Finite Group Databasebeta

← #390258

Subgroups — #390258

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1955C212yesmaximal, complemented
2C2×#1923C212yesmaximal, complemented
3C4×#392C211yesmaximal, characteristic
4#11147C211yesmaximal, characteristic, complemented
5#10964C212yesmaximal, complemented
6#10964C212yesmaximal, complemented
7#10922C212yesmaximal, complemented
8#10921C211yesmaximal, characteristic, complemented
9#10877C212yesmaximal, complemented
10C22×C4×D4C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C4⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#392C411yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1955—28no—
26#1923—28no—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1444C411yescharacteristic
34#1439C412yes—
35#1439C411yescharacteristic
36#1438C411yescharacteristic
37#1438C412yes—
38C4×#95C2211yescharacteristic
39C23×D4C2×C412yes—
40C23×D4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2312yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4—42no—
50C2×C4×D4—42no—
51C2×C4×D4—42no—
52C2×C4×D4—42no—
53C2×C4×D4—42no—
54C2×C4×D4—42no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2312yes—
69C2×C4⋊C8—42no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8D418yescomplemented
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—41no—
91C2×#97—24no—
92C2×#97—42no—
93C2×#97—42no—
94C2×C4×C8—24no—
95C2×C4×C8—41no—
96#392—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—41no—
100C4×C22⋊C4—41no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108#336—24no—
109#336—24no—
110#336—24no—
111#336—42no—
112#336—42no—
113#336—41no—
114C25C22×C411yescharacteristic
115C25C22×C412yes—
116C22×D4—24no—
117C22×D4—24no—
118C22×D4—24no—
119C22×D4—24no—
120C22×D4—22no—
121C22×D4—41no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4—41no—
127C22×D4—42no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C2×D414yes—
185C22×C8C2×D414yes—
186C22×C8#4114yes—
187C4×D4—48no—
188C4×D4—48no—
189C4×D4—48no—
190C4×D4—48no—
191C4×D4—48no—
192C4×D4—44no—
193C4×D4—44no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4M1614yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42C2×D414yes—
233C2×C42#4114yes—
234C4⋊C8—48no—
235C22⋊C8—48no—
236C22⋊C8—44no—
237C22⋊C8—48no—
238C22⋊C8—48no—
239C22⋊C8—48no—
240C22⋊C8—48no—
241C22⋊C8—44no—
242#97—48no—
243C4×C8—44no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—41no—
248#95—42no—
249#95—41no—
250#95—41no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—21no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—22no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24C2×M1612yes—
276C24#13114yes—
277C24C23×C411yescharacteristic
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—44no—
283C2×D4—48no—
284C2×D4—44no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—48no—
288C2×D4—44no—
289C2×D4—48no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4C4×D414yes—
376C22×C4C4×D414yes—
377C22×C4C2×M1612yes—
378C22×C4C2×M1612yes—
379C22×C4#13112yes—
380C22×C4C22×D412yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C4H(5,𝔽2)12yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C22⋊C4—216no—
419C22⋊C4—44no—
420C22⋊C4—48no—
421C22⋊C4—44no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—41no—
430C42—42no—
431C42—41no—
432C42—42no—
433C42—41no—
434C42—41no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—24no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23C2×C4×D412yes—
494C23C4×#4111yescharacteristic
495C23#51811yescharacteristic
496C23#51812yes—
497C23#51811yescharacteristic
498C23C22×M1611yescharacteristic
499C23C2×#13111yescharacteristic
500C23C2×#13112yes—
501C23#56811yescharacteristic
502C23#56812yes—
503C23#56811yescharacteristic
504D4—416no—
505D4—432no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#53412yes—
565C2×C4#53512yes—
566C2×C4#53812yes—
567C2×C4#53812yes—
568C2×C4#53812yes—
569C2×C4#54012yes—
570C2×C4D4211yescharacteristic
571C2×C4#54612yes—
572C2×C4#55012yes—
573C2×C4#55012yes—
574C2×C4#55312yes—
575C2×C4#55512yes—
576C2×C4#55911yescharacteristic
577C2×C4#56012yes—
578C2×C4#56011yescharacteristic
579C2×C4H(3,𝔽4)11yescharacteristic
580C8—44no—
581C8—44no—
582C8—44no—
583C8—82no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—42no—
595C22—44no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—81no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—81no—
606C22—82no—
607C22—81no—
608C22—81no—
609C22—82no—
610C22#232112yes—
611C22#233311yescharacteristic
612C22#236312yes—
613C22#236611yescharacteristic
614C22#237511yescharacteristic
615C22#292912yes—
616C22#293212yes—
617C22#293312yes—
618C22D4×M1612yes—
619C22#296511yescharacteristic
620C22#296711yescharacteristic
621C22#296812yes—
622C22#297311yescharacteristic
623C22#297412yes—
624C22#297511yescharacteristic
625C22#297612yes—
626C22#297612yes—
627C22#297611yescharacteristic
628C22#297711yescharacteristic
629C22#297812yes—
630C22#298011yescharacteristic
631C22#298111yescharacteristic
632C22#298112yes—
633C4—42no—
634C4—44no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—81no—
639C4—81no—
640C4—82no—
641C4#267312yes—
642C4#271412yes—
643C4#273312yes—
644C4#280312yes—
645C2—84no—
646C2—82no—
647C2#1456711yescharacteristic
648C2#2203112yes—
649C2#2203812yes—
650C2#2233711yescharacteristic
651C2#2234011yescharacteristic
652C2#2308512yes—
653C2#2309112yes—
654C2#2317511yescharacteristic
655C2#2317711yescharacteristic
656C2#2330711yescharacteristic
657C2#2331311yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1955C2semidirect
2. C2×#1923C2semidirect
4. #11147C2semidirect
5. #10964C2semidirect
6. #10964C2semidirect
7. #10922C2semidirect
8. #10921C2semidirect
9. #10877C2semidirect
71. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#19552
C2×#19232
C4×#3922
#111472
#109642
#109222
#109212
#108772