Finite Group Databasebeta

← #6433834

Subgroups — #6433834

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#35257C211yesmaximal, characteristic
3#30302C214yesmaximal, complemented
4#30299C214yesmaximal, complemented
5#30298C212yesmaximal, complemented
6#30295C212yesmaximal, complemented
7#30294C212yesmaximal, complemented
8#30293C212yesmaximal, complemented
9#23662C212yesmaximal, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23656C211yesmaximal, characteristic, complemented
12#23556C212yesmaximal, complemented
13#23552C211yesmaximal, characteristic, complemented
14C2×#2058C211yesmaximal, characteristic, complemented
15#20664C212yesmaximal
16#20662C212yesmaximal
17#20336C211yesmaximal, characteristic
18C22×#131C2212yes—
19C2×#544C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C2×#428C2212yes—
24C2×#427C2212yes—
25C22×#106C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2851C2212yes—
30#2629C2212yes—
31#2628C2214yes—
32#2628C2212yes—
33#2568—21no—
34#2567—21no—
35#2563—21no—
36#2542—21no—
37#2487—22no—
38#2454—21no—
39#2453—21no—
40#2452—21no—
41#2451—21no—
42#2450—22no—
43#2449—22no—
44#2409—22no—
45#2317C2212yes—
46#2317C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2212yes—
49#2311C2211yescharacteristic
50#2296C2211yescharacteristic
51C2×#389C2211yescharacteristic
52#2285C2211yescharacteristic
53#2085C2214yes—
54#2085C2214yes—
55#2085C2212yes—
56#2085C2214yes—
57#2085C2212yes—
58#2084C2211yescharacteristic
59#2084C2212yes—
60#2084C2214yes—
61#2084C2211yescharacteristic
62#2083C2212yes—
63#2083C2211yescharacteristic
64#2083C2211yescharacteristic
65#2083C2214yes—
66#2059C2214yes—
67#2059C2214yes—
68#2058C2212yes—
69#2058C2212yes—
70#2058C2214yes—
71#2058C2214yes—
72#2058C2212yes—
73#2058C2212yes—
74#2058C2214yes—
75#2057C2212yes—
76#2057C2212yes—
77#1809C2214yes—
78#1809C2214yes—
79#1809C2214yes—
80#1809C2214yes—
81#1808C2214yes—
82#1807C2214yes—
83#1807C2214yes—
84#1807C2214yes—
85#1736C2212yes—
86#1736C2212yes—
87#1735C2212yes—
88C2×#340C2212yes—
89#1729C2212yes—
90#1729C2214yes—
91#1728C2211yescharacteristic
92#1728C2212yes—
93C2×#336C2212yes—
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C2×#131—44no—
98C2×#131—44no—
99C2×#131Q8116yescomplemented
100C22×M16C2312yes—
101C22×M16C2314yes—
102C23×C8C2312yes—
103C23×C8C2314yes—
104#544—44no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—44no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C4×#41—44no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D412yes—
123C2×C4×D4D412yes—
124C2×C4×D4D411yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#428—44no—
138#428—44no—
139#427—44no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106—44no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149C2×#106C2312yes—
150C2×#106C2314yes—
151#400—24no—
152#400—22no—
153#399—22no—
154#399—24no—
155#398—22no—
156#398—22no—
157#396—22no—
158#396—22no—
159#395—24no—
160#395—22no—
161#393—24no—
162#393—22no—
163#391—22no—
164#391—21no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391—22no—
169#390—22no—
170#390—21no—
171#390—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389C2311yescharacteristic
180#389C2311yescharacteristic
181#389C2312yes—
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2314yes—
186#389C2314yes—
187#387—22no—
188#387—22no—
189#386—24no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—21no—
195#385—21no—
196#385—24no—
197#385—22no—
198#384—22no—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382—22no—
203#382C2312yes—
204#382C2312yes—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—21no—
209#380—21no—
210#380C2312yes—
211#380C2314yes—
212#380C2312yes—
213#380C2312yes—
214#380C2314yes—
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C2×#95—21no—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41C2×Q818yes—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—44no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2×D411yescharacteristic
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260#131—432no—
261C2×M16—24no—
262C2×M16—24no—
263C2×M16—44no—
264C2×M16—44no—
265C2×M16C2×Q818yes—
266C2×M16C2×Q8116yescomplemented
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—44no—
270C22×C8—44no—
271C22×C8C2×Q818yes—
272C22×C8C2×Q8116yescomplemented
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282#125—44no—
283#125—44no—
284C22⋊Q8—48no—
285C22⋊Q8—84no—
286C22⋊Q8—84no—
287C22⋊Q8—84no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294#117—24no—
295#117—24no—
296#117—22no—
297#117—24no—
298#117—22no—
299#117—24no—
300#117—44no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—44no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—42no—
368#107—42no—
369#107—44no—
370#107—44no—
371#106—44no—
372#106—44no—
373#106—42no—
374#106—42no—
375#106—44no—
376#106—44no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397#41—216no—
398#41—216no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—816no—
411C2×Q8C4⋊Q814yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4C4⋊Q818yes—
417C2×D4C4⋊Q814yes—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×Q818yes—
460C22×C4C22×Q814yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+412yes—
464C22×C4C2511yescharacteristic
465M16—216no—
466M16—416no—
467C2×C8—216no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—416no—
475C2×C8C22×Q814yes—
476C2×C8C22×Q814yes—
477C2×C8C22×Q818yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—84no—
528C22⋊C4—82no—
529C22⋊C4—84no—
530C22⋊C4—82no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—42no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23C2×C4⋊Q814yes—
552C23C2×C4⋊Q812yes—
553C23#53412yes—
554C23#53712yes—
555C23C23×D411yescharacteristic
556Q8—28no—
557Q8—88no—
558Q8—88no—
559Q8—164no—
560D4—216no—
561D4—28no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4—162no—
616C2×C4C2×C4⋊Q814yes—
617C2×C4C2×C4⋊Q812yes—
618C2×C4#53412yes—
619C2×C4#53712yes—
620C2×C4#54318yes—
621C2×C4#54418yes—
622C2×C4#57712yes—
623C2×C4#57712yes—
624C2×C4C23×Q812yes—
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4C2×2−1+411yescharacteristic
628C8—24no—
629C8—24no—
630C8—28no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22#265914yes—
645C22C22×C4⋊Q811yescharacteristic
646C22C2×#53411yescharacteristic
647C22C2×#53711yescharacteristic
648C22C2×#57711yescharacteristic
649C22C2×#57711yescharacteristic
650C22#358511yescharacteristic
651C22#358611yescharacteristic
652C4—24no—
653C4—22no—
654C4—22no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4—84no—
666C4—82no—
667C4—81no—
668C4#265914yes—
669C4C2×#54312yes—
670C4C2×#54412yes—
671C2—24no—
672C2—44no—
673C2C2×#265911yescharacteristic
674C2#6066511yescharacteristic
675C2#6066811yescharacteristic
676C2#6075711yescharacteristic
677C2#6075811yescharacteristic
678C2#6092911yescharacteristic
679C2#6093111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30302C2semidirect
4. #30299C2semidirect
5. #30298C2semidirect
6. #30295C2semidirect
7. #30294C2semidirect
8. #30293C2semidirect
9. #23662C2semidirect
10. #23657C2semidirect
11. #23656C2semidirect
12. #23556C2semidirect
13. #23552C2semidirect
14. C2×#2058C2semidirect
99. C2×#131Q8semidirect
266. C2×M16C2×Q8semidirect
272. C22×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352792
#352572
#303022
#302992
#302982
#302952
#302942
#302932
#236622
#236572
#236562
#235562
#235522
C2×#20582
#206642
#206622
#203362