Finite Group Databasebeta

← #363065

Subgroups — #363065

Order 512.

1205 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2C2×#1996C211yesmaximal, characteristic
3C2×#1996C211yesmaximal, characteristic
4#11351C212yesmaximal
5#11351C212yesmaximal
6#10854C211yesmaximal, characteristic
7#10854C211yesmaximal, characteristic
8#10846C211yesmaximal, characteristic
9#10829C212yesmaximal
10#10829C212yesmaximal
11C4×#342C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13#2368—22no—
14#2360—22no—
15#2345C412yescomplemented
16#2345C414yescomplemented
17Q8×C22⋊C4C412yescomplemented
18#2329—22no—
19#2315C2212yes—
20C4×C22⋊Q8—22no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#2273C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1996—22no—
29#1996—22no—
30#1996—22no—
31#1996—22no—
32C2×#342C2212yes—
33C2×#342C2211yescharacteristic
34C2×#342C2212yes—
35C2×#342C2212yes—
36C2×#342C2212yes—
37C2×#342C2212yes—
38C2×#342C2212yes—
39C2×#342C2212yes—
40C2×#342C2211yescharacteristic
41C2×#342C2211yescharacteristic
42C2×#342C2212yes—
43C2×#342C2212yes—
44C2×#342C2212yes—
45C2×#342C2211yescharacteristic
46C23×Q8C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8D412yes—
57C2×C22⋊Q8D412yes—
58C2×C22⋊Q8D412yes—
59C2×C22⋊Q8D412yes—
60C2×C4×Q8—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2312yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2311yescharacteristic
71C22×C22⋊C4C2311yescharacteristic
72C22×C22⋊C4C2311yescharacteristic
73C22×C22⋊C4C2311yescharacteristic
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79#391—24no—
80#391—22no—
81#389—24no—
82#389—22no—
83#387—24no—
84#387—24no—
85#387—24no—
86#385—24no—
87#385—24no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—21no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380C2×C414yes—
99#380C2×C414yes—
100#380C2×C412yes—
101#380C2×C412yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—24no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—24no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4C2×C412yes—
117C4×C22⋊C4C2×C414yes—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4C2×C414yes—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126#342—24no—
127#342—24no—
128#342—24no—
129#342—24no—
130#342—22no—
131#342—22no—
132#342—22no—
133#342—24no—
134#342—24no—
135#342—24no—
136#342—42no—
137#342—42no—
138#342—42no—
139#342Q814yescomplemented
140C25C2411yescharacteristic
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—82no—
146C22×Q8C22⋊C412yes—
147C22×Q8C22⋊C412yes—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—82no—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4#4111yescharacteristic
169C23×C4#4111yescharacteristic
170C23×C4#4111yescharacteristic
171C23×C4#4111yescharacteristic
172C23×C4#4111yescharacteristic
173C23×C4#4111yescharacteristic
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C4×Q8—84no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C22⋊C414yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C414yes—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4C22×C412yes—
264C2×C22⋊C4C22×C414yes—
265C2×C22⋊C4C22×C412yes—
266C2×C22⋊C4C2×D414yes—
267C2×C22⋊C4C2×D414yes—
268C2×C22⋊C4C2×Q812yes—
269C2×C22⋊C4C2×Q812yes—
270C2×C22⋊C4C2×Q812yes—
271C2×C22⋊C4#4112yes—
272C2×C22⋊C4#4112yes—
273C2×C22⋊C4#4112yes—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42—82no—
283C2×C42—82no—
284C2×C42—82no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—41no—
294#95—41no—
295#95—41no—
296#95—41no—
297#95—41no—
298#95—41no—
299#95—41no—
300#95—41no—
301C24—81no—
302C24—81no—
303C24C23×C411yescharacteristic
304C24C22×D411yescharacteristic
305C24C22×D411yescharacteristic
306C24C22×Q811yescharacteristic
307C24C2×#4111yescharacteristic
308C24C2×#4111yescharacteristic
309C24C2×#4111yescharacteristic
310C24C2×#4111yescharacteristic
311C24C2×#4111yescharacteristic
312C24C2×#4111yescharacteristic
313C24C2×#4111yescharacteristic
314C24H(5,𝔽2)11yescharacteristic
315C242−1+411yescharacteristic
316C242−1+411yescharacteristic
317C242−1+411yescharacteristic
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—44no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×Q8—82no—
326C2×Q8—82no—
327C2×Q8—82no—
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—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
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—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
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—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C414yes—
395C22×C4C2×C22⋊C414yes—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4#11711yescharacteristic
400C22×C4#11711yescharacteristic
401C22×C4#11711yescharacteristic
402C22×C4#11711yescharacteristic
403C22×C4#11711yescharacteristic
404C22×C4#11711yescharacteristic
405C22×C4#11711yescharacteristic
406C22×C4#11711yescharacteristic
407C22×C4#12311yescharacteristic
408C22×C4#12311yescharacteristic
409C22×C4#12311yescharacteristic
410C22×C4#12311yescharacteristic
411C22×C4#12311yescharacteristic
412C22×C4#12311yescharacteristic
413C22×C4#12311yescharacteristic
414C22×C4#12311yescharacteristic
415C22×C4#12311yescharacteristic
416C22×C4#12311yescharacteristic
417C22×C4#12311yescharacteristic
418C22×C4#12311yescharacteristic
419C22×C4#12311yescharacteristic
420C22×C4#12311yescharacteristic
421C22×C4#12311yescharacteristic
422C22×C4#12311yescharacteristic
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—44no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C4⋊C4—82no—
436C4⋊C4—84no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C22⋊C4—22no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—42no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4C4×Q814yes—
479C42—44no—
480C42—82no—
481C42—82no—
482C42—84no—
483C42—82no—
484C42—84no—
485C42—82no—
486C42—82no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23C22×C22⋊C411yescharacteristic
510C23C2×#11711yescharacteristic
511C23C2×#11711yescharacteristic
512C23C2×C4×Q811yescharacteristic
513C23C4×#4111yescharacteristic
514C23#51911yescharacteristic
515C23#52011yescharacteristic
516C23C2×#12311yescharacteristic
517C23C2×#12311yescharacteristic
518C23C2×#12311yescharacteristic
519C23C2×#12311yescharacteristic
520C23#52911yescharacteristic
521C23#52911yescharacteristic
522C23#53311yescharacteristic
523C23#53311yescharacteristic
524C23#54111yescharacteristic
525C23#54111yescharacteristic
526C23#54211yescharacteristic
527C23#54211yescharacteristic
528C23#54811yescharacteristic
529C23#54811yescharacteristic
530C23#54811yescharacteristic
531C23#54811yescharacteristic
532C23D4×Q811yescharacteristic
533C23D4×Q811yescharacteristic
534C23#55011yescharacteristic
535C23#55011yescharacteristic
536C23#55211yescharacteristic
537C23#55211yescharacteristic
538C23#55411yescharacteristic
539C23#55611yescharacteristic
540C23#55611yescharacteristic
541C23#55611yescharacteristic
542C23#55711yescharacteristic
543C23#55911yescharacteristic
544Q8—44no—
545Q8—44no—
546Q8—82no—
547Q8—82no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—162no—
624C2×C4—162no—
625C2×C4—162no—
626C2×C4—162no—
627C2×C4—162no—
628C2×C4—162no—
629C2×C4#38111yescharacteristic
630C2×C4#38111yescharacteristic
631C2×C4#38111yescharacteristic
632C2×C4#38111yescharacteristic
633C2×C4#38111yescharacteristic
634C2×C4#38111yescharacteristic
635C2×C4#38111yescharacteristic
636C2×C4#38111yescharacteristic
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22C2×#38111yescharacteristic
680C22#229511yescharacteristic
681C22#232011yescharacteristic
682C22Q8×C22⋊C411yescharacteristic
683C22#234111yescharacteristic
684C22#235511yescharacteristic
685C22#235611yescharacteristic
686C22#242111yescharacteristic
687C22#242111yescharacteristic
688C22#242311yescharacteristic
689C22#242311yescharacteristic
690C22#244611yescharacteristic
691C22#244611yescharacteristic
692C22#244811yescharacteristic
693C22#244811yescharacteristic
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—82no—
705C4—84no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—162no—
711C4—162no—
712C4#182812yes—
713C2—21no—
714C2—21no—
715C2—21no—
716C2—21no—
717C2—21no—
718C2—21no—
719C2—21no—
720C2—21no—
721C2—21no—
722C2—21no—
723C2—21no—
724C2—21no—
725C2—21no—
726C2—21no—
727C2#1393111yescharacteristic
728C2C2×#182811yescharacteristic
729C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2345C4semidirect
16. #2345C4semidirect
17. Q8×C22⋊C4C4semidirect
139. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#342122
C2×#19962
#113512
#108542
#108462
#108292
C4×#3422