Finite Group Databasebeta

← #198394

Subgroups — #198394

Order 512.

1317 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3#21759C211yesmaximal, characteristic
4C2×#1996C211yesmaximal, characteristic
5C2×#1520C211yesmaximal, characteristic
6#9258C214yesmaximal, complemented
7#9257C214yesmaximal, complemented
8C2×#1289C211yesmaximal, characteristic
9C2×#1282C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12#3189—22no—
13#3189—22no—
14#3189—22no—
15#3189—22no—
16C2×#484C412yescomplemented
17C2×#484C2212yes—
18C2×#467C414yescomplemented
19C2×#451C412yescomplemented
20C2×#451C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2368—22no—
24#2360—22no—
25#2345—22no—
26#2345C2212yes—
27Q8×C22⋊C4C2212yes—
28#2329—22no—
29#2315—21no—
30C4×C22⋊Q8—22no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33#2273—21no—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1996—22no—
36#1996C2214yes—
37#1892—22no—
38#1872—22no—
39#1824C2212yes—
40#1787C2212yes—
41C2×#342—21no—
42C2×#342—21no—
43#1520—22no—
44#1520C2214yes—
45C2×#328—22no—
46#1289C2214yes—
47#1282C2214yes—
48C23×Q8C2×C411yescharacteristic
49C23×Q8C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C22×Q16C2×C414yescomplemented
52C23×C8—21no—
53C23×C8C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2×C412yes—
71C2×C22⋊Q8C2×C412yes—
72C2×C22⋊Q8C2×C412yes—
73C2×C22⋊Q8D412yes—
74C2×C22⋊Q8D412yes—
75C2×C22⋊Q8D412yes—
76C2×C22⋊Q8D412yes—
77C2×C22⋊Q8C2312yes—
78C2×C4×Q8—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C42—21no—
92#484—24no—
93#484—24no—
94#484—24no—
95#484—24no—
96#467—28no—
97#467—28no—
98#451—24no—
99#451—24no—
100#451—24no—
101#451—24no—
102C2×#107C2×C414yescomplemented
103C2×#103—24no—
104C2×#103C2×C414yescomplemented
105C2×#103C2×C414yescomplemented
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8C2×C412yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2312yes—
118#391—22no—
119#391—24no—
120#389—24no—
121#389—22no—
122#387—24no—
123#387—24no—
124#387—24no—
125#385—24no—
126#385—24no—
127#380—21no—
128#380—21no—
129#380—22no—
130#380—22no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380D412yes—
135#380C2312yes—
136#380C2312yes—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—22no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4D412yes—
146C4×C22⋊C4C2312yes—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95D412yes—
150#342—24no—
151#342—24no—
152#340—24no—
153#328—28no—
154C25C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—82no—
164C22×Q8—82no—
165C22×Q8C22×C412yes—
166C22×Q8C22×C412yes—
167C22×Q8C2×D412yes—
168C22×Q8C2×D412yes—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—41no—
177C23×C4—82no—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4#4111yescharacteristic
189C23×C4#4111yescharacteristic
190C23×C4#4111yescharacteristic
191C23×C4C2411yescharacteristic
192C2×Q16—28no—
193C2×Q16—84no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—41no—
198C22×C8—41no—
199C22×C8—82no—
200C22×C8—82no—
201C22×C8C22×C414yes—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C22⋊Q8—41no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230C22⋊Q8C22⋊C414yes—
231C22⋊Q8C22⋊C414yes—
232C22⋊Q8C22⋊C414yes—
233C22⋊Q8C22⋊C414yes—
234C4×Q8—84no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
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—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4Q1614yescomplemented
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—82no—
300#107—28no—
301#103—28no—
302#103—28no—
303#103—84no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—41no—
311C22⋊C8—41no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—41no—
321#95C2×D414yes—
322C24—81no—
323C24—81no—
324C24C4×D411yescharacteristic
325C24C4×D411yescharacteristic
326C24#12011yescharacteristic
327C24#12011yescharacteristic
328C24#12111yescharacteristic
329C24#12111yescharacteristic
330C24C2×Q1612yes—
331C24#13512yes—
332C24#13712yes—
333C24#13712yes—
334C24C22×D411yescharacteristic
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—82no—
347C2×Q8—82no—
348C2×Q8—82no—
349C2×Q8—82no—
350C2×Q8—82no—
351C2×Q8—82no—
352C2×Q8—82no—
353C2×Q8—82no—
354C2×Q8C2×C22⋊C414yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—82no—
392C22×C4—81no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4#12011yescharacteristic
412C22×C4#12011yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4#12111yescharacteristic
416C22×C4#12111yescharacteristic
417C22×C4#12111yescharacteristic
418C22×C4#12111yescharacteristic
419C22×C4#12311yescharacteristic
420C22×C4#12311yescharacteristic
421C22×C4#12311yescharacteristic
422C22×C4#12311yescharacteristic
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429Q16—88no—
430C2×C8—24no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—42no—
434C2×C8—82no—
435C2×C8—82no—
436C2×C8—82no—
437C2×C8—82no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—84no—
456C4⋊C4—82no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4C2×C22⋊C414yes—
460C22⋊C4—22no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—82no—
479C22⋊C4C2×Q1614yes—
480C42—44no—
481C42—84no—
482C42—82no—
483C42—84no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23#38611yescharacteristic
503C23#38611yescharacteristic
504C23#38611yescharacteristic
505C23#38611yescharacteristic
506C23C4×Q1612yes—
507C23C4×Q1612yes—
508C23#44112yes—
509C23#44112yes—
510C23#45112yes—
511C23#45112yes—
512C23#45212yes—
513C23#45212yes—
514C23#46212yes—
515C23#46212yes—
516C23#46412yes—
517C23#46412yes—
518C23C2×C4×D411yescharacteristic
519C23C2×#12011yescharacteristic
520C23C2×#12111yescharacteristic
521C23C22×Q1611yescharacteristic
522C23C2×#13511yescharacteristic
523C23C2×#13711yescharacteristic
524C23C2×#13711yescharacteristic
525Q8—28no—
526Q8—44no—
527Q8—44no—
528Q8—82no—
529Q8—82no—
530Q8—82no—
531Q8—82no—
532Q8—82no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—164no—
592C2×C4—164no—
593C2×C4#38611yescharacteristic
594C2×C4#38611yescharacteristic
595C2×C4#38611yescharacteristic
596C2×C4#38611yescharacteristic
597C2×C4C22×C22⋊C411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12311yescharacteristic
601C8—44no—
602C8—82no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22#187812yes—
634C22#187812yes—
635C22#187812yes—
636C22#187812yes—
637C22C2×#38611yescharacteristic
638C22C2×C4×Q1611yescharacteristic
639C22C2×#44111yescharacteristic
640C22C2×#45111yescharacteristic
641C22C2×#45211yescharacteristic
642C22C2×#46211yescharacteristic
643C22C2×#46411yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—162no—
659C4#288312yes—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—22no—
664C2—22no—
665C2—22no—
666C2—22no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2C2×#187811yescharacteristic
671C2#2098411yescharacteristic
672C2#2099811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9258C2semidirect
7. #9257C2semidirect
16. C2×#484C4semidirect
18. C2×#467C4semidirect
19. C2×#451C4semidirect
51. C22×Q16C2×C4semidirect
102. C2×#107C2×C4semidirect
104. C2×#103C2×C4semidirect
105. C2×#103C2×C4semidirect
292. C2×C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#342122
#217592
C2×#19962
C2×#15202
#92582
#92572
C2×#12892
C2×#12822