Finite Group Databasebeta

← #6514147

Subgroups — #6514147

Order 512.

2034 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3245C211yesmaximal, characteristic
2C2×#3112C211yesmaximal, characteristic, complemented
3#35249C211yesmaximal, characteristic, complemented
4#34398C211yesmaximal, characteristic, complemented
5#28288C212yesmaximal, complemented
6#28257C212yesmaximal, complemented
7#28211C212yesmaximal, complemented
8#28167C212yesmaximal, complemented
9#25200C214yesmaximal, complemented
10#25090C214yesmaximal, complemented
11#24833C212yesmaximal, complemented
12#24749C212yesmaximal, complemented
13#24696C212yesmaximal, complemented
14#24647C212yesmaximal, complemented
15#21977C211yesmaximal, characteristic, complemented
16#21865C211yesmaximal, characteristic, complemented
17C2×#1805C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3245—28no—
22#3112—28no—
23C2×#480C2212yes—
24C2×#477C2211yescharacteristic
25C2×#477C2211yescharacteristic
26C2×#474C2212yes—
27C2×#474C2212yes—
28C2×#466C2214yes—
29C2×#460C2211yescharacteristic
30C2×#460C2211yescharacteristic
31C2×#463C2214yes—
32C2×#459C2212yes—
33C2×#459C2212yes—
34C2×#448C2212yes—
35C2×#440C2212yes—
36C2×#440C2212yes—
37C2×#432C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C22×#103C2212yes—
40#2854—22no—
41#2632—22no—
42#2580—22no—
43#2567C2211yescharacteristic
44#2567C2211yescharacteristic
45#2548C2211yescharacteristic
46#2548C2211yescharacteristic
47#2501C2212yes—
48#2436C2212yes—
49#2406—22no—
50#2379—22no—
51#2361C2214yes—
52#2357C2212yes—
53#2349C2212yes—
54#2339—22no—
55#2331C2212yes—
56#2328C2212yes—
57#2326C2212yes—
58#2326C2212yes—
59#2325—22no—
60#2318—22no—
61#2283C2211yescharacteristic
62C2×#382C2211yescharacteristic
63#2083C2214yes—
64#2071C2214yes—
65#2068C2214yes—
66#2067C2214yes—
67#2055C2214yes—
68#2054C2214yes—
69#2042C2214yes—
70#2012C2214yes—
71#1993C2212yes—
72#1983C2214yes—
73#1961C2212yes—
74#1947C2214yes—
75#1940C2212yes—
76#1940C2212yes—
77#1928C2214yes—
78#1925C2214yes—
79#1879C2214yes—
80#1877C2214yes—
81#1874C2214yes—
82#1850C2214yes—
83#1843C2212yes—
84#1832C2212yes—
85#1805C2212yes—
86#1805C2212yes—
87#1805C2212yes—
88#1805C2212yes—
89#1804C2212yes—
90#1804C2212yes—
91#1802C2214yes—
92C2×#340C2212yes—
93C22×#41—21no—
94C22×#41—21no—
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97C22×SD16C2312yes—
98C22×D8C2314yes—
99C23×C8C2312yes—
100#532—24no—
101#532—24no—
102#532—24no—
103#532D418yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#124—21no—
109C2×#124—21no—
110C2×#124C2312yes—
111C2×#124C2312yes—
112C2×#123—24no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—24no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2314yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136#480—28no—
137#477—24no—
138#477—24no—
139#474—28no—
140#474—28no—
141#466—216no—
142#463—48no—
143#460—24no—
144#460—24no—
145#459—28no—
146#459—28no—
147#448—28no—
148#440—28no—
149#440—28no—
150#432—24no—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#107C2314yes—
154C2×C4⋊C8—24no—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2314yes—
161C2×#103—44no—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2312yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×C22⋊C8C2312yes—
173C2×#97C2312yes—
174#399—24no—
175#395—22no—
176#395—22no—
177#394—22no—
178#394—22no—
179#393—24no—
180#391—24no—
181#391C2312yes—
182#391C2312yes—
183#390—22no—
184#390—22no—
185#390C2312yes—
186#390C2314yes—
187#390C2312yes—
188#389—22no—
189#389—22no—
190#389C2314yes—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#387—24no—
196#386—24no—
197#386—24no—
198#386—24no—
199#386—24no—
200#385—24no—
201#385—24no—
202#385—24no—
203#385—24no—
204#384C2312yes—
205#384C2312yes—
206#383D412yes—
207#382—21no—
208#382—21no—
209#382D412yes—
210#382C2312yes—
211#382C2312yes—
212#381—22no—
213#381D412yes—
214#380—22no—
215#380D412yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4C2312yes—
221#376C2312yes—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95—22no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2314yes—
230#336C2314yes—
231C2×#41—28no—
232C2×#41—44no—
233C2×#41—84no—
234C2×#41—84no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—41no—
243C22×Q8—41no—
244C22×Q8C2411yescharacteristic
245C22×Q8C2411yescharacteristic
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2412yes—
253C22×D4C2412yes—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—24no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—24no—
267C2×SD16—42no—
268C2×SD16—42no—
269C2×D8—28no—
270C2×D8—44no—
271C2×D8—44no—
272C22×C8—24no—
273C22×C8—42no—
274C22×C8—42no—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8C2×D414yes—
282#127—22no—
283#127—22no—
284#127—22no—
285#127C2×D414yes—
286#124—24no—
287#124—24no—
288#124—28no—
289#124—42no—
290#124—42no—
291#124—44no—
292#123—88no—
293#121—28no—
294#121—24no—
295#121—24no—
296#121—24no—
297#121—44no—
298#121C2×D418yes—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×D4—28no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—44no—
314C4×D4—44no—
315C4×D4—44no—
316C4×D4—88no—
317C4×D4C2×D418yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4C2×D414yes—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—24no—
358C2×C42—44no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D414yes—
362C2×C42#4114yes—
363C2×C42C2412yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—24no—
368#107—24no—
369#107—28no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374C4⋊C8—44no—
375C4⋊C8—44no—
376#103—28no—
377#103—24no—
378#103—24no—
379#103—44no—
380#102—24no—
381#102—24no—
382#102—28no—
383#102—28no—
384#102—44no—
385#102—44no—
386C22⋊C8—24no—
387#97—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95C2×D412yes—
397#95C2×D414yes—
398#95C2×D414yes—
399#95C2×D412yes—
400#95C2×D414yes—
401C24—22no—
402C24—22no—
403C24C22×D411yescharacteristic
404#41—816no—
405#41—816no—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—28no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8H(5,𝔽2)12yes—
428C2×Q82−1+412yes—
429C2×D4—28no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—44no—
444C2×D4—48no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4C22×D414yes—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4#12114yes—
482C22×C4#12114yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C2×#4112yes—
492C22×C4C2511yescharacteristic
493SD16—48no—
494SD16—48no—
495D8—88no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—84no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C22×D414yes—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—84no—
550C42#12114yes—
551C42#12114yes—
552C42C22×D412yes—
553C42C22×D412yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C23—22no—
557C23—21no—
558C23—21no—
559C23—24no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23C2×#12112yes—
570C23#57614yes—
571C23C23×D411yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—22no—
575Q8—22no—
576Q8—48no—
577Q8—41no—
578Q8—41no—
579Q8—41no—
580Q8—41no—
581Q8—84no—
582Q8—84no—
583D4—24no—
584D4—24no—
585D4—416no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—84no—
593D4—84no—
594D4—84no—
595D4—84no—
596D4—88no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4—84no—
636C2×C4—168no—
637C2×C4—164no—
638C2×C4C2×#12112yes—
639C2×C4C2×#12112yes—
640C2×C4C2×#12112yes—
641C2×C4C2×#12112yes—
642C2×C4C2×#12112yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C2×C4#58511yescharacteristic
648C2×C4#58511yescharacteristic
649C8—44no—
650C8—42no—
651C8—42no—
652C22—22no—
653C22—24no—
654C22—22no—
655C22—24no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—84no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—82no—
665C22—84no—
666C22#304114yes—
667C22C22×#12111yescharacteristic
668C22#348811yescharacteristic
669C22#348911yescharacteristic
670C22C2×#57612yes—
671C22#358612yes—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—44no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—81no—
683C4—81no—
684C4—82no—
685C4—84no—
686C4#348811yescharacteristic
687C4#348911yescharacteristic
688C4#349211yescharacteristic
689C4#349311yescharacteristic
690C2—24no—
691C2—84no—
692C2—162no—
693C2#3551811yescharacteristic
694C2C2×#304111yescharacteristic
695C2#6042711yescharacteristic
696C2#6202212yes—
697C2#6217012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3245C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3112C2semidirect
3. #35249C2semidirect
4. #34398C2semidirect
5. #28288C2semidirect
6. #28257C2semidirect
7. #28211C2semidirect
8. #28167C2semidirect
9. #25200C2semidirect
10. #25090C2semidirect
11. #24833C2semidirect
12. #24749C2semidirect
13. #24696C2semidirect
14. #24647C2semidirect
15. #21977C2semidirect
16. #21865C2semidirect
17. C2×#1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32452
C2×#31122
#352492
#343982
#282882
#282572
#282112
#281672
#252002
#250902
#248332
#247492
#246962
#246472
#219772
#218652
C2×#18052