Finite Group Databasebeta

← #880159

Subgroups — #880159

Order 512.

2094 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17080C212yesmaximal
2#17076C212yesmaximal
3#16120C214yesmaximal
4#15691C214yesmaximal
5#15201C212yesmaximal
6#15191C212yesmaximal
7#15146C212yesmaximal
8#14714C211yesmaximal, characteristic
9#14714C211yesmaximal, characteristic
10#14471C212yesmaximal
11#14471C212yesmaximal
12#14024C212yesmaximal
13#14017C212yesmaximal
14#13976C212yesmaximal
15C4×#391C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#400C2214yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#389C2212yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1446C2214yes—
36#1446C2214yes—
37#1446C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1436C2212yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2211yescharacteristic
73C4×#95C2212yes—
74C4×#95C2212yes—
75C23×Q8C2312yes—
76C23×Q8C2312yes—
77C23×Q8C2311yescharacteristic
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8Q818yescomplemented
84C2×C4×Q8—24no—
85C2×C4×Q8—42no—
86C2×C4×Q8—42no—
87C2×C4×Q8—44no—
88C2×C4×Q8—44no—
89C2×C4×Q8—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#400—48no—
117#391—28no—
118#391—28no—
119#391—24no—
120#391—28no—
121#391—44no—
122#391—44no—
123#391—44no—
124#391—44no—
125#391Q818yescomplemented
126#389—44no—
127#384—42no—
128#384—42no—
129C4×C4⋊C4—42no—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C43—42no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8C2×Q818yes—
181C22×Q8C2×Q814yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C4⋊Q8—416no—
198C4⋊Q8—416no—
199C4×Q8—416no—
200C4×Q8—432no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×Q818yes—
247C2×C4⋊C4C2×Q818yes—
248C2×C4⋊C4C2×Q814yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42C2×Q814yes—
275C2×C42C2×Q814yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—44no—
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#95—44no—
307#95C2×Q818yes—
308#95C2×Q818yes—
309C24C2511yescharacteristic
310C2×Q8—28no—
311C2×Q8—216no—
312C2×Q8—216no—
313C2×Q8—216no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—84no—
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—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4C22×Q814yes—
411C22×C4C22×Q814yes—
412C22×C4C22×Q818yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q812yes—
418C22×C4C22×Q812yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+412yes—
434C22×C42−1+412yes—
435C22×C42−1+412yes—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—416no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—88no—
452C42—28no—
453C42—28no—
454C42—28no—
455C42—44no—
456C42—44no—
457C42—44no—
458C42—44no—
459C42—416no—
460C42—48no—
461C42—48no—
462C23C23×Q811yescharacteristic
463C23C23×Q811yescharacteristic
464C23C22×#4111yescharacteristic
465C23C22×#4111yescharacteristic
466C23C22×#4111yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×H(5,𝔽2)11yescharacteristic
470C23C2×H(5,𝔽2)11yescharacteristic
471C23C2×H(5,𝔽2)11yescharacteristic
472C23C2×2−1+411yescharacteristic
473C23C2×2−1+411yescharacteristic
474C23C2×2−1+411yescharacteristic
475C23#58511yescharacteristic
476C23#58511yescharacteristic
477Q8—416no—
478Q8—432no—
479Q8—816no—
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—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4#53314yes—
574C2×C4#53814yes—
575C2×C4#53814yes—
576C2×C4#54314yes—
577C2×C4#55414yes—
578C2×C4#55414yes—
579C2×C4#55614yes—
580C2×C4#55714yes—
581C2×C4#55714yes—
582C2×C4Q8214yes—
583C2×C4#55914yes—
584C2×C4H(3,𝔽4)14yes—
585C22C2×#53311yescharacteristic
586C22C2×#53811yescharacteristic
587C22C2×#53811yescharacteristic
588C22#345511yescharacteristic
589C22C2×#54311yescharacteristic
590C22C2×#55411yescharacteristic
591C22C2×#55411yescharacteristic
592C22C2×#55611yescharacteristic
593C22C2×#55711yescharacteristic
594C22C2×#55711yescharacteristic
595C22C2×Q8211yescharacteristic
596C22C2×#55911yescharacteristic
597C22#350111yescharacteristic
598C22#350111yescharacteristic
599C22#350211yescharacteristic
600C22#350211yescharacteristic
601C22#350311yescharacteristic
602C22#350311yescharacteristic
603C22#350511yescharacteristic
604C22#350611yescharacteristic
605C22#350611yescharacteristic
606C22#350711yescharacteristic
607C22#350811yescharacteristic
608C22#350911yescharacteristic
609C22#351111yescharacteristic
610C22#351111yescharacteristic
611C22C2×H(3,𝔽4)11yescharacteristic
612C22#353611yescharacteristic
613C22#353811yescharacteristic
614C22#354811yescharacteristic
615C22#355011yescharacteristic
616C22#355811yescharacteristic
617C22#356111yescharacteristic
618C22#356111yescharacteristic
619C22#356111yescharacteristic
620C4—24no—
621C4—24no—
622C4—24no—
623C4—24no—
624C4—24no—
625C4—48no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—84no—
630C4—84no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4#280418yes—
635C2#3501411yescharacteristic
636C2#3656311yescharacteristic
637C2#3659811yescharacteristic
638C2#3732411yescharacteristic
639C2#3951111yescharacteristic
640C2#3962411yescharacteristic
641C2#3962911yescharacteristic
642C2#3964011yescharacteristic
643C2#3975211yescharacteristic
644C2#3976511yescharacteristic
645C2#4600411yescharacteristic
646C2#4601611yescharacteristic
647C2C2×#280411yescharacteristic
648C2#5987011yescharacteristic
649C2#5987111yescharacteristic
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
83. C2×C4⋊Q8Q8semidirect
125. #391Q8semidirect

Maximal subgroups

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

subgroupindex
#170802
#170762
#161202
#156912
#152012
#151912
#151462
#147142
#144712
#140242
#140172
#139762
C4×#3912