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—164no—
585C2×C4—164no—
586C2×C4#38112yes—
587C2×C4#38112yes—
588C2×C4#38112yes—
589C2×C4#38112yes—
590C2×C4C2×#10412yes—
591C2×C4#42112yes—
592C2×C4C2×#11711yescharacteristic
593C2×C4C2×#11711yescharacteristic
594C2×C4C2×#12311yescharacteristic
595C2×C4C2×#12311yescharacteristic
596C8—81no—
597C8—81no—
598C8—82no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22C2×#38111yescharacteristic
630C22#288311yescharacteristic
631C22#288311yescharacteristic
632C22#289711yescharacteristic
633C22#289811yescharacteristic
634C22C22×#10411yescharacteristic
635C22C2×#42111yescharacteristic
636C22#309611yescharacteristic
637C22#309711yescharacteristic
638C22#309811yescharacteristic
639C22#309911yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—42no—
649C4—84no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—81no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—81no—
658C4#182614yes—
659C2—21no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—44no—
664C2—82no—
665C2#1754711yescharacteristic
666C2#1754811yescharacteristic
667C2#1759911yescharacteristic
668C2#1760011yescharacteristic
669C2#2069011yescharacteristic
670C2#2070311yescharacteristic
671C2C2×#182611yescharacteristic
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. #8715C2semidirect
5. #8715C2semidirect
6. #8713C2semidirect
7. #8712C2semidirect
8. #8022C2semidirect
9. #8021C2semidirect
10. #8014C2semidirect
11. #8013C2semidirect
15. #2354C4semidirect
16. #2354C4semidirect
17. #2352C4semidirect
18. #2352C4semidirect
19. D4×C4⋊C4C4semidirect
20. D4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#341942
#217882
#137852
#87152
#87132
#87122
#80222
#80212
#80142
#80132