Finite Group Databasebeta

← #805549

Subgroups — #805549

Order 512.

2380 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2429C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#17066C212yesmaximal
4#15810C214yesmaximal
5#15806C214yesmaximal, complemented
6#15197C212yesmaximal, complemented
7#15195C211yesmaximal, characteristic
8#15192C211yesmaximal, characteristic, complemented
9#15185C211yesmaximal, characteristic, complemented
10#15111C212yesmaximal, complemented
11#14249C212yesmaximal
12#14024C212yesmaximal, complemented
13#13997C212yesmaximal
14#13969C212yesmaximal
15#13933C212yesmaximal, complemented
16C4×#397C211yesmaximal, characteristic
17#13817C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2212yes—
23#2429—28no—
24C2×#398C2212yes—
25C2×#397C2211yescharacteristic
26C2×#396C2212yes—
27Q8×C4⋊C4—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45#1445C2211yescharacteristic
46#1445C2212yes—
47#1445C2214yes—
48#1445C2214yes—
49#1443C2212yes—
50#1443C2214yes—
51#1442C2212yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2211yescharacteristic
55#1442C2212yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2212yes—
63#1439C2211yescharacteristic
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2214yes—
80#1434C2212yes—
81#1434C2214yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—28no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C2×C4×Q8D418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2314yes—
129#398—28no—
130#397Q818yescomplemented
131#396—28no—
132#391—28no—
133#391—24no—
134#391—44no—
135#391—44no—
136#389—216no—
137#389—28no—
138#389—24no—
139#389—28no—
140#388—28no—
141#384—24no—
142#384—28no—
143#380—44no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—44no—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8C2×D414yes—
191C22×Q8#4118yes—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
214C22⋊Q8—416no—
215C22⋊Q8—416no—
216C4×Q8—232no—
217C4×Q8—432no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×Q814yes—
255C2×C4⋊C4#4118yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C2×Q814yes—
274C2×C22⋊C4C2×Q818yes—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42C2×D418yes—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95—42no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C2×Q818yes—
324#95C2×Q814yes—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24C22×Q812yes—
336C24C2511yescharacteristic
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—216no—
340C2×Q8—216no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8#124116yes—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4C22×D412yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q812yes—
436C22×C4C2×#4118yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C22×C42−1+412yes—
444C4⋊C4—216no—
445C4⋊C4—28no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—416no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—88no—
458C4⋊C4—84no—
459C4⋊C4#123116yes—
460C22⋊C4—28no—
461C22⋊C4—48no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—88no—
467C42—216no—
468C42—28no—
469C42—416no—
470C42—44no—
471C42—48no—
472C42—48no—
473C42—88no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23C23×D411yescharacteristic
498C23C23×Q811yescharacteristic
499C23C22×#4112yes—
500C23C22×#4112yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×2−1+411yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23#58512yes—
506C23#58512yes—
507C23#58511yescharacteristic
508Q8—232no—
509Q8—88no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—88no—
595C2×C4C2×#12318yes—
596C2×C4C2×#12314yes—
597C2×C4C2×#12418yes—
598C2×C4C2×#12414yes—
599C2×C4#53514yes—
600C2×C4D4×Q814yes—
601C2×C4#55318yes—
602C2×C4#55318yes—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22C22×#12311yescharacteristic
615C22C22×#12411yescharacteristic
616C22C2×#53511yescharacteristic
617C22C2×D4×Q811yescharacteristic
618C22C2×#55312yes—
619C22C2×#55312yes—
620C22Q8×#4112yes—
621C22Q8×#4112yes—
622C22#348711yescharacteristic
623C22#348911yescharacteristic
624C22#350011yescharacteristic
625C22#350111yescharacteristic
626C22#350311yescharacteristic
627C22#351512yes—
628C22#351612yes—
629C22#351612yes—
630C22#351812yes—
631C22#352112yes—
632C22#352412yes—
633C22#353912yes—
634C22#355011yescharacteristic
635C22#355411yescharacteristic
636C22#356212yes—
637C4—24no—
638C4—28no—
639C4—28no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—84no—
649C4—81no—
650C4—82no—
651C4—84no—
652C4#250418yes—
653C2—82no—
654C2C2×#250411yescharacteristic
655C2#3585411yescharacteristic
656C2#3606711yescharacteristic
657C2Q8×#12311yescharacteristic
658C2Q8×#12411yescharacteristic
659C2#3959611yescharacteristic
660C2#4288011yescharacteristic
661C2#4529812yes—
662C2#4534812yes—
663C2#5939312yes—
664C2#5969412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
5. #15806C2semidirect
6. #15197C2semidirect
8. #15192C2semidirect
9. #15185C2semidirect
10. #15111C2semidirect
12. #14024C2semidirect
15. #13933C2semidirect
17. #13817C2semidirect
130. #397Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
C2×Q8×C4⋊C42
#170662
#158102
#158062
#151972
#151952
#151922
#151852
#151112
#142492
#140242
#139972
#139692
#139332
C4×#3972
#138172