Finite Group Databasebeta

← #406803

Subgroups — #406803

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C211yesmaximal, characteristic
2C2×#1854C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5#14340C211yesmaximal, characteristic
6#13795C211yesmaximal, characteristic
7#11379C211yesmaximal, characteristic, complemented
8#11378C211yesmaximal, characteristic, complemented
9#11378C211yesmaximal, characteristic, complemented
10#11377C211yesmaximal, characteristic, complemented
11#11373C211yesmaximal, characteristic, complemented
12#11371C211yesmaximal, characteristic, complemented
13#11325C211yesmaximal, characteristic, complemented
14#11325C211yesmaximal, characteristic, complemented
15#10713C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29#1854—24no—
30#1854—24no—
31#1824—24no—
32#1823—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49#1441—22no—
50#1439—22no—
51C4⋊#95—22no—
52C4⋊#95C2211yescharacteristic
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C22⋊#95—22no—
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—41no—
74C2×#117—41no—
75C2×#117—41no—
76C2×#117—41no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4Q814yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4D412yes—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—22no—
104C2×#107—22no—
105C2×#107—41no—
106C2×#107—41no—
107C2×#106—22no—
108C2×#106—22no—
109C2×#106—22no—
110C2×#106—22no—
111C2×#106—41no—
112C2×#106—41no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8D414yes—
118C2×C22⋊C8D414yes—
119#384—22no—
120#384—22no—
121#382—41no—
122#382—41no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#336—22no—
149#336—22no—
150#336—41no—
151#336—41no—
152C25C2×D411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×Q812yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4#4112yes—
183C23×C4C2411yescharacteristic
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212#117—44no—
213#117—44no—
214#117—44no—
215#117—44no—
216#117—44no—
217#117—44no—
218#117—44no—
219#117—44no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4C2×Q812yes—
278C2×C4⋊C4C2×Q812yes—
279C2×C4⋊C4#4112yes—
280C2×C4⋊C4#4112yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—82no—
290C2×C22⋊C4—82no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—24no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42#4112yes—
305C2×C42#4112yes—
306#107—44no—
307#107—44no—
308#106—44no—
309#106—44no—
310C22⋊C8—28no—
311C22⋊C8—28no—
312#95—44no—
313#95—42no—
314#95—42no—
315#95—82no—
316#95—82no—
317#95—82no—
318#95—82no—
319#95—82no—
320#95—82no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24C22⋊Q812yes—
330C24#12312yes—
331C24#12712yes—
332C24C22×D411yescharacteristic
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—48no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4C22⋊Q814yes—
420C22×C4C22⋊Q814yes—
421C22×C4C22⋊Q812yes—
422C22×C4#12314yes—
423C22×C4#12314yes—
424C22×C4#12314yes—
425C22×C4#12312yes—
426C22×C4#12712yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×Q811yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C42−1+411yescharacteristic
440C22×C42−1+411yescharacteristic
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—48no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—48no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—88no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—48no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23#39914yes—
540C23C2×C22⋊Q811yescharacteristic
541C23C2×#12311yescharacteristic
542C23C2×#12311yescharacteristic
543C23#52511yescharacteristic
544C23C2×#12711yescharacteristic
545C23#53411yescharacteristic
546C23#53611yescharacteristic
547C23#57612yes—
548C23#57712yes—
549C23#57712yes—
550C23#57812yes—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—816no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4#39914yes—
621C2×C4C2×C22⋊Q811yescharacteristic
622C2×C4C2×C22⋊Q811yescharacteristic
623C2×C4C2×#12311yescharacteristic
624C2×C4C2×#12311yescharacteristic
625C2×C4#52911yescharacteristic
626C2×C4#52911yescharacteristic
627C2×C4#53311yescharacteristic
628C2×C4#53311yescharacteristic
629C2×C4#53911yescharacteristic
630C2×C4#53911yescharacteristic
631C2×C4#54411yescharacteristic
632C2×C4#54411yescharacteristic
633C2×C4#54611yescharacteristic
634C2×C4#54611yescharacteristic
635C2×C4#54611yescharacteristic
636C2×C4#54611yescharacteristic
637C2×C4#54711yescharacteristic
638C2×C4#54711yescharacteristic
639C2×C4#54711yescharacteristic
640C2×C4#54711yescharacteristic
641C2×C4#55111yescharacteristic
642C2×C4#55111yescharacteristic
643C2×C4#55111yescharacteristic
644C2×C4#55111yescharacteristic
645C2×C4#55211yescharacteristic
646C2×C4#55211yescharacteristic
647C2×C4#55211yescharacteristic
648C2×C4#55211yescharacteristic
649C8—42no—
650C8—42no—
651C8—42no—
652C8—42no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22C2×#39911yescharacteristic
665C22#248311yescharacteristic
666C22#248411yescharacteristic
667C22#250111yescharacteristic
668C22#253911yescharacteristic
669C22#262711yescharacteristic
670C22#285511yescharacteristic
671C22#308112yes—
672C22#308212yes—
673C22#309612yes—
674C22#309612yes—
675C22#309712yes—
676C22#309712yes—
677C22#312412yes—
678C22#312512yes—
679C22#315411yescharacteristic
680C22#315411yescharacteristic
681C22#315511yescharacteristic
682C22#315611yescharacteristic
683C22#320312yes—
684C22#320412yes—
685C22#324112yes—
686C22#324212yes—
687C4—24no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—161no—
695C4—161no—
696C4—161no—
697C4—161no—
698C4#241911yescharacteristic
699C4#241911yescharacteristic
700C4#241911yescharacteristic
701C4#241911yescharacteristic
702C4#243811yescharacteristic
703C4#243811yescharacteristic
704C4#243811yescharacteristic
705C4#243811yescharacteristic
706C2—44no—
707C2#1493611yescharacteristic
708C2#2373511yescharacteristic
709C2#2373611yescharacteristic
710C2#2760712yes—
711C2#2762212yes—
712C2#2792211yescharacteristic
713C2#2792311yescharacteristic
714C2#2832711yescharacteristic
715C2#2832811yescharacteristic
716C2#2997812yes—
717C2#3315912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11379C2semidirect
8. #11378C2semidirect
9. #11378C2semidirect
10. #11377C2semidirect
11. #11373C2semidirect
12. #11371C2semidirect
13. #11325C2semidirect
14. #11325C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#18242
C2×#18232
#143402
#137952
#113792
#113782
#113772
#113732
#113712
#113252
#107132