Finite Group Databasebeta

← #205761

Subgroups — #205761

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3190C211yesmaximal, characteristic
2#21941C211yesmaximal, characteristic, complemented
3#21874C211yesmaximal, characteristic
4#21825C211yesmaximal, characteristic, complemented
5#20528C211yesmaximal, characteristic, complemented
6#17572C211yesmaximal, characteristic
7C2×#1520C211yesmaximal, characteristic, complemented
8#9786C212yesmaximal, complemented
9#9784C212yesmaximal, complemented
10#9782C212yesmaximal, complemented
11#9781C212yesmaximal, complemented
12C2×#525C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14#3190—21no—
15#3190—21no—
16#3190—22no—
17#3190—22no—
18#3190—21no—
19#3190—21no—
20C2×#485C412yescomplemented
21C2×#483C411yescharacteristic, complemented
22C2×#480C411yescharacteristic, complemented
23C2×#467C2212yes—
24C2×#465C2212yes—
25C2×#452C412yescomplemented
26C2×#451C411yescharacteristic, complemented
27C2×#448C411yescharacteristic, complemented
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30#2280C2211yescharacteristic
31#2280C2211yescharacteristic
32C2×#380C2211yescharacteristic
33#1996—22no—
34#1996C2212yes—
35#1996C2212yes—
36#1996C2212yes—
37#1964C2212yes—
38#1898—22no—
39#1854—22no—
40#1853—21no—
41#1852—21no—
42#1819—22no—
43#1808C2212yes—
44#1807C2212yes—
45#1799—22no—
46#1752—21no—
47#1752C2212yes—
48C8×C22⋊C4—21no—
49C8×C22⋊C4C2212yes—
50#1750—22no—
51#1520—22no—
52#1520C2212yes—
53#1520C2211yescharacteristic
54#1520C2211yescharacteristic
55#1520C2212yes—
56#1519C2211yescharacteristic
57#1519C2211yescharacteristic
58#1516—22no—
59#1501—22no—
60C2×#328—22no—
61C22×#41C2×C411yescharacteristic
62C23×Q8C2×C411yescharacteristic
63C24×C4C2311yescharacteristic
64C22×Q16C2×C412yescomplemented
65C22×SD16C2×C412yescomplemented
66C23×C8—21no—
67C23×C8C2312yes—
68#525—21no—
69#525—21no—
70#525—21no—
71#525—21no—
72#525—21no—
73#525—21no—
74#525—21no—
75#525—21no—
76C2×#123—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8C2×C411yescharacteristic
88C2×C22⋊Q8C2×C412yes—
89C2×C22⋊Q8C2312yes—
90C2×C22⋊Q8C2311yescharacteristic
91C2×#121C2×C411yescharacteristic
92C2×#121C2311yescharacteristic
93C2×#120—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×#117—21no—
97C2×#117—21no—
98C2×#117C2×C411yescharacteristic
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104#485—22no—
105#485—22no—
106#485—22no—
107#485—22no—
108#483—21no—
109#483—21no—
110#483—22no—
111#480—21no—
112#480—21no—
113#480—22no—
114#467—22no—
115#467—22no—
116#467—22no—
117#467—22no—
118#465—22no—
119#465—22no—
120#465—22no—
121#465—22no—
122#452—22no—
123#452—22no—
124#452—22no—
125#452—22no—
126#451—21no—
127#451—21no—
128#451—22no—
129#448—21no—
130#448—21no—
131#448—22no—
132C2×#107—22no—
133C2×#107C2×C412yescomplemented
134C2×#106—22no—
135C2×#106C2×C412yescomplemented
136C2×#103C2×C412yescomplemented
137C2×#103C2×C412yescomplemented
138C2×#103C2×C412yescomplemented
139C2×#102C2×C412yescomplemented
140C2×#99—24no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8—21no—
144C2×C22⋊C8C2×C411yescharacteristic
145C2×C22⋊C8C2×C411yescharacteristic
146C2×C22⋊C8C2×C412yes—
147C2×C22⋊C8D412yes—
148C2×C22⋊C8D412yes—
149C2×C22⋊C8D411yescharacteristic
150C2×C22⋊C8D411yescharacteristic
151C2×C22⋊C8D412yes—
152C2×C22⋊C8D412yes—
153C2×C22⋊C8D411yescharacteristic
154C2×C22⋊C8D411yescharacteristic
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2311yescharacteristic
157C2×#97—22no—
158C2×C4×C8—22no—
159#381—21no—
160#381—21no—
161#380—21no—
162#380—21no—
163#380—21no—
164#380—21no—
165#380—21no—
166#380D412yes—
167#380C2311yescharacteristic
168#380C2312yes—
169#380C2311yescharacteristic
170#379—21no—
171#379D412yes—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4C2312yes—
176C2×#95—21no—
177#342—24no—
178#342—24no—
179#342—24no—
180#340—22no—
181#340—22no—
182#336—22no—
183#336—22no—
184#328—24no—
185#328—24no—
186#328—22no—
187#328—22no—
188#327—22no—
189#327—22no—
190C25C2×D411yescharacteristic
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—82no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×Q8—21no—
201C22×Q8—82no—
202C22×Q8C22×C412yes—
203C22×Q8C22×C411yescharacteristic
204C22×D4—21no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4C22×C411yescharacteristic
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—41no—
212C23×C4—82no—
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4#4111yescharacteristic
223C23×C4#4111yescharacteristic
224C23×C4#4111yescharacteristic
225C23×C4#4111yescharacteristic
226C23×C4C2411yescharacteristic
227C2×Q16—22no—
228C2×Q16—22no—
229C2×Q16—82no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×SD16—82no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—82no—
238C22×C8—82no—
239C22×C8C22×C412yes—
240C22×C8C22×C412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245#123—41no—
246#123—41no—
247#123—41no—
248#123—41no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—21no—
252C22⋊Q8—21no—
253C22⋊Q8—22no—
254C22⋊Q8—22no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—21no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263C22⋊Q8—22no—
264C22⋊Q8—21no—
265C22⋊Q8—41no—
266C22⋊Q8—41no—
267C22⋊Q8—41no—
268C22⋊Q8—41no—
269C22⋊Q8—41no—
270C22⋊Q8—41no—
271C22⋊Q8—41no—
272C22⋊Q8—41no—
273#121—21no—
274#121—21no—
275#121—21no—
276#121—21no—
277#121—21no—
278#121—21no—
279#121—21no—
280#121—21no—
281#120—41no—
282#120—41no—
283#120—41no—
284#120—41no—
285C4×D4—41no—
286C4×D4—41no—
287C4×D4—41no—
288C4×D4—41no—
289C4×D4—41no—
290C4×D4—41no—
291C4×D4—41no—
292C4×D4—41no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—82no—
298#117—82no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—41no—
313C2×C4⋊C4—41no—
314C2×C4⋊C4—41no—
315C2×C4⋊C4—41no—
316C2×C4⋊C4—41no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C22×C412yes—
319C2×C4⋊C4C22×C411yescharacteristic
320C2×C4⋊C4C22×C411yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—41no—
334C2×C22⋊C4—41no—
335C2×C22⋊C4—41no—
336C2×C22⋊C4—41no—
337C2×C22⋊C4—41no—
338C2×C22⋊C4—41no—
339C2×C22⋊C4—41no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4C2×D414yes—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—41no—
345#107—22no—
346#107—22no—
347#107—82no—
348#106—22no—
349#106—22no—
350#106—82no—
351#103—22no—
352#103—22no—
353#103—22no—
354#103—22no—
355#103—22no—
356#103—22no—
357#102—22no—
358#102—22no—
359#99—84no—
360C22⋊C8—22no—
361C22⋊C8—22no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364C22⋊C8C22⋊C411yescharacteristic
365C22⋊C8C22⋊C411yescharacteristic
366C22⋊C8C22⋊C412yes—
367C22⋊C8C22⋊C411yescharacteristic
368C22⋊C8C22⋊C411yescharacteristic
369C22⋊C8C22⋊C412yes—
370C22⋊C8C22⋊C412yes—
371C22⋊C8C22⋊C412yes—
372C22⋊C8C22⋊C412yes—
373C22⋊C8C22⋊C412yes—
374C22⋊C8C2×D412yes—
375C22⋊C8C2×D412yes—
376#97—82no—
377C4×C8—82no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—41no—
384#95—41no—
385C24—22no—
386C24—22no—
387C24—21no—
388C24—42no—
389C24—81no—
390C24C4×D411yescharacteristic
391C24C4×D411yescharacteristic
392C24#12111yescharacteristic
393C24#12111yescharacteristic
394C24#12711yescharacteristic
395C24#12711yescharacteristic
396C24C22×D411yescharacteristic
397#41—82no—
398#41—82no—
399#41—82no—
400#41—82no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—41no—
409C2×Q8—41no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—82no—
427C2×D4—82no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C4×D411yescharacteristic
468C22×C4C4×D411yescharacteristic
469C22×C4C4×D411yescharacteristic
470C22×C4C4×D411yescharacteristic
471C22×C4C4×D411yescharacteristic
472C22×C4C4×D411yescharacteristic
473C22×C4#12111yescharacteristic
474C22×C4#12111yescharacteristic
475C22×C4#12111yescharacteristic
476C22×C4#12111yescharacteristic
477C22×C4#12111yescharacteristic
478C22×C4#12111yescharacteristic
479C22×C4#12411yescharacteristic
480C22×C4#12411yescharacteristic
481C22×C4#12411yescharacteristic
482C22×C4#12411yescharacteristic
483C22×C4#12711yescharacteristic
484C22×C4#12711yescharacteristic
485C22×C4C23×C411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4111yescharacteristic
490C22×C4C2×#4111yescharacteristic
491Q16—82no—
492Q16—82no—
493SD16—82no—
494SD16—82no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8—42no—
504C2×C8—42no—
505C2×C8—82no—
506C2×C8—82no—
507C2×C8C2×C22⋊C412yes—
508C2×C8C2×C22⋊C412yes—
509C2×C8C2×C22⋊C412yes—
510C2×C8C2×C22⋊C412yes—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—41no—
523C4⋊C4—42no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—41no—
538C22⋊C4—42no—
539C22⋊C4—41no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—44no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C42—42no—
558C42—42no—
559C42—41no—
560C42—41no—
561C42—82no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—21no—
570C23—21no—
571C23—22no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23—81no—
587C23—81no—
588C23—84no—
589C23—84no—
590C23#39011yescharacteristic
591C23#39011yescharacteristic
592C23#39011yescharacteristic
593C23#39011yescharacteristic
594C23C2×C4×D411yescharacteristic
595C23C2×#12111yescharacteristic
596C23C2×#12711yescharacteristic
597C23#57712yes—
598C23#57812yes—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—82no—
603Q8—81no—
604Q8—82no—
605Q8—82no—
606D4—42no—
607D4—81no—
608D4—82no—
609D4—82no—
610D4—82no—
611D4—81no—
612D4—81no—
613D4—82no—
614D4—82no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
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—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4—82no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—81no—
660C2×C4—82no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—82no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—164no—
672C2×C4#39011yescharacteristic
673C2×C4#39011yescharacteristic
674C2×C4#39011yescharacteristic
675C2×C4#39011yescharacteristic
676C2×C4C22×C22⋊C411yescharacteristic
677C2×C4C2×C4×D411yescharacteristic
678C2×C4C2×#12111yescharacteristic
679C2×C4C2×#12411yescharacteristic
680C8—22no—
681C8—22no—
682C8—22no—
683C8—22no—
684C8—42no—
685C8—42no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—44no—
707C22—42no—
708C22—44no—
709C22—82no—
710C22—81no—
711C22—81no—
712C22—82no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—162no—
717C22#197612yes—
718C22C2×#39011yescharacteristic
719C22#294712yes—
720C22#306312yes—
721C22#315411yescharacteristic
722C22#315611yescharacteristic
723C4—21no—
724C4—21no—
725C4—21no—
726C4—21no—
727C4—21no—
728C4—21no—
729C4—21no—
730C4—84no—
731C4—81no—
732C4—82no—
733C4—81no—
734C4—82no—
735C4—81no—
736C4—161no—
737C4—161no—
738C4—162no—
739C4#288312yes—
740C2—22no—
741C2—21no—
742C2—21no—
743C2—22no—
744C2—22no—
745C2—41no—
746C2—41no—
747C2—41no—
748C2—161no—
749C2#2121311yescharacteristic
750C2#2123411yescharacteristic
751C2C2×#197611yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21941C2semidirect
4. #21825C2semidirect
5. #20528C2semidirect
7. C2×#1520C2semidirect
8. #9786C2semidirect
9. #9784C2semidirect
10. #9782C2semidirect
11. #9781C2semidirect
20. C2×#485C4semidirect
21. C2×#483C4semidirect
22. C2×#480C4semidirect
25. C2×#452C4semidirect
26. C2×#451C4semidirect
27. C2×#448C4semidirect
64. C22×Q16C2×C4semidirect
65. C22×SD16C2×C4semidirect
133. C2×#107C2×C4semidirect
135. C2×#106C2×C4semidirect
136. C2×#103C2×C4semidirect
137. C2×#103C2×C4semidirect
138. C2×#103C2×C4semidirect
139. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31902
#219412
#218742
#218252
#205282
#175722
C2×#15202
#97862
#97842
#97822
#97812