Finite Group Databasebeta

← #6484780

Subgroups — #6484780

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C211yesmaximal, characteristic, complemented
2C2×#3238C211yesmaximal, characteristic, complemented
3#35343C211yesmaximal, characteristic
4#34216C211yesmaximal, characteristic, complemented
5#33119C212yesmaximal, complemented
6#33118C212yesmaximal
7#29768C212yesmaximal, complemented
8#29767C212yesmaximal, complemented
9#28754C212yesmaximal, complemented
10#28692C212yesmaximal, complemented
11#28659C212yesmaximal, complemented
12#28658C212yesmaximal, complemented
13#28291C212yesmaximal, complemented
14#28290C212yesmaximal, complemented
15#28212C212yesmaximal, complemented
16#28211C212yesmaximal, complemented
17#21868C211yesmaximal, characteristic, complemented
18#21868C211yesmaximal, characteristic, complemented
19#20811C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C2×#529C2211yescharacteristic
22C2×#529C2211yescharacteristic
23#3239—28no—
24#3238—28no—
25C2×#501C2212yes—
26C2×#501C2212yes—
27C2×#489C2212yes—
28C2×#489C2212yes—
29C2×#484C2212yes—
30C2×#481C2212yes—
31C2×#483C2212yes—
32C2×#480C2212yes—
33C2×#477C2212yes—
34C2×#475C2212yes—
35C2×#476C2212yes—
36C2×#474C2212yes—
37C2×#432C2211yescharacteristic
38C2×#432C2211yescharacteristic
39#2861C2211yescharacteristic
40#2861C2211yescharacteristic
41#2859C2211yescharacteristic
42#2858C2211yescharacteristic
43#2621C2211yescharacteristic
44#2620C2211yescharacteristic
45#2620C2211yescharacteristic
46#2619C2211yescharacteristic
47#2602—22no—
48#2602—22no—
49#2601—22no—
50#2601—22no—
51#2369—22no—
52#2365—22no—
53#2345—22no—
54Q8×C22⋊C4—22no—
55#2331C2212yes—
56#2331C2212yes—
57#2328C2211yescharacteristic
58#2327C2211yescharacteristic
59#2326C2211yescharacteristic
60#2326C2211yescharacteristic
61#2316C2212yes—
62C4×C4⋊Q8C2212yes—
63C4×C4⋊Q8C2212yes—
64#2285C2211yescharacteristic
65#2274C2212yes—
66C4×#117C2211yescharacteristic
67#2084C2214yes—
68#2083C2214yes—
69#2068C2214yes—
70#2067C2214yes—
71#2066C2214yes—
72#2065C2214yes—
73#2042C2214yes—
74#2042C2214yes—
75#1992C2212yes—
76#1992C2212yes—
77#1982C2214yes—
78#1981C2214yes—
79#1959C2212yes—
80#1959C2212yes—
81#1949C2214yes—
82#1949C2214yes—
83#1928C2214yes—
84#1927C2214yes—
85#1926C2214yes—
86#1925C2214yes—
87#1875C2212yes—
88#1875C2212yes—
89#1873C2212yes—
90#1873C2212yes—
91#1843C2212yes—
92#1832C2212yes—
93#1805C2212yes—
94#1805C2212yes—
95#1804C2212yes—
96#1804C2212yes—
97#1798C2212yes—
98#1798C2212yes—
99#1767C2212yes—
100#1767C2212yes—
101C23×Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8D418yescomplemented
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#127C2311yescharacteristic
110#529—24no—
111#529—24no—
112#529—24no—
113#529—24no—
114C2×#126—21no—
115C2×#126—21no—
116C2×#125C2311yescharacteristic
117C2×#125C2311yescharacteristic
118C2×#124C2311yescharacteristic
119C2×#124C2311yescharacteristic
120C2×#124C2311yescharacteristic
121C2×#124C2311yescharacteristic
122C2×#123—21no—
123C2×#123—21no—
124C2×C22⋊Q8C2311yescharacteristic
125C2×C22⋊Q8C2311yescharacteristic
126C2×#121C2311yescharacteristic
127C2×#121C2311yescharacteristic
128C2×C4×Q8—22no—
129C2×C4×Q8—22no—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2311yescharacteristic
135C2×C4×D4C2311yescharacteristic
136C2×#117—21no—
137C2×#117—21no—
138C2×#117C2312yes—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2312yes—
143C22×C42C2311yescharacteristic
144#501—28no—
145#501—28no—
146#489—28no—
147#489—28no—
148#484—28no—
149#483—28no—
150#481—28no—
151#480—28no—
152#477—28no—
153#476—28no—
154#475—28no—
155#474—28no—
156#432—24no—
157#432—24no—
158C2×#107C2314yes—
159C2×#107C2314yes—
160C2×#106C2314yes—
161C2×#106C2314yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2314yes—
165C2×C4⋊C8C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×#102C2314yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2312yes—
176C2×#97C2312yes—
177C2×#97C2312yes—
178#398—22no—
179#397—22no—
180#396—22no—
181#396—22no—
182#394—22no—
183#394—22no—
184#394—22no—
185#392—22no—
186#391—22no—
187#391—22no—
188#391—24no—
189#391—24no—
190#391C2311yescharacteristic
191#391C2311yescharacteristic
192#391C2311yescharacteristic
193#391C2311yescharacteristic
194#390—22no—
195#390—22no—
196#390C2311yescharacteristic
197#390C2311yescharacteristic
198#390C2311yescharacteristic
199#390C2311yescharacteristic
200#389—24no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389—22no—
205#389C2314yes—
206#389C2314yes—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#387—22no—
214#387—22no—
215#387—24no—
216#387—24no—
217#386—22no—
218#386—22no—
219#385—22no—
220#385—22no—
221#384C2312yes—
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#382C2312yes—
225#382C2311yescharacteristic
226#382C2311yescharacteristic
227#380—21no—
228#380—21no—
229#380—24no—
230C4×C4⋊C4—24no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4D412yes—
236C4×C4⋊C4D412yes—
237C4×C4⋊C4C2314yes—
238C4×C4⋊C4C2314yes—
239C4×C22⋊C4—24no—
240C4×C22⋊C4D412yes—
241C4×C22⋊C4D412yes—
242#376C2311yescharacteristic
243#376C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2314yes—
252C22×Q8—28no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—84no—
256C22×Q8C2411yescharacteristic
257C22×Q8C2411yescharacteristic
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4C2411yescharacteristic
262C22×D4C2411yescharacteristic
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C22×C8C2414yes—
270C22×C8C2414yes—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8C2×D418yes—
277C4⋊Q8C2×D418yes—
278#127—24no—
279#126—44no—
280#126—44no—
281#125—22no—
282#125—22no—
283#125—22no—
284#125—22no—
285#124—22no—
286#124—22no—
287#124—22no—
288#124—22no—
289#124—42no—
290#124—42no—
291#123—44no—
292#123—44no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—22no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—84no—
306C4×Q8—84no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—44no—
312C4×D4—44no—
313#117—24no—
314#117—42no—
315#117—44no—
316#117—44no—
317#117—44no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D418yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2414yes—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D414yes—
378C2×C42#4114yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—28no—
387#106—28no—
388#106—28no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391C4⋊C8—28no—
392C4⋊C8—44no—
393#103—28no—
394#103—28no—
395#103—44no—
396#103—44no—
397#102—28no—
398#102—28no—
399#102—44no—
400#102—44no—
401C22⋊C8—24no—
402C22⋊C8—24no—
403#97—24no—
404#97—24no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95C2×D412yes—
418#95C2×D412yes—
419#95C2×D412yes—
420#95C2×D412yes—
421C24—21no—
422C24—21no—
423C24C22×D411yescharacteristic
424C2×Q8—28no—
425C2×Q8—28no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—84no—
437C2×Q8—816no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4—82no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—84no—
471C22×C4#12114yes—
472C22×C4#12114yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2511yescharacteristic
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4C22×D418yes—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—42no—
552C42—84no—
553C42—84no—
554C42#12114yes—
555C42#12114yes—
556C42C22×D412yes—
557C42C22×D412yes—
558C42C2×#4112yes—
559C42C2×#4112yes—
560C422−1+414yes—
561C23—22no—
562C23—22no—
563C23—21no—
564C23—21no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C2×#12112yes—
576C23C23×D411yescharacteristic
577Q8—416no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—88no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—84no—
588D4—84no—
589D4—164no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4#53614yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×2−1+412yes—
643C2×C4#58512yes—
644C8—44no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—82no—
657C22—81no—
658C22—81no—
659C22—82no—
660C22—82no—
661C22C22×#12111yescharacteristic
662C22C2×#53611yescharacteristic
663C22#344911yescharacteristic
664C22#358512yes—
665C22#358612yes—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—24no—
672C4—48no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—84no—
681C4—81no—
682C4—81no—
683C4#349712yes—
684C4#350712yes—
685C2—42no—
686C2—161no—
687C2—161no—
688C2#3517811yescharacteristic
689C2#6042311yescharacteristic
690C2#6042411yescharacteristic
691C2#6088512yes—
692C2#6102112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3239C2semidirect
2. C2×#3238C2semidirect
4. #34216C2semidirect
5. #33119C2semidirect
7. #29768C2semidirect
8. #29767C2semidirect
9. #28754C2semidirect
10. #28692C2semidirect
11. #28659C2semidirect
12. #28658C2semidirect
13. #28291C2semidirect
14. #28290C2semidirect
15. #28212C2semidirect
16. #28211C2semidirect
17. #21868C2semidirect
18. #21868C2semidirect
19. #20811C2semidirect
105. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
C2×#32382
#353432
#342162
#331192
#331182
#297682
#297672
#287542
#286922
#286592
#286582
#282912
#282902
#282122
#282112
#218682
#208112