Finite Group Databasebeta

← #193549

Subgroups — #193549

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2988C211yesmaximal, characteristic, complemented
2C2×C4×#135C211yesmaximal, characteristic
3#21847C211yesmaximal, characteristic, complemented
4C2×#1477C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic, complemented
6#8948C212yesmaximal, complemented
7#8947C212yesmaximal, complemented
8#8946C212yesmaximal, complemented
9#8945C212yesmaximal, complemented
10C2×#1277C211yesmaximal, characteristic, complemented
11#7849C211yesmaximal, characteristic, complemented
12C22×#135C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#2988—24no—
16#2988—24no—
17C2×C8×Q8C2212yes—
18C4×#135—24no—
19C4×#135—24no—
20C2×C4×Q16C411yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×SD16C411yescharacteristic
23C2×C4×SD16C411yescharacteristic
24C2×C4×SD16C2211yescharacteristic
25C2×C4×SD16C2211yescharacteristic
26C2×C4×D8C411yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×C8×D4—22no—
29C2×C8×D4C2212yes—
30C2×#431—21no—
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#427C411yescharacteristic
34C22×C4⋊C8—21no—
35C22×C4⋊C8C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37C2×#416C411yescharacteristic
38C2×#416C411yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#1984C2212yes—
41#1954C2212yes—
42#1869—22no—
43#1797—21no—
44#1797—21no—
45#1477—24no—
46#1477—24no—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50C2×D4⋊C8C2212yes—
51#1277C2212yes—
52#1277C2212yes—
53#1273C2212yes—
54#1273C2212yes—
55C22×#41C2×C411yescharacteristic
56C22×#41C2×C411yescharacteristic
57C2×#135—24no—
58C2×#135—24no—
59C2×#135C818yescomplemented
60C22×Q16C2×C411yescharacteristic
61C22×SD16C2×C411yescharacteristic
62C22×SD16C2×C411yescharacteristic
63C22×D8C2×C411yescharacteristic
64C23×C8—21no—
65C23×C8—22no—
66C23×C8—21no—
67C23×C8C2×C411yescharacteristic
68C23×C8C2×C411yescharacteristic
69C4×#41—22no—
70C4×#41—22no—
71C4×#41—22no—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41—22no—
75C4×#41D414yes—
76C4×#41D414yes—
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C2×#117—21no—
90C2×#117—21no—
91C2×#117C2×C411yescharacteristic
92C2×#117C2×C411yescharacteristic
93C22×C42—21no—
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96C8×Q8—24no—
97C8×Q8—24no—
98C4×Q16—24no—
99C4×Q16—22no—
100C4×Q16—22no—
101C4×SD16—24no—
102C4×SD16—22no—
103C4×SD16—22no—
104C4×SD16—24no—
105C4×SD16—22no—
106C4×SD16—22no—
107C4×D8—24no—
108C4×D8—22no—
109C4×D8—22no—
110C8×D4—216no—
111C8×D4—24no—
112C8×D4—24no—
113#431—28no—
114#431—22no—
115#431—22no—
116#431—22no—
117#431—22no—
118#427—24no—
119C2×#107C2×C411yescharacteristic
120C2×#107C2×C411yescharacteristic
121C2×#106C2×C411yescharacteristic
122C2×#106C2×C411yescharacteristic
123C2×C4⋊C8—28no—
124C2×C4⋊C8—22no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—22no—
127C2×C4⋊C8—22no—
128C2×C4⋊C8—22no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8D411yescharacteristic
132C2×C4⋊C8D411yescharacteristic
133C2×C4⋊C8D414yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136#416—24no—
137#416—24no—
138C2×#103C2×C411yescharacteristic
139C2×#103C2×C411yescharacteristic
140C2×#103C2×C411yescharacteristic
141C2×#103C2×C411yescharacteristic
142C2×#102C2×C411yescharacteristic
143C2×#102C2×C411yescharacteristic
144C2×#102C2×C411yescharacteristic
145C2×#102C2×C411yescharacteristic
146C2×C22⋊C8—22no—
147C2×C22⋊C8—21no—
148C2×C22⋊C8—22no—
149C2×C22⋊C8—22no—
150C2×C22⋊C8D412yes—
151C2×C4×C8—22no—
152C2×C4×C8—22no—
153C2×C4×C8—22no—
154C2×C4×C8C2×C411yescharacteristic
155C2×C4×C8C2×C411yescharacteristic
156C2×C4×C8C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160#336—22no—
161#336—22no—
162Q8⋊C8—24no—
163Q8⋊C8—24no—
164D4⋊C8—24no—
165D4⋊C8—24no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—22no—
174C2×#41C22⋊C414yes—
175C2×#41C2×C814yes—
176C2×#41C2×C814yescomplemented
177C2×#41M1614yes—
178C22×Q8C22×C411yescharacteristic
179C22×Q8C22×C411yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C22×D4C22×C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4111yescharacteristic
193#135—232no—
194C2×Q16—22no—
195C2×Q16—22no—
196C2×Q16C2×C814yescomplemented
197C2×SD16—22no—
198C2×SD16—22no—
199C2×SD16—22no—
200C2×SD16—22no—
201C2×SD16C2×C814yescomplemented
202C2×SD16C2×C814yescomplemented
203C2×D8—22no—
204C2×D8—22no—
205C2×D8C2×C814yescomplemented
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—28no—
213C22×C8—24no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—42no—
217C22×C8—82no—
218C22×C8—82no—
219C22×C8—81no—
220C22×C8—81no—
221C22×C8—81no—
222C22×C8—81no—
223C22×C8C2×C814yescomplemented
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226C22×C8C22×C411yescharacteristic
227C22×C8C22×C411yescharacteristic
228C22×C8C2×D412yes—
229C22×C8C2×D412yes—
230C4×Q8—22no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8C22⋊C414yes—
240C4×Q8C2×D412yes—
241C4×Q8C2×D412yes—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4C22⋊C414yes—
260C4×D4C2×D412yes—
261C4×D4C2×D412yes—
262#117—22no—
263#117—24no—
264#117—24no—
265#117—22no—
266#117—22no—
267#117—42no—
268#117—42no—
269#117C22⋊C414yes—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C22×C411yescharacteristic
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42#4112yes—
300C2×C42#4111yescharacteristic
301C2×C42C2411yescharacteristic
302#107—22no—
303#107—22no—
304#106—22no—
305#106—22no—
306C4⋊C8—28no—
307C4⋊C8—22no—
308C4⋊C8—24no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315C4⋊C8C2×D414yes—
316#103—22no—
317#103—22no—
318#103—22no—
319#103—22no—
320#102—22no—
321#102—22no—
322#102—22no—
323#102—22no—
324C22⋊C8—28no—
325C22⋊C8—24no—
326C22⋊C8—42no—
327C22⋊C8—82no—
328C22⋊C8—82no—
329C4×C8—28no—
330C4×C8—22no—
331C4×C8—22no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—22no—
337C4×C8—22no—
338C4×C8—22no—
339C4×C8—22no—
340C24—21no—
341C24—21no—
342C24C4×D411yescharacteristic
343#41—28no—
344#41—28no—
345#41—28no—
346#41—28no—
347#41—28no—
348#41—44no—
349#41—44no—
350#41C22⋊C8116yescomplemented
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8C2×C22⋊C412yes—
360C2×Q8C22×C812yes—
361C2×Q8C22×C812yes—
362C2×Q8C2×M1612yes—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4C2×C22⋊C412yes—
378C2×D4C22×C812yes—
379C2×D4C22×C812yes—
380C2×D4C2×M1612yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4C2×C22⋊C412yes—
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D412yes—
421C22×C4C4×D411yescharacteristic
422C22×C4#12012yes—
423C22×C4#12112yes—
424C22×C4#12112yes—
425C22×C4#12312yes—
426C22×C4C22×C812yes—
427C22×C4C2×M1612yes—
428C22×C4#13112yes—
429C22×C4#13112yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2×#4111yescharacteristic
433Q16—28no—
434SD16—28no—
435SD16—28no—
436D8—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—28no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—44no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—84no—
465C2×C8—88no—
466C2×C8—84no—
467C2×C8C22×C812yes—
468C2×C8C22×C812yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4C2×C22⋊C412yes—
482C4⋊C4C2×C22⋊C412yes—
483C22⋊C4—24no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—24no—
487C22⋊C4—42no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—42no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—41no—
496C42—41no—
497C42—42no—
498C42—42no—
499C42C2×C22⋊C412yes—
500C42C4×D412yes—
501C42#12012yes—
502C42#12112yes—
503C42#12112yes—
504C42#12312yes—
505C42C22×D411yescharacteristic
506C42C22×D411yescharacteristic
507C42C22×D411yescharacteristic
508C42C2×#4111yescharacteristic
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23#38612yes—
528C23C8×D412yes—
529C23#43512yes—
530C23C2×C4×D411yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8C2×C22⋊C814yes—
539D4—216no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4—22no—
544D4—22no—
545D4—48no—
546D4—41no—
547D4—41no—
548D4C2×C22⋊C814yes—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—216no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4#38612yes—
592C2×C4#38612yes—
593C2×C4#38612yes—
594C2×C4C2×C22⋊C814yes—
595C2×C4#40814yes—
596C2×C4C8×D412yes—
597C2×C4C8×D414yes—
598C2×C4#43512yes—
599C2×C4#43514yes—
600C2×C4C22×C22⋊C411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×#12011yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12311yescharacteristic
606C2×C4C23×C811yescharacteristic
607C2×C4C22×M1611yescharacteristic
608C2×C4C2×#13111yescharacteristic
609C2×C4C2×#13111yescharacteristic
610C2×C4#57611yescharacteristic
611C2×C4#57711yescharacteristic
612C2×C4#57711yescharacteristic
613C2×C4#57811yescharacteristic
614C8—22no—
615C8—22no—
616C8—41no—
617C8—41no—
618C8—42no—
619C8—44no—
620C8—42no—
621C8—82no—
622C8—82no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#186914yes—
644C22C2×#38611yescharacteristic
645C22C2×C8×D411yescharacteristic
646C22C2×#43511yescharacteristic
647C22#295111yescharacteristic
648C22#295211yescharacteristic
649C22#295711yescharacteristic
650C22#295711yescharacteristic
651C4—24no—
652C4—21no—
653C4—22no—
654C4—21no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—81no—
660C4—81no—
661C4#186914yes—
662C4C22×C22⋊C811yescharacteristic
663C4C2×#40811yescharacteristic
664C4C2×C8×D411yescharacteristic
665C4C2×#43511yescharacteristic
666C4#301211yescharacteristic
667C4#301311yescharacteristic
668C4#304511yescharacteristic
669C4#304611yescharacteristic
670C2—24no—
671C2—44no—
672C2—82no—
673C2#1785711yescharacteristic
674C2#1785811yescharacteristic
675C2#1788111yescharacteristic
676C2#1788211yescharacteristic
677C2C2×#186911yescharacteristic
678C2#2091911yescharacteristic
679C2#2093711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#135C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2988C2semidirect
3. #21847C2semidirect
4. C2×#1477C2semidirect
5. C2×#1477C2semidirect
6. #8948C2semidirect
7. #8947C2semidirect
8. #8946C2semidirect
9. #8945C2semidirect
10. C2×#1277C2semidirect
11. #7849C2semidirect
59. C2×#135C8semidirect
176. C2×#41C2×C8semidirect
196. C2×Q16C2×C8semidirect
201. C2×SD16C2×C8semidirect
202. C2×SD16C2×C8semidirect
205. C2×D8C2×C8semidirect
223. C22×C8C2×C8semidirect
350. #41C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
C2×C4×#1352
#218472
C2×#14772
#89482
#89472
#89462
#89452
C2×#12772
#78492