Finite Group Databasebeta

← #6971334

Subgroups — #6971334

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36477C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic
3#32724C212yesmaximal, complemented
4#31292C214yesmaximal, complemented
5#30449C214yesmaximal, complemented
6#30220C211yesmaximal, characteristic, complemented
7#30109C212yesmaximal, complemented
8#29627C212yesmaximal, complemented
9#29123C211yesmaximal, characteristic, complemented
10#28440C211yesmaximal, characteristic, complemented
11#24732C212yesmaximal, complemented
12#24632C212yesmaximal, complemented
13#24243C212yesmaximal
14#24243C212yesmaximal
15#24212C212yesmaximal, complemented
16#23801C211yesmaximal, characteristic, complemented
17#22008C211yesmaximal, characteristic
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C2×#494C2212yes—
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×#462C2214yes—
24C2×#451C2214yes—
25C2×#451C2214yes—
26C2×C4×Q16C2212yes—
27C2×C4×Q16C2212yes—
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2212yes—
31#2855C2212yes—
32#2608—22no—
33#2587—22no—
34#2570—21no—
35#2566—21no—
36#2551—21no—
37#2551—21no—
38#2546C2212yes—
39#2538C2211yescharacteristic
40#2538C2211yescharacteristic
41#2497C2212yes—
42#2440—22no—
43#2440—22no—
44#2434C2214yes—
45#2429C2212yes—
46#2370—21no—
47#2357—21no—
48#2355C2211yescharacteristic
49Q8×C4⋊C4C2211yescharacteristic
50D4×C4⋊C4—21no—
51D4×C4⋊C4—21no—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C22⋊Q8C2212yes—
55C42×D4C2211yescharacteristic
56#2100C2212yes—
57#2095C2212yes—
58#2095C2211yescharacteristic
59#2095C2211yescharacteristic
60#2068C2214yes—
61#2056C2214yes—
62#2043C2214yes—
63#2043C2212yes—
64#2043C2212yes—
65#2043C2214yes—
66#2043C2212yes—
67#2036C2214yes—
68#2036C2214yes—
69#2016C2212yes—
70#2016C2212yes—
71#2016C2214yes—
72#2016C2211yescharacteristic
73#2016C2211yescharacteristic
74#2016C2212yes—
75#2001C2214yes—
76#2001C2214yes—
77#1981C2214yes—
78#1956C2212yes—
79#1956C2212yes—
80#1956C2212yes—
81#1954C2212yes—
82#1939C2212yes—
83#1933C2214yes—
84#1928C2214yes—
85#1923C2214yes—
86#1878C2214yes—
87#1817C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91#1787C2214yes—
92#1787C2214yes—
93C4×C4⋊C8C2211yescharacteristic
94#1744C2211yescharacteristic
95C23×Q8C2314yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2312yes—
99C22×Q16C2314yes—
100D4×Q8—44no—
101D4×Q8—44no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—48no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×#125—21no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2314yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C42C2312yes—
128#494—44no—
129#462—48no—
130#462—44no—
131#462—44no—
132#451—48no—
133#451—48no—
134C4×Q16—44no—
135C4×Q16—44no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#398—22no—
158#398—24no—
159#396—22no—
160#396—22no—
161#396—22no—
162#396—22no—
163#395—24no—
164#395—24no—
165#391C2311yescharacteristic
166#391C2312yes—
167#391C2311yescharacteristic
168#391C2312yes—
169#391C2314yes—
170#390—21no—
171#390—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—22no—
197#385—22no—
198#385—24no—
199#385—24no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384—21no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382—21no—
210#382—21no—
211#381—22no—
212#380—22no—
213#380—22no—
214#380C2312yes—
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C22⋊C4—22no—
228C4×C22⋊C4D412yes—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4C2312yes—
231C43C2311yescharacteristic
232#340C2314yes—
233#340C2314yes—
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2311yescharacteristic
237#336C2314yes—
238#336C2312yes—
239C22×Q8—24no—
240C22×Q8—28no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2414yes—
248C22×Q8C2411yescharacteristic
249C22×Q8C2412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—24no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—28no—
259C2×Q16—42no—
260C2×Q16—42no—
261C2×Q16—42no—
262C2×Q16—44no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—28no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—41no—
278C4⋊Q8—42no—
279C4⋊Q8—42no—
280C4⋊Q8—48no—
281C4⋊Q8—48no—
282C4⋊Q8—48no—
283#125—44no—
284C22⋊Q8—24no—
285C22⋊Q8—416no—
286C22⋊Q8—42no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—44no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—48no—
298C4×D4Q1618yescomplemented
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—42no—
364#107—42no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#103—28no—
368#103—44no—
369#103—42no—
370#103—42no—
371#103—42no—
372#103—42no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379C22⋊C8—44no—
380C22⋊C8—44no—
381C4×C8—24no—
382C4×C8—42no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24C22×D412yes—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—21no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—84no—
424C2×Q8—88no—
425C2×Q8—88no—
426C2×Q8—88no—
427C2×Q8—88no—
428C2×Q8H(5,𝔽2)14yes—
429C2×Q8H(5,𝔽2)12yes—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4C2×Q1614yes—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—84no—
454C22×C4C2×Q1618yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465Q16—48no—
466Q16—48no—
467Q16—88no—
468Q16—88no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4C2×Q1614yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4C2×Q1618yes—
526C42—21no—
527C42—24no—
528C42—22no—
529C42—21no—
530C42—42no—
531C42—42no—
532C42—41no—
533C42—41no—
534C42—44no—
535C42—42no—
536C42—42no—
537C42—82no—
538C42—82no—
539C42C2×Q1614yes—
540C42#13714yes—
541C42H(5,𝔽2)14yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23C22×Q1614yes—
549C23C23×D411yescharacteristic
550Q8—24no—
551Q8—24no—
552Q8—24no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—44no—
558Q8—44no—
559Q8—42no—
560Q8—44no—
561Q8—44no—
562Q8—816no—
563D4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
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—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—816no—
614C2×C4#53512yes—
615C2×C4#53512yes—
616C2×C4C22×Q1612yes—
617C2×C4C22×Q1612yes—
618C2×C4C22×Q1614yes—
619C2×C4C22×Q1612yes—
620C2×C4C2×#13712yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4#58511yescharacteristic
627C8—42no—
628C8—44no—
629C8—42no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22C2×#53511yescharacteristic
638C22#348411yescharacteristic
639C22#348811yescharacteristic
640C22C23×Q1611yescharacteristic
641C22C22×#13711yescharacteristic
642C22#358611yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—81no—
658C4—82no—
659C4—84no—
660C4—81no—
661C4#320914yes—
662C4#353712yes—
663C4#354212yes—
664C2—44no—
665C2#4248011yescharacteristic
666C2C2×#320911yescharacteristic
667C2#6083111yescharacteristic
668C2#6156311yescharacteristic
669C2#6158411yescharacteristic
670C2#6195211yescharacteristic
671C2#6196411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32724C2semidirect
4. #31292C2semidirect
5. #30449C2semidirect
6. #30220C2semidirect
7. #30109C2semidirect
8. #29627C2semidirect
9. #29123C2semidirect
10. #28440C2semidirect
11. #24732C2semidirect
12. #24632C2semidirect
15. #24212C2semidirect
16. #23801C2semidirect
298. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#364772
D4×C4⋊Q82
#327242
#312922
#304492
#302202
#301092
#296272
#291232
#284402
#247322
#246322
#242432
#242122
#238012
#220082