Finite Group Databasebeta

← #222217

Subgroups — #222217

Order 512.

1051 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3#22402C211yesmaximal, characteristic
4C2×#1590C211yesmaximal, characteristic, complemented
5C2×#1582C211yesmaximal, characteristic, complemented
6#9661C211yesmaximal, characteristic, complemented
7#9655C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic, complemented
9#8952C211yesmaximal, characteristic, complemented
10#8949C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8598C211yesmaximal, characteristic, complemented
13#8313C211yesmaximal, characteristic
14#8311C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#3103—24no—
19#3103—24no—
20C2×#481C411yescharacteristic
21C2×#481C411yescharacteristic
22C2×#475C411yescharacteristic
23C2×#475C411yescharacteristic
24C2×#460C411yescharacteristic
25C2×#460C411yescharacteristic
26C2×#448C411yescharacteristic
27C2×#448C411yescharacteristic
28C2×#446C2211yescharacteristic
29C2×#446C2211yescharacteristic
30C2×C4×SD16C2211yescharacteristic
31C2×C4×SD16C2211yescharacteristic
32C2×C4×SD16C2211yescharacteristic
33C2×C4×SD16C2211yescharacteristic
34C2×#436C2211yescharacteristic
35C2×C8×D4C2211yescharacteristic
36C2×#431C2211yescharacteristic
37C2×#408—21no—
38#1985—21no—
39#1985—21no—
40#1984C2211yescharacteristic
41#1984C2211yescharacteristic
42#1954C2211yescharacteristic
43#1954C2211yescharacteristic
44#1870—21no—
45#1869—21no—
46#1797—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1780—21no—
50#1749—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1590—24no—
54#1582—24no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66C2×C82C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×#41—21no—
72C23×Q8C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C22×SD16C2×C411yescharacteristic
76C22×SD16C2×C411yescharacteristic
77C22×M16—21no—
78C23×C8—21no—
79#532—22no—
80#532—22no—
81#532—41no—
82#532—41no—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×#127C2×C411yescharacteristic
85C2×#124—21no—
86C2×#121—21no—
87C2×#121C2×C411yescharacteristic
88C2×#121C2×C411yescharacteristic
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—41no—
94C2×C4×Q8—41no—
95C2×C4×Q8C2×C411yescharacteristic
96C2×C4×Q8C2×C411yescharacteristic
97C2×C4×Q8C2311yescharacteristic
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4C2311yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106#481—24no—
107#481—24no—
108#475—24no—
109#475—24no—
110#460—24no—
111#460—24no—
112#448—24no—
113#448—24no—
114#446—22no—
115#446—22no—
116#446—41no—
117#446—41no—
118C4×SD16—22no—
119C4×SD16—22no—
120C4×SD16—22no—
121C4×SD16—22no—
122C4×SD16—22no—
123C4×SD16—22no—
124C4×SD16D414yes—
125C4×SD16D414yes—
126#436—22no—
127#436—41no—
128C8×D4—22no—
129C8×D4—41no—
130#431—22no—
131#431—22no—
132C2×#106C2×C411yescharacteristic
133C2×#106C2×C411yescharacteristic
134C2×#106C2×C411yescharacteristic
135C2×#106C2×C411yescharacteristic
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2×C411yescharacteristic
139C2×C4⋊C8C2×C411yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#103C2×C411yescharacteristic
147C2×#103C2×C411yescharacteristic
148C2×#102C2×C411yescharacteristic
149C2×#102C2×C411yescharacteristic
150C2×#102C2×C411yescharacteristic
151C2×#102C2×C411yescharacteristic
152#408—44no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155C2×C22⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8—21no—
158C2×C22⋊C8—21no—
159C2×C22⋊C8—21no—
160C2×C22⋊C8—21no—
161C2×C22⋊C8—21no—
162C2×C22⋊C8C2×C411yescharacteristic
163C2×C22⋊C8C2×C411yescharacteristic
164C2×C22⋊C8D411yescharacteristic
165C2×C22⋊C8D411yescharacteristic
166C2×#97—21no—
167C2×#97—21no—
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336D411yescharacteristic
182#336D411yescharacteristic
183#334—22no—
184#334—22no—
185Q8⋊C8—22no—
186Q8⋊C8—22no—
187Q8⋊C8—41no—
188Q8⋊C8—41no—
189D4⋊C8—22no—
190D4⋊C8—22no—
191D4⋊C8—41no—
192D4⋊C8—41no—
193#322—22no—
194C82—22no—
195C2×#41—42no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41—42no—
199C2×#41—81no—
200C2×#41—81no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—41no—
206C22×Q8—41no—
207C22×Q8—41no—
208C22×Q8—41no—
209C22×Q8C22×C411yescharacteristic
210C22×Q8C22×C411yescharacteristic
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C22×C411yescharacteristic
216C22×D4C22×C411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4C22×C411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4#4111yescharacteristic
224C2×SD16—22no—
225C2×SD16—22no—
226C2×SD16—22no—
227C2×SD16—22no—
228C2×SD16—22no—
229C2×SD16—41no—
230C2×SD16—41no—
231C2×SD16—41no—
232C2×SD16—41no—
233C2×SD16—41no—
234C2×SD16—41no—
235C2×SD16C22⋊C414yes—
236C2×M16—42no—
237C2×M16—42no—
238C2×M16—81no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—42no—
246C22×C8—42no—
247C22×C8—81no—
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C22×C411yescharacteristic
251C22×C8C22×C411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C22×C8C2×D411yescharacteristic
256C4⋊Q8—22no—
257C4⋊Q8—41no—
258#127—22no—
259#127—41no—
260#124—42no—
261#124—42no—
262#121—22no—
263#121—22no—
264#121—42no—
265#121—42no—
266#121—41no—
267#121—41no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8—41no—
279C4×Q8—41no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8C2×D412yes—
283C4×Q8C2×D412yes—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4C2×D412yes—
293C4×D4C2×D412yes—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42C22×C411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42#4111yescharacteristic
325C2×C42C2411yescharacteristic
326#106—22no—
327#106—22no—
328#106—22no—
329#106C22⋊C414yes—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342#103—22no—
343#103—22no—
344#103—22no—
345#103C22⋊C414yes—
346#102—22no—
347#102—22no—
348#102—22no—
349#102C22⋊C414yes—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—42no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360C22⋊C8—81no—
361C22⋊C8—81no—
362C22⋊C8SD1614yescomplemented
363#97—42no—
364#97—42no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—21no—
370C4×C8—21no—
371C4×C8—21no—
372C4×C8—21no—
373C4×C8—21no—
374C4×C8—21no—
375C4×C8C2×D412yes—
376C4×C8C2×D412yes—
377C24—21no—
378C24—21no—
379C24C4×D411yescharacteristic
380#41—84no—
381#41—84no—
382#41—84no—
383#41—84no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—81no—
404C2×Q8C2×C22⋊C412yes—
405C2×Q8#13112yes—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4C2×C22⋊C412yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—81no—
468C22×C4C4×D411yescharacteristic
469C22×C4C4×D411yescharacteristic
470C22×C4C4×D411yescharacteristic
471C22×C4C2×SD1612yes—
472C22×C4#13512yes—
473C22×C4C23×C411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4111yescharacteristic
476SD16—44no—
477SD16—44no—
478SD16—44no—
479SD16—44no—
480M16—84no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—22no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—84no—
520C2×C8C2×C22⋊C412yes—
521C2×C8C2×C22⋊C412yes—
522C2×C8C2×SD1612yes—
523C2×C8C2×SD1612yes—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4C2×C22⋊C412yes—
539C4⋊C4C2×C22⋊C412yes—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C42—21no—
547C42—21no—
548C42—22no—
549C42—22no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—42no—
555C42—42no—
556C42—41no—
557C42—41no—
558C42C22×D411yescharacteristic
559C42C22×D411yescharacteristic
560C42H(5,𝔽2)11yescharacteristic
561C42H(5,𝔽2)11yescharacteristic
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C4×SD1612yes—
583C23C2×C4×D411yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8—22no—
588Q8—48no—
589Q8—41no—
590Q8—41no—
591Q8—41no—
592Q8—41no—
593Q8—84no—
594D4—22no—
595D4—22no—
596D4—41no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—41no—
602D4—84no—
603D4—84no—
604D4—84no—
605D4—84no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—48no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
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—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—84no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4C4×SD1612yes—
655C2×C4C22×C22⋊C411yescharacteristic
656C2×C4#51811yescharacteristic
657C2×C4D4211yescharacteristic
658C2×C4D4211yescharacteristic
659C2×C4#54611yescharacteristic
660C2×C4#54611yescharacteristic
661C2×C4C2×#13111yescharacteristic
662C2×C4#56811yescharacteristic
663C2×C4#56811yescharacteristic
664C2×C4#56811yescharacteristic
665C2×C4C22×SD1611yescharacteristic
666C2×C4C2×#13511yescharacteristic
667C2×C4#57711yescharacteristic
668C2×C4#57711yescharacteristic
669C8—22no—
670C8—22no—
671C8—22no—
672C8—22no—
673C8—42no—
674C8—41no—
675C8—41no—
676C8—41no—
677C8—41no—
678C8—81no—
679C8—81no—
680C8—81no—
681C8—81no—
682C8—81no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22D4×C22⋊C411yescharacteristic
705C22#293011yescharacteristic
706C22#293311yescharacteristic
707C22C2×C4×SD1611yescharacteristic
708C22#294911yescharacteristic
709C22#295511yescharacteristic
710C22#295611yescharacteristic
711C4—22no—
712C4—21no—
713C4—21no—
714C4—41no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—41no—
719C4—42no—
720C4—81no—
721C4—81no—
722C4#287911yescharacteristic
723C4#288011yescharacteristic
724C4#297611yescharacteristic
725C4#297811yescharacteristic
726C4D4×SD1611yescharacteristic
727C4D4×SD1611yescharacteristic
728C4#328211yescharacteristic
729C4#328211yescharacteristic
730C2—42no—
731C2—82no—
732C2—161no—
733C2#1792911yescharacteristic
734C2#1795111yescharacteristic
735C2#1799711yescharacteristic
736C2#1803011yescharacteristic
737C2#2240011yescharacteristic
738C2C22⋊C4×SD1611yescharacteristic
739C2#2245311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
4. C2×#1590C2semidirect
5. C2×#1582C2semidirect
6. #9661C2semidirect
7. #9655C2semidirect
8. #8952C2semidirect
9. #8952C2semidirect
10. #8949C2semidirect
11. #8945C2semidirect
12. #8598C2semidirect
14. #8311C2semidirect
15. #8148C2semidirect
362. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
#225102
#224022
C2×#15902
C2×#15822
#96612
#96552
#89522
#89492
#89452
#85982
#83132
#83112
#81482