Finite Group Databasebeta

← #516441

Subgroups — #516441

Order 512.

2794 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C212yesmaximal, complemented
2C2×#2293C211yesmaximal, characteristic
3#14701C214yesmaximal, complemented
4#14280C212yesmaximal, complemented
5#14278C212yesmaximal
6#14158C214yesmaximal, complemented
7#14157C212yesmaximal
8#14156C212yesmaximal, complemented
9#13909C212yesmaximal, complemented
10#13905C212yesmaximal
11#13876C212yesmaximal, complemented
12#13875C212yesmaximal, complemented
13#13777C212yesmaximal, complemented
14C4×#382C211yesmaximal, characteristic, complemented
15C42×C4⋊C4C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2212yes—
18#2314—216no—
19#2293C4116yescomplemented
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#388C2214yes—
27C2×#388C2212yes—
28C2×#388C2212yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#384C2212yes—
32C2×#382C2212yes—
33C2×#382C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C43C2212yes—
46C2×C43C2211yescharacteristic
47#1445C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2212yes—
71#1434C2212yes—
72C4×#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4×D4—28no—
80C2×C4×D4C2×C418yes—
81C2×#117—28no—
82C2×#117—28no—
83C2×#117C2×C4116yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C42C2314yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2314yes—
109C22×C42C2314yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#390C2×C418yes—
118#390C2×C418yes—
119#389C2×C418yes—
120#389C2×C418yes—
121#388—216no—
122#388—216no—
123#388—48no—
124#388C2×C4116yes—
125#388C2×C4116yes—
126#387—216no—
127#387—216no—
128#384—28no—
129#382—28no—
130#382—28no—
131#380—28no—
132#380—28no—
133#380—28no—
134#380—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4C2×C418yes—
140C4×C4⋊C4D418yescomplemented
141C4×C22⋊C4C2×C4116yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C43—44no—
167C43C2×C418yes—
168C25C2412yes—
169C22×D4—24no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—42no—
173C22×D4C22×C414yes—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4C22×C418yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C4×D4—232no—
196#117—264no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
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—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C22×C418yes—
222C2×C4⋊C4C22×C418yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C2×D418yes—
227C2×C4⋊C4#4118yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4C22×C418yes—
251C2×C22⋊C4C22×C418yes—
252C2×C22⋊C4C22×C418yes—
253C2×C22⋊C4C22×C418yes—
254C2×C22⋊C4C22×C418yes—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42C22×C418yes—
279C2×C42C22×C414yes—
280C2×C42C22×C414yes—
281C2×C42C22×C414yes—
282C2×C42C22×C414yes—
283C2×C42C22×C414yes—
284C2×C42C22×C414yes—
285C2×C42C22×C414yes—
286C2×C42C2×D418yes—
287C2×C42C2×D414yes—
288C2×C42#4118yes—
289C2×C42#4114yes—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C22×C418yes—
311#95C22×C418yes—
312#95C22×C418yes—
313#95C2×D418yes—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24C23×C414yes—
329C24C2511yescharacteristic
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—48no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4C23×C412yes—
405C22×C4C23×C414yes—
406C22×C4C23×C414yes—
407C22×C4C23×C414yes—
408C22×C4C23×C412yes—
409C22×C4C23×C412yes—
410C22×C4C23×C412yes—
411C22×C4C23×C412yes—
412C22×C4C23×C412yes—
413C22×C4C23×C412yes—
414C22×C4C22×D418yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C2×#4118yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4114yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+412yes—
428C4⋊C4—28no—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—216no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C22⋊C4—216no—
440C22⋊C4—216no—
441C22⋊C4—216no—
442C22⋊C4—216no—
443C22⋊C4—48no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C42—216no—
449C42—28no—
450C42—28no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42—48no—
455C42—416no—
456C42—416no—
457C42—48no—
458C42—88no—
459C42#117116yes—
460C42C4×D4116yes—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23C24×C411yescharacteristic
497C23C23×D411yescharacteristic
498C23C22×#4112yes—
499C23C22×#4111yescharacteristic
500C23C22×#4112yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×2−1+411yescharacteristic
503C23#58511yescharacteristic
504C23#58511yescharacteristic
505C23#58512yes—
506C23#58512yes—
507D4—416no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—432no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4C2×#11714yes—
586C2×C4C2×#11714yes—
587C2×C4C2×#11714yes—
588C2×C4C2×C4×D414yes—
589C2×C4C2×C4×D414yes—
590C2×C4C2×C4×D414yes—
591C2×C4C4×#4114yes—
592C2×C4#52014yes—
593C2×C4#52918yes—
594C2×C4#53218yes—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C22×#11711yescharacteristic
611C22C22×C4×D411yescharacteristic
612C22C2×C4×#4111yescharacteristic
613C22C2×#52011yescharacteristic
614C22#342811yescharacteristic
615C22#342811yescharacteristic
616C22#342811yescharacteristic
617C22C2×#52912yes—
618C22C2×#53212yes—
619C22#345512yes—
620C22#345512yes—
621C22#346012yes—
622C22#346112yes—
623C22#349611yescharacteristic
624C22#349711yescharacteristic
625C22#350012yes—
626C22#350611yescharacteristic
627C22#350711yescharacteristic
628C22#351212yes—
629C22#351312yes—
630C22#351612yes—
631C22#352112yes—
632C22#352412yes—
633C4—24no—
634C4—24no—
635C4—24no—
636C4—24no—
637C4—24no—
638C4—48no—
639C4—48no—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—44no—
644C4—84no—
645C4—84no—
646C4#229318yes—
647C2—84no—
648C2C2×#229311yescharacteristic
649C2#3404811yescharacteristic
650C2#3406011yescharacteristic
651C2#3420611yescharacteristic
652C2#3420811yescharacteristic
653C2#3424111yescharacteristic
654C2#3424411yescharacteristic
655C2#3516912yes—
656C2#3518312yes—
657C2#3521812yes—
658C2#3522012yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2314C2semidirect
3. #14701C2semidirect
4. #14280C2semidirect
6. #14158C2semidirect
8. #14156C2semidirect
9. #13909C2semidirect
11. #13876C2semidirect
12. #13875C2semidirect
13. #13777C2semidirect
14. C4×#382C2semidirect
15. C42×C4⋊C4C2semidirect
19. #2293C4semidirect
140. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#22932
#147012
#142802
#142782
#141582
#141572
#141562
#139092
#139052
#138762
#138752
#137772
C4×#3822
C42×C4⋊C42