Finite Group Databasebeta

← #368339

Subgroups — #368339

Order 512.

1628 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1981C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic
5C2×#1865C212yesmaximal
6C4×#389C211yesmaximal, characteristic
7#11122C212yesmaximal
8#11067C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10832C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C4×C4⋊Q8—24no—
22C4×C4⋊Q8—24no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1981—24no—
36#1981C418yescomplemented
37#1948—24no—
38#1865—28no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2211yescharacteristic
48C4×#95—21no—
49C23×Q8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D414yes—
57C2×C4⋊Q8Q814yes—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×Q8—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—22no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8C2×C414yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103—42no—
105C2×#103C2×C418yes—
106C2×#103C2×C418yes—
107#391—24no—
108#391—24no—
109#389—28no—
110#389—28no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389D414yes—
115#384—22no—
116#384D414yes—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—28no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—44no—
123C4×C4⋊C4C2×C414yes—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C43—22no—
135C43—22no—
136#340—24no—
137#340—24no—
138#340—42no—
139#336—24no—
140#336—42no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C22×C414yes—
153C22×Q8C2×D414yes—
154C22×Q8C2×Q814yes—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
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×C8C22×C414yes—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—28no—
185C4⋊Q8C4⋊C4116yescomplemented
186C4×Q8—48no—
187C4×Q8—48no—
188C4×Q8—48no—
189C4×Q8—48no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4#4112yes—
240C2×C4⋊C4#4112yes—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—28no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42SD1614yes—
272C2×C42C22×C412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×Q812yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280#106—48no—
281C4⋊C8—28no—
282C4⋊C8—44no—
283#103—216no—
284#103—48no—
285#103—48no—
286#103—48no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297C24C22×D411yescharacteristic
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8C2×C4⋊C418yes—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4#12012yes—
397C22×C4#12012yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4C22⋊Q812yes—
401C22×C4C22⋊Q812yes—
402C22×C4C2×SD1612yes—
403C22×C4C2×SD1612yes—
404C22×C4C2×SD1612yes—
405C22×C4#13512yes—
406C22×C4#13712yes—
407C22×C4#13712yes—
408C22×C4#13712yes—
409C22×C4#13712yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×Q811yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C42−1+411yescharacteristic
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—216no—
442C4⋊C4—216no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—416no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4C4×D414yes—
465C42—28no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—28no—
471C42—44no—
472C42—416no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—48no—
478C42—44no—
479C42—84no—
480C42C2×C4⋊C414yes—
481C42C4×D414yes—
482C42C22⋊Q814yes—
483C42#12314yes—
484C23C2×C4×D411yescharacteristic
485C23C2×#12011yescharacteristic
486C23C2×#12111yescharacteristic
487C23C2×C22⋊Q811yescharacteristic
488C23#52511yescharacteristic
489C23#53511yescharacteristic
490C23#53711yescharacteristic
491C23C22×SD1611yescharacteristic
492C23C2×#13511yescharacteristic
493C23C2×#13711yescharacteristic
494C23C2×#13711yescharacteristic
495C23C2×#13711yescharacteristic
496C23C2×#13711yescharacteristic
497C23#57511yescharacteristic
498C23#57511yescharacteristic
499Q8—28no—
500Q8—28no—
501Q8—44no—
502Q8—44no—
503Q8—44no—
504Q8—44no—
505Q8—44no—
506Q8—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#38514yes—
583C2×C4#38514yes—
584C2×C4C4×SD1614yes—
585C2×C4#44114yes—
586C2×C4#44814yes—
587C2×C4#45214yes—
588C2×C4#46114yes—
589C2×C4#47514yes—
590C2×C4#47914yes—
591C2×C4C22×C4⋊C411yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C4×#4111yescharacteristic
595C2×C4#51811yescharacteristic
596C2×C4#52011yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4C2×#12311yescharacteristic
599C2×C4#52911yescharacteristic
600C2×C4#53211yescharacteristic
601C2×C4#54011yescharacteristic
602C2×C4#54211yescharacteristic
603C2×C4D4211yescharacteristic
604C2×C4D4211yescharacteristic
605C2×C4#54611yescharacteristic
606C2×C4#54611yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54711yescharacteristic
609C2×C4#54811yescharacteristic
610C2×C4D4×Q811yescharacteristic
611C2×C4#55011yescharacteristic
612C2×C4#55011yescharacteristic
613C2×C4#55411yescharacteristic
614C2×C4#55411yescharacteristic
615C2×C4#55411yescharacteristic
616C2×C4#55611yescharacteristic
617C2×C4#55911yescharacteristic
618C2×C4#55911yescharacteristic
619C8—44no—
620C8—82no—
621C22C2×#38511yescharacteristic
622C22C4×#12111yescharacteristic
623C22#232411yescharacteristic
624C22#240711yescharacteristic
625C22#240811yescharacteristic
626C22#253811yescharacteristic
627C22#258011yescharacteristic
628C22C2×C4×SD1611yescharacteristic
629C22C2×#44111yescharacteristic
630C22C4×#13711yescharacteristic
631C22C4×#13711yescharacteristic
632C22C2×#44811yescharacteristic
633C22C2×#45211yescharacteristic
634C22#300411yescharacteristic
635C22#300511yescharacteristic
636C22C2×#46111yescharacteristic
637C22#303511yescharacteristic
638C22#303711yescharacteristic
639C22#303911yescharacteristic
640C22C2×#47511yescharacteristic
641C22C2×#47911yescharacteristic
642C22#307811yescharacteristic
643C22#307811yescharacteristic
644C22#310111yescharacteristic
645C22#310711yescharacteristic
646C22#311311yescharacteristic
647C22#311611yescharacteristic
648C22#320611yescharacteristic
649C22#321211yescharacteristic
650C22#321611yescharacteristic
651C22#321811yescharacteristic
652C22#324611yescharacteristic
653C22#324811yescharacteristic
654C22#325211yescharacteristic
655C22#325511yescharacteristic
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—48no—
662C4—44no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4#186518yescomplemented
668C4D4×C4⋊C411yescharacteristic
669C4#234911yescharacteristic
670C4#236611yescharacteristic
671C4#236711yescharacteristic
672C4#244511yescharacteristic
673C4#244711yescharacteristic
674C4#245211yescharacteristic
675C4#245411yescharacteristic
676C2#1401811yescharacteristic
677C2C2×#186511yescharacteristic
678C2#2087411yescharacteristic
679C2C4×#46111yescharacteristic
680C2#2150611yescharacteristic
681C2#2214611yescharacteristic
682C2#2215411yescharacteristic
683C2#2415111yescharacteristic
684C2#2415711yescharacteristic
685C2#2420911yescharacteristic
686C2#2421311yescharacteristic
687C2#2828311yescharacteristic
688C2#2828711yescharacteristic
689C2#2925511yescharacteristic
690C2#2925711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
36. #1981C4semidirect
185. C4⋊Q8C4⋊C4semidirect
667. C4#1865semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#19812
C2×#19482
C2×#18652
C4×#3892
#111222
#110672
#108492
#108322
#107892