Finite Group Databasebeta

← #354422

Subgroups — #354422

Order 512.

2258 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C4⋊C4C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic
4C2×#1776C212yesmaximal, complemented
5C2×#1749C211yesmaximal, characteristic
6C4×#380C211yesmaximal, characteristic
7#10908C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10703C212yesmaximal, complemented
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42—21no—
15C23×C42C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4—28no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4—21no—
31C22×#95—21no—
32#1853C418yescomplemented
33#1852C418yescomplemented
34#1776—28no—
35#1749C418yescomplemented
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45#1434—21no—
46C22⋊#95—21no—
47C4×#95—21no—
48C24×C4—21no—
49C24×C4—21no—
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4C2×C414yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C42—24no—
74C22×C42—24no—
75C22×C42—28no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—44no—
81C22×C42—44no—
82C22×C42C2×C414yes—
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42Q814yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#107—24no—
89C2×#107—24no—
90C2×#107C2×C418yescomplemented
91C2×#106—24no—
92C2×#106—24no—
93C2×#106C2×C418yescomplemented
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2×C414yes—
97C2×C22⋊C8C2×C414yes—
98C2×C22⋊C8C2×C414yes—
99C2×C22⋊C8C2×C414yes—
100C2×#97—24no—
101C2×#97—24no—
102C2×#97C2×C418yescomplemented
103#380—22no—
104#380—22no—
105#380C2×C414yes—
106#380C2×C414yes—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—28no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4D418yes—
115C4×C22⋊C4—44no—
116C2×#95—28no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340C2×C418yescomplemented
132#340C2×C418yescomplemented
133#336C2×C418yescomplemented
134C25C2×D411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—41no—
162C23×C4—41no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4C4⋊C414yes—
172C23×C4C4⋊C414yes—
173C23×C4C22×C412yes—
174C23×C4C22×C412yes—
175C23×C4C22×C412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×Q812yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C22×C414yes—
196C22×C8C22×C414yes—
197C22×C8C22×C414yes—
198C22×C8C22×C414yes—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—216no—
202C2×C4⋊C4—216no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4#4114yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—216no—
250C2×C42—216no—
251C2×C42—28no—
252C2×C42—28no—
253C2×C42—432no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42C4⋊C414yes—
265C2×C42C4⋊C414yes—
266C2×C42C22×C412yes—
267C2×C42C22×C412yes—
268C2×C42C2×D414yes—
269C2×C42C2×Q812yes—
270C2×C42C2×Q812yes—
271C2×C42#4114yes—
272#107—216no—
273#106—216no—
274C22⋊C8—28no—
275C22⋊C8C42116yescomplemented
276#97—216no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—42no—
282#95—42no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24C2×C4⋊C412yes—
296C24C4×D412yes—
297C24C4×D412yes—
298C24AGL(1,ℤ/8)12yes—
299C24AGL(1,ℤ/8)12yes—
300C24#13712yes—
301C24#13712yes—
302C24C22×D411yescharacteristic
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—28no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—216no—
339C22×C4—216no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
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×C4C2×C4214yes—
408C22×C4C2×C4⋊C412yes—
409C22×C4C2×C4⋊C412yes—
410C22×C4C2×C4⋊C412yes—
411C22×C4C2×C4⋊C412yes—
412C22×C4C2×C4⋊C412yes—
413C22×C4#11714yes—
414C22×C4C4×D412yes—
415C22×C4C4×D412yes—
416C22×C4C4×Q814yes—
417C22×C4C4×Q814yes—
418C22×C4#12012yes—
419C22×C4#12012yes—
420C22×C4#12312yes—
421C22×C4#12312yes—
422C22×C4AGL(1,ℤ/8)12yes—
423C22×C4#13712yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C23×C411yescharacteristic
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×Q811yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8C2×C4218yescomplemented
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4C4×D418yes—
465C4⋊C4C4×D418yes—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C42—24no—
469C42—24no—
470C42—28no—
471C42—28no—
472C42—416no—
473C42—416no—
474C42—84no—
475C42—84no—
476C42C4×D418yes—
477C42C22⋊Q818yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23C4×C4⋊C414yes—
507C23#42814yes—
508C23#42814yes—
509C23#44014yes—
510C23#44014yes—
511C23#44114yes—
512C23#44214yes—
513C23C22×C4⋊C411yescharacteristic
514C23C2×C4×D411yescharacteristic
515C23C2×C4×D411yescharacteristic
516C23C2×#12011yescharacteristic
517C23C2×#12311yescharacteristic
518C23#52511yescharacteristic
519C23#52511yescharacteristic
520C23C2×AGL(1,ℤ/8)11yescharacteristic
521C23C2×AGL(1,ℤ/8)11yescharacteristic
522C23C2×AGL(1,ℤ/8)11yescharacteristic
523C23C2×#13711yescharacteristic
524C23C2×#13711yescharacteristic
525C23C2×#13711yescharacteristic
526C23#57511yescharacteristic
527C23#57511yescharacteristic
528C2×C4—232no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—416no—
563C2×C4—432no—
564C2×C4—432no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
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—81no—
608C2×C4—81no—
609C2×C4C4×C4⋊C414yes—
610C2×C4#38514yes—
611C2×C4#38514yes—
612C2×C4#42814yes—
613C2×C4C22×C4211yescharacteristic
614C2×C4C2×#11711yescharacteristic
615C2×C4C2×C4×D412yes—
616C2×C4C2×C4×D412yes—
617C2×C4C2×C4×D412yes—
618C2×C4C2×C4×Q811yescharacteristic
619C2×C4C2×C4×Q811yescharacteristic
620C2×C4C4×#4112yes—
621C2×C4C4×#4112yes—
622C2×C4C4×#4112yes—
623C2×C4C2×C22⋊Q812yes—
624C2×C4#52512yes—
625C2×C4#52512yes—
626C2×C4#52912yes—
627C2×C4#52912yes—
628C2×C4#53312yes—
629C8—28no—
630C8—44no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22#177618yescomplemented
646C22C2×C4×C4⋊C411yescharacteristic
647C22C2×#38511yescharacteristic
648C22#229611yescharacteristic
649C22C4×#12011yescharacteristic
650C22C4×#12011yescharacteristic
651C22C4×#12311yescharacteristic
652C22C4×#12311yescharacteristic
653C22C2×#42811yescharacteristic
654C22C2×#42811yescharacteristic
655C22C2×#42811yescharacteristic
656C22#291111yescharacteristic
657C22C2×#44011yescharacteristic
658C22C2×#44011yescharacteristic
659C22C2×#44111yescharacteristic
660C22C2×#44211yescharacteristic
661C22C4×AGL(1,ℤ/8)11yescharacteristic
662C22C4×AGL(1,ℤ/8)11yescharacteristic
663C22C4×#13711yescharacteristic
664C22C4×#13711yescharacteristic
665C22#300311yescharacteristic
666C22#300411yescharacteristic
667C22#300611yescharacteristic
668C22#300711yescharacteristic
669C22#309211yescharacteristic
670C22#309211yescharacteristic
671C22#309411yescharacteristic
672C22#309511yescharacteristic
673C22#310911yescharacteristic
674C22#311011yescharacteristic
675C22#311211yescharacteristic
676C22#311311yescharacteristic
677C22#311411yescharacteristic
678C22#311411yescharacteristic
679C22#311711yescharacteristic
680C22#311711yescharacteristic
681C4—24no—
682C4—24no—
683C4—24no—
684C4—24no—
685C4—42no—
686C4—48no—
687C4—48no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4C42×D412yes—
693C4#229012yes—
694C4C4×C22⋊Q812yes—
695C4C4×C22⋊Q812yes—
696C2—28no—
697C2C4×#38511yescharacteristic
698C2C2×#177611yescharacteristic
699C2C4×#42811yescharacteristic
700C2#2084411yescharacteristic
701C2#2084611yescharacteristic
702C2#2136311yescharacteristic
703C2#2136511yescharacteristic
704C2#2145211yescharacteristic
705C2#2145311yescharacteristic
706C2#2145411yescharacteristic
707C2#2145511yescharacteristic
708C2#2157811yescharacteristic
709C2#2157911yescharacteristic
710C2#2158011yescharacteristic
711C2#2158111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1776C2semidirect
7. #10908C2semidirect
8. #10907C2semidirect
11. #10703C2semidirect
32. #1853C4semidirect
33. #1852C4semidirect
35. #1749C4semidirect
90. C2×#107C2×C4semidirect
93. C2×#106C2×C4semidirect
102. C2×#97C2×C4semidirect
131. #340C2×C4semidirect
132. #340C2×C4semidirect
133. #336C2×C4semidirect
275. C22⋊C8C42semidirect
447. C2×C8C2×C42semidirect
645. C22#1776semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C4⋊C42
C2×#18532
C2×#18522
C2×#17762
C2×#17492
C4×#3802
#109082
#109072
#107302
#107032