Finite Group Databasebeta

← #360449

Subgroups — #360449

Order 512.

1480 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1934C212yesmaximal
2C2×#1802C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic, complemented
6#13906C211yesmaximal, characteristic
7#11292C212yesmaximal, complemented
8#11066C212yesmaximal
9#10880C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×C4×C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#384—21no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×#380—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1934—28no—
30#1802—24no—
31C8×C4⋊C4—24no—
32C4×#102C418yescomplemented
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#336C2212yes—
41#1444—22no—
42#1439—22no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8D414yes—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—42no—
83C2×#102C2×C414yes—
84C2×#102C2×C414yes—
85C2×#102D414yes—
86C2×#102Q814yes—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—24no—
90C2×C4×C8—24no—
91C2×C4×C8—42no—
92C2×C4×C8C2×C414yes—
93#390—24no—
94#390—42no—
95#389—24no—
96#389—24no—
97#389—42no—
98#389—42no—
99#384—22no—
100#384—22no—
101#384—22no—
102#384—22no—
103#384—44no—
104#380—44no—
105#380—44no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C414yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D412yes—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—42no—
121#340C2×C414yes—
122#340C2×C414yes—
123#336—24no—
124C25—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C22×C412yes—
137C22×D4C2×D412yes—
138C22×D4C2×Q812yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8C22×C412yes—
173C22×C8C22×C412yes—
174C22×C8C2×D414yes—
175C22×C8C2×D412yes—
176C22×C8C2×Q812yes—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—44no—
180C4×D4—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4D814yes—
227C2×C4⋊C4SD1614yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×Q812yes—
233C2×C4⋊C4#4112yes—
234C2×C4⋊C4#4112yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C22×C412yes—
254C2×C42C2×D412yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C4⋊C8—28no—
260#102—28no—
261#102—48no—
262#102C4⋊C4116yescomplemented
263C4×C8—28no—
264C4×C8—48no—
265#95—24no—
266#95—24no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24C22×D411yescharacteristic
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4C2×C4⋊C414yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
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—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4C4×D412yes—
374C22×C4C4×D412yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4#12412yes—
378C22×C4#12412yes—
379C22×C4C2×D814yes—
380C22×C4C2×D812yes—
381C22×C4C2×SD1614yes—
382C22×C4C2×SD1612yes—
383C22×C4#13512yes—
384C22×C4#13512yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—42no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8C2×C4⋊C414yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4C2×C4⋊C414yes—
447C4⋊C4#11714yes—
448C22⋊C4—48no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—48no—
456C42#11714yes—
457C42C4×D414yes—
458C42#12314yes—
459C42#12614yes—
460C23—24no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23C2×C4×D411yescharacteristic
496C23C2×#12111yescharacteristic
497C23C2×#12411yescharacteristic
498C23D4211yescharacteristic
499C23#54611yescharacteristic
500C23#54811yescharacteristic
501C23D4×Q811yescharacteristic
502C23C22×D811yescharacteristic
503C23C22×SD1611yescharacteristic
504C23C2×#13511yescharacteristic
505C23C2×#13511yescharacteristic
506C23#57611yescharacteristic
507C23#57711yescharacteristic
508C23#57711yescharacteristic
509C23#57811yescharacteristic
510D4—24no—
511D4—24no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
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—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—816no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4#38814yes—
575C2×C4#38814yes—
576C2×C4C4×D814yes—
577C2×C4C4×SD1614yes—
578C2×C4#46514yes—
579C2×C4#46614yes—
580C2×C4#48614yes—
581C2×C4#48814yes—
582C2×C4C22×C4⋊C411yescharacteristic
583C2×C4C2×#11711yescharacteristic
584C2×C4C2×#11711yescharacteristic
585C2×C4C2×C4×D411yescharacteristic
586C2×C4#52011yescharacteristic
587C2×C4#52011yescharacteristic
588C2×C4C2×#12311yescharacteristic
589C2×C4C2×#12611yescharacteristic
590C2×C4#53211yescharacteristic
591C2×C4#53211yescharacteristic
592C2×C4#53311yescharacteristic
593C2×C4#53311yescharacteristic
594C2×C4#54011yescharacteristic
595C2×C4#54111yescharacteristic
596C2×C4#54211yescharacteristic
597C2×C4#54211yescharacteristic
598C2×C4D4211yescharacteristic
599C2×C4#54711yescharacteristic
600C2×C4#54711yescharacteristic
601C2×C4#54711yescharacteristic
602C2×C4D4×Q811yescharacteristic
603C2×C4#55011yescharacteristic
604C2×C4#55211yescharacteristic
605C2×C4#55311yescharacteristic
606C2×C4#55411yescharacteristic
607C2×C4#55411yescharacteristic
608C2×C4#55511yescharacteristic
609C2×C4#55611yescharacteristic
610C8—24no—
611C8—42no—
612C8—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×#38811yescharacteristic
629C22D4×C4⋊C411yescharacteristic
630C22#235211yescharacteristic
631C22#242811yescharacteristic
632C22#242911yescharacteristic
633C22#249111yescharacteristic
634C22#249211yescharacteristic
635C22C2×C4×D811yescharacteristic
636C22C2×C4×SD1611yescharacteristic
637C22#295111yescharacteristic
638C22#295211yescharacteristic
639C22C2×#46511yescharacteristic
640C22C2×#46611yescharacteristic
641C22#306211yescharacteristic
642C22#306311yescharacteristic
643C22C2×#48611yescharacteristic
644C22C2×#48811yescharacteristic
645C22#313611yescharacteristic
646C22#313711yescharacteristic
647C22D4×D811yescharacteristic
648C22D4×SD1611yescharacteristic
649C22#328311yescharacteristic
650C22#328411yescharacteristic
651C22#329411yescharacteristic
652C22#329811yescharacteristic
653C22#330311yescharacteristic
654C22#330511yescharacteristic
655C22#335811yescharacteristic
656C22#336011yescharacteristic
657C22#336211yescharacteristic
658C22#336411yescharacteristic
659C22Q8×D811yescharacteristic
660C22Q8×SD1611yescharacteristic
661C22#338011yescharacteristic
662C22#338111yescharacteristic
663C4—24no—
664C4—44no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—82no—
669C4—162no—
670C4—162no—
671C4#193418yescomplemented
672C4#228511yescharacteristic
673C4#231411yescharacteristic
674C4D4×C4⋊C411yescharacteristic
675C4#235411yescharacteristic
676C4#242211yescharacteristic
677C4#242311yescharacteristic
678C4#249611yescharacteristic
679C4#249711yescharacteristic
680C2—84no—
681C2#1387611yescharacteristic
682C2C2×#193411yescharacteristic
683C2#2108711yescharacteristic
684C2C4⋊C4×D811yescharacteristic
685C2C4⋊C4×SD1611yescharacteristic
686C2#2276611yescharacteristic
687C2#2276711yescharacteristic
688C2#2538411yescharacteristic
689C2#2538511yescharacteristic
690C2#2546911yescharacteristic
691C2#2547011yescharacteristic
692C2#2776711yescharacteristic
693C2#2776811yescharacteristic
694C2#2781311yescharacteristic
695C2#2781411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C8×C4⋊C4C2semidirect
5. #13910C2semidirect
7. #11292C2semidirect
9. #10880C2semidirect
10. #10849C2semidirect
11. #10841C2semidirect
32. C4×#102C4semidirect
262. #102C4⋊C4semidirect
671. C4#1934semidirect

Maximal subgroups

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

subgroupindex
C2×#19342
C2×#18022
C2×C8×C4⋊C42
C2×C4×#1022
#139102
#139062
#112922
#110662
#108802
#108492
#108412