Finite Group Databasebeta

← #355358

Subgroups — #355358

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1749C211yesmaximal, characteristic
4C4×#380C211yesmaximal, characteristic
5C2×C4⋊#95C211yesmaximal, characteristic
6#10908C212yesmaximal, complemented
7#10907C212yesmaximal, complemented
8#10863C212yesmaximal, complemented
9#10821C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10730C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×#97C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4—21no—
25C22×#95—21no—
26C22×#95—21no—
27#1853C418yescomplemented
28#1852C418yescomplemented
29#1749C418yescomplemented
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43#1434—21no—
44C22⋊#95—21no—
45C4×#95—21no—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—42no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4—42no—
61C22×C4⋊C4C2×C414yes—
62C22×C4⋊C4C2×C414yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42—42no—
74C22×C42—42no—
75C22×C42C2×C414yes—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42Q814yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—42no—
82C2×#107—42no—
83C2×#107C2×C418yescomplemented
84C2×#106—42no—
85C2×#106—42no—
86C2×#106C2×C418yescomplemented
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8C2×C414yes—
90C2×C22⋊C8C2×C414yes—
91C2×C22⋊C8C2×C414yes—
92C2×C22⋊C8C2×C414yes—
93C2×#97—42no—
94C2×#97—42no—
95C2×#97C2×C418yescomplemented
96#380—22no—
97#380—22no—
98#380C2×C414yes—
99#380C2×C414yes—
100C4×C4⋊C4—42no—
101C4×C4⋊C4—42no—
102C4×C22⋊C4—44no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—44no—
116C2×#95—44no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#340C2×C418yescomplemented
122#340C2×C418yescomplemented
123#336C2×C418yescomplemented
124C25C2×D411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C4⋊C414yes—
162C23×C4C4⋊C414yes—
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×Q812yes—
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8C22×C414yes—
186C22×C8C22×C414yes—
187C22×C8C22×C414yes—
188C22×C8C22×C414yes—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
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—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—48no—
238C2×C42—48no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42C4⋊C414yes—
250C2×C42C4⋊C414yes—
251C2×C42C22×C412yes—
252C2×C42C22×C412yes—
253C2×C42C2×Q812yes—
254C2×C42C2×Q812yes—
255#107—48no—
256#106—48no—
257C22⋊C8—28no—
258C22⋊C8C42116yescomplemented
259#97—48no—
260#95—24no—
261#95—24no—
262#95—42no—
263#95—42no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—44no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24C2×C4⋊C412yes—
281C24C4×D412yes—
282C24C4×D412yes—
283C24AGL(1,ℤ/8)12yes—
284C24AGL(1,ℤ/8)12yes—
285C24#13712yes—
286C24#13712yes—
287C24C22×D411yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
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—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
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—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4C2×C4214yes—
397C22×C4C2×C4⋊C412yes—
398C22×C4C2×C4⋊C412yes—
399C22×C4C2×C4⋊C412yes—
400C22×C4C2×C4⋊C412yes—
401C22×C4C2×C4⋊C412yes—
402C22×C4#11714yes—
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4C4×Q814yes—
406C22×C4C4×Q814yes—
407C22×C4AGL(1,ℤ/8)12yes—
408C22×C4#13712yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C23×C411yescharacteristic
411C22×C4C23×C411yescharacteristic
412C22×C4C22×Q811yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+412yes—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8C2×C4218yes—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—24no—
457C42—44no—
458C42—44no—
459C42—84no—
460C42—84no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23C4×C4⋊C414yes—
490C23#42814yes—
491C23#42814yes—
492C23#44014yes—
493C23#44014yes—
494C23#44114yes—
495C23#44214yes—
496C23C22×C4⋊C411yescharacteristic
497C23C2×C4×D411yescharacteristic
498C23C2×C4×D411yescharacteristic
499C23#53411yescharacteristic
500C23#53411yescharacteristic
501C23#53511yescharacteristic
502C23#53611yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23C2×AGL(1,ℤ/8)11yescharacteristic
505C23C2×AGL(1,ℤ/8)11yescharacteristic
506C23C2×#13711yescharacteristic
507C23C2×#13711yescharacteristic
508C23C2×#13711yescharacteristic
509C23#57511yescharacteristic
510C23#57511yescharacteristic
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—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—416no—
537C2×C4—416no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
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—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—816no—
569C2×C4—816no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4C4×C4⋊C414yes—
595C2×C4#42814yes—
596C2×C4C22×C4211yescharacteristic
597C2×C4C2×#11711yescharacteristic
598C2×C4C2×C4×Q811yescharacteristic
599C2×C4C2×C4×Q811yescharacteristic
600C2×C4#51812yes—
601C2×C4#51812yes—
602C2×C4#51812yes—
603C2×C4#52012yes—
604C2×C4#52012yes—
605C2×C4#52012yes—
606C2×C4#53812yes—
607C2×C4#53812yes—
608C2×C4#53912yes—
609C2×C4#53912yes—
610C2×C4#54312yes—
611C2×C4#54412yes—
612C8—28no—
613C8—44no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22#177618yes—
629C22C2×C4×C4⋊C411yescharacteristic
630C22#231511yescharacteristic
631C22#231711yescharacteristic
632C22#232111yescharacteristic
633C22#232111yescharacteristic
634C22#232211yescharacteristic
635C22#232211yescharacteristic
636C22C2×#42811yescharacteristic
637C22C2×#42811yescharacteristic
638C22C2×#42811yescharacteristic
639C22#291111yescharacteristic
640C22C2×#44011yescharacteristic
641C22C2×#44011yescharacteristic
642C22C2×#44111yescharacteristic
643C22C2×#44211yescharacteristic
644C22C4×AGL(1,ℤ/8)11yescharacteristic
645C22C4×AGL(1,ℤ/8)11yescharacteristic
646C22C4×#13711yescharacteristic
647C22C4×#13711yescharacteristic
648C22#319211yescharacteristic
649C22#319311yescharacteristic
650C22#319311yescharacteristic
651C22#319411yescharacteristic
652C22#319411yescharacteristic
653C22#319511yescharacteristic
654C22#319611yescharacteristic
655C22#319611yescharacteristic
656C22#321311yescharacteristic
657C22#321511yescharacteristic
658C22#321711yescharacteristic
659C22#321811yescharacteristic
660C22#323411yescharacteristic
661C22#323611yescharacteristic
662C22#323811yescharacteristic
663C22#323911yescharacteristic
664C4—24no—
665C4—24no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4#227512yes—
676C4#231812yes—
677C4#232912yes—
678C4#232912yes—
679C2—28no—
680C2#1378211yescharacteristic
681C2C2×#177611yescharacteristic
682C2C4×#42811yescharacteristic
683C2#2194211yescharacteristic
684C2#2194411yescharacteristic
685C2#2198611yescharacteristic
686C2#2198711yescharacteristic
687C2#2204611yescharacteristic
688C2#2204711yescharacteristic
689C2#2204911yescharacteristic
690C2#2205011yescharacteristic
691C2#2208111yescharacteristic
692C2#2208211yescharacteristic
693C2#2208411yescharacteristic
694C2#2208511yescharacteristic
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
6. #10908C2semidirect
7. #10907C2semidirect
8. #10863C2semidirect
9. #10821C2semidirect
27. #1853C4semidirect
28. #1852C4semidirect
29. #1749C4semidirect
83. C2×#107C2×C4semidirect
86. C2×#106C2×C4semidirect
95. C2×#97C2×C4semidirect
121. #340C2×C4semidirect
122. #340C2×C4semidirect
123. #336C2×C4semidirect
258. C22⋊C8C42semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C2×#18522
C2×#17492
C4×#3802
C2×C4⋊#952
#109082
#109072
#108632
#108212
#107302