Finite Group Databasebeta

← #880168

Subgroups — #880168

Order 512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17080C212yesmaximal
2#17073C212yesmaximal
3#15682C214yesmaximal
4#15333C214yesmaximal
5#15202C212yesmaximal
6#15137C212yesmaximal
7#15137C212yesmaximal
8#14714C211yesmaximal, characteristic
9#14712C211yesmaximal, characteristic
10#14471C212yesmaximal
11#14467C212yesmaximal
12#14053C212yesmaximal
13#14048C212yesmaximal
14#14045C212yesmaximal
15C4×#391C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#398C2214yes—
19C2×#398C2214yes—
20C2×#391C2212yes—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#391C2214yes—
24C2×#391C2211yescharacteristic
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C43C2211yescharacteristic
38#1446C2214yes—
39#1446C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1436C2212yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2211yescharacteristic
73C4×#95C2212yes—
74C4×#95C2212yes—
75C23×Q8C2311yescharacteristic
76C23×Q8C2312yes—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—42no—
79C2×C4⋊Q8Q818yescomplemented
80C2×C4×Q8—24no—
81C2×C4×Q8—42no—
82C2×C4×Q8—42no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#398—48no—
114#398—48no—
115#391—24no—
116#391—28no—
117#391—44no—
118#391—48no—
119#391Q818yescomplemented
120#389—28no—
121#389—44no—
122#389—44no—
123#389—44no—
124#389—44no—
125#389—44no—
126#389—44no—
127#384—42no—
128#384—42no—
129#384Q818yescomplemented
130C4×C4⋊C4—42no—
131C4×C4⋊C4—44no—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C43—42no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8C2×Q818yes—
173C22×Q8C2×Q814yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C4⋊Q8—416no—
190C4×Q8—416no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C2×Q818yes—
246C2×C4⋊C4C2×Q818yes—
247C2×C4⋊C4C2×Q818yes—
248C2×C4⋊C4C2×Q818yes—
249C2×C4⋊C4C2×Q818yes—
250C2×C4⋊C4C2×Q814yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—41no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C2×Q814yes—
277C2×C42C2×Q814yes—
278C2×C42C2×Q814yes—
279C2×C42#4114yes—
280C2×C42#4114yes—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95C2×Q818yes—
307#95C2×Q818yes—
308C24C2511yescharacteristic
309C2×Q8—28no—
310C2×Q8—216no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—84no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4C22×Q814yes—
402C22×C4C22×Q814yes—
403C22×C4C22×Q818yes—
404C22×C4C22×Q818yes—
405C22×C4C22×Q812yes—
406C22×C4C22×Q812yes—
407C22×C4C22×Q812yes—
408C22×C4C22×Q812yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q812yes—
411C22×C4C22×Q812yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q812yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+412yes—
427C22×C42−1+412yes—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C42—28no—
449C42—28no—
450C42—28no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42—44no—
455C42—48no—
456C42—48no—
457C42—88no—
458C23C23×Q811yescharacteristic
459C23C23×Q811yescharacteristic
460C23C23×Q811yescharacteristic
461C23C22×#4111yescharacteristic
462C23C22×#4111yescharacteristic
463C23C2×H(5,𝔽2)11yescharacteristic
464C23C2×H(5,𝔽2)11yescharacteristic
465C23C2×H(5,𝔽2)11yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×2−1+411yescharacteristic
470C23C2×2−1+411yescharacteristic
471C23#58511yescharacteristic
472C23#58511yescharacteristic
473Q8—416no—
474Q8—816no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4#53314yes—
569C2×C4#53814yes—
570C2×C4#54314yes—
571C2×C4#54414yes—
572C2×C4#55414yes—
573C2×C4#55414yes—
574C2×C4#55714yes—
575C2×C4#55714yes—
576C2×C4Q8214yes—
577C2×C4Q8214yes—
578C2×C4Q8214yes—
579C2×C4#56214yes—
580C22C2×#53311yescharacteristic
581C22C2×#53811yescharacteristic
582C22#345511yescharacteristic
583C22C2×#54311yescharacteristic
584C22C2×#54411yescharacteristic
585C22C2×#55411yescharacteristic
586C22C2×#55411yescharacteristic
587C22C2×#55711yescharacteristic
588C22C2×#55711yescharacteristic
589C22C2×Q8211yescharacteristic
590C22C2×Q8211yescharacteristic
591C22C2×Q8211yescharacteristic
592C22#350111yescharacteristic
593C22#350111yescharacteristic
594C22#350111yescharacteristic
595C22#350111yescharacteristic
596C22#350111yescharacteristic
597C22#350211yescharacteristic
598C22#350311yescharacteristic
599C22#350311yescharacteristic
600C22#350511yescharacteristic
601C22#350611yescharacteristic
602C22#350811yescharacteristic
603C22#350811yescharacteristic
604C22#351111yescharacteristic
605C22#351111yescharacteristic
606C22C2×#56211yescharacteristic
607C22#353611yescharacteristic
608C22#353711yescharacteristic
609C22#354811yescharacteristic
610C22#354911yescharacteristic
611C22#355811yescharacteristic
612C22#355811yescharacteristic
613C22#355911yescharacteristic
614C22#356311yescharacteristic
615C4—24no—
616C4—24no—
617C4—24no—
618C4—24no—
619C4—24no—
620C4—48no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—48no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4—84no—
629C4#280618yes—
630C2#3501411yescharacteristic
631C2#3656311yescharacteristic
632C2#3656311yescharacteristic
633C2#3731611yescharacteristic
634C2#3949511yescharacteristic
635C2#3962311yescharacteristic
636C2#3962411yescharacteristic
637C2#3962811yescharacteristic
638C2#3963711yescharacteristic
639C2#3975111yescharacteristic
640C2#4600511yescharacteristic
641C2#4601511yescharacteristic
642C2C2×#280611yescharacteristic
643C2#5987111yescharacteristic
644C2#5992411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
79. C2×C4⋊Q8Q8semidirect
119. #391Q8semidirect
129. #384Q8semidirect

Maximal subgroups

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

subgroupindex
#170802
#170732
#156822
#153332
#152022
#151372
#147142
#147122
#144712
#144672
#140532
#140482
#140452
C4×#3912