Finite Group Databasebeta

← #192933

Subgroups — #192933

Order 512.

967 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60127C211yesmaximal, characteristic
2C2×#1806C211yesmaximal, characteristic
3#17817C211yesmaximal, characteristic, complemented
4#17809C211yesmaximal, characteristic, complemented
5#17443C211yesmaximal, characteristic
6#8905C212yesmaximal, complemented
7#8904C212yesmaximal, complemented
8#8903C212yesmaximal, complemented
9#8902C212yesmaximal, complemented
10#8098C211yesmaximal, characteristic, complemented
11#7892C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13#2925—21no—
14#2925—21no—
15#2925C2211yescharacteristic
16#2925C2211yescharacteristic
17#2924—22no—
18#2924—21no—
19#2922—22no—
20#2922—21no—
21#2920—21no—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431—21no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2906C2211yescharacteristic
28#2906C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C22×#104—21no—
31#2880—21no—
32C4×#131—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1806—21no—
35#1806—21no—
36#1806C2212yes—
37#1806C2212yes—
38C2×#339—21no—
39#1574—21no—
40#1574—21no—
41#1574—22no—
42#1573—22no—
43#1571C2212yescomplemented
44#1570C2212yescomplemented
45#1567C2212yescomplemented
46#1566C2212yescomplemented
47#1565—21no—
48#1565—21no—
49#1488C412yescomplemented
50#1488C412yescomplemented
51#1487C411yescharacteristic, complemented
52#1487C411yescharacteristic, complemented
53#1486C411yescharacteristic, complemented
54#1486C411yescharacteristic, complemented
55#1476C2212yes—
56#1476C2212yes—
57#1303C2212yescomplemented
58#1303C2212yescomplemented
59#1279C2212yescomplemented
60#1279C2212yescomplemented
61C22×#41D411yescharacteristic
62C2×#131—21no—
63C2×#131—21no—
64C22×M16—21no—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41—21no—
68C4×#41—21no—
69C4×#41C2×C411yescharacteristic
70C4×#41C2×C411yescharacteristic
71C4×#41D411yescharacteristic
72C4×#41D411yescharacteristic
73C4×#41Q811yescharacteristic
74C4×#41Q811yescharacteristic
75C4×#41C2311yescharacteristic
76C4×#41C2311yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8Q811yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4Q811yescharacteristic
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C22×C4⋊C4—21no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#433—22no—
90#433—22no—
91#433—22no—
92#433—22no—
93#433—22no—
94#433—22no—
95#432—22no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432C2×C412yes—
101#432D412yes—
102#431—22no—
103#431—22no—
104#431—22no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431C2×C411yescharacteristic
114#431C2×C411yescharacteristic
115#431C2×C412yes—
116#431D411yescharacteristic
117#431D411yescharacteristic
118#431D411yescharacteristic
119#431D411yescharacteristic
120#431D412yes—
121#431C2312yes—
122#424—21no—
123#424—21no—
124#424—22no—
125#424—22no—
126#423—21no—
127#423—21no—
128#423—21no—
129#423—21no—
130#423—21no—
131#423C2×C411yescharacteristic
132#423C2×C411yescharacteristic
133#423C2312yes—
134#423C2311yescharacteristic
135#423C2311yescharacteristic
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8—21no—
139C2×C4⋊C8—21no—
140C2×C4⋊C8—21no—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×#104—22no—
144C2×#104—22no—
145C2×#104—44no—
146#408—22no—
147#408—22no—
148#408—21no—
149#408—21no—
150#407—22no—
151#407—21no—
152#407—21no—
153C2×C22⋊C8—22no—
154C2×C22⋊C8—21no—
155C2×C22⋊C8—21no—
156#405—22no—
157#405—22no—
158#405—21no—
159#405—21no—
160C4×M16—22no—
161C4×M16—21no—
162C4×M16—21no—
163C2×C4×C8—22no—
164C2×C4×C8—21no—
165C2×C4×C8C2312yes—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4C2312yes—
170C2×#95—21no—
171#339—22no—
172#339—22no—
173#335—22no—
174#335—22no—
175#335—24no—
176#334—22no—
177#334—22no—
178#334—24no—
179Q8⋊C8C2×C412yescomplemented
180Q8⋊C8C2×C412yescomplemented
181Q8⋊C8C2×C412yescomplemented
182Q8⋊C8C2×C412yescomplemented
183D4⋊C8C2×C412yescomplemented
184D4⋊C8C2×C412yescomplemented
185D4⋊C8C2×C412yescomplemented
186D4⋊C8C2×C412yescomplemented
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—41no—
192C2×#41—41no—
193C2×#41C4⋊C411yescharacteristic
194C2×#41C4⋊C411yescharacteristic
195C2×#41C2×D411yescharacteristic
196C2×#41C2×D411yescharacteristic
197C22×Q8C4⋊C411yescharacteristic
198C22×D4—21no—
199C22×D4C4⋊C411yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—41no—
204C23×C4C2×D411yescharacteristic
205C23×C4#4111yescharacteristic
206C23×C4#4111yescharacteristic
207#131—42no—
208#131—42no—
209#131—42no—
210#131—42no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—22no—
218C2×M16—41no—
219C2×M16—42no—
220C2×M16—42no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—22no—
226C22×C8—41no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8C4⋊C411yescharacteristic
232C4×Q8C4⋊C411yescharacteristic
233C4×Q8C4⋊C411yescharacteristic
234C4×Q8C4⋊C411yescharacteristic
235C4×Q8C22×C411yescharacteristic
236C4×Q8C22×C411yescharacteristic
237C4×Q8C2×Q811yescharacteristic
238C4×Q8C2×Q811yescharacteristic
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4C4⋊C411yescharacteristic
260C4×D4C4⋊C411yescharacteristic
261C4×D4C4⋊C411yescharacteristic
262C4×D4C4⋊C411yescharacteristic
263C4×D4C22×C411yescharacteristic
264C4×D4C22×C411yescharacteristic
265C4×D4C2×Q811yescharacteristic
266C4×D4C2×Q811yescharacteristic
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117C4⋊C411yescharacteristic
278#117C4⋊C411yescharacteristic
279#117C2×D411yescharacteristic
280#117C2×D411yescharacteristic
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4C4⋊C411yescharacteristic
288C2×C4⋊C4C4⋊C411yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—41no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×Q811yescharacteristic
315C2×C42C2×Q811yescharacteristic
316C2×C42#4111yescharacteristic
317C2×C42#4111yescharacteristic
318C2×C42C2411yescharacteristic
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—41no—
335C4⋊C8—41no—
336C4⋊C8C22×C412yes—
337C4⋊C8C22×C412yes—
338C4⋊C8C2×D411yescharacteristic
339C4⋊C8C2×D411yescharacteristic
340C4⋊C8C2×D412yes—
341C4⋊C8C2×D411yescharacteristic
342C4⋊C8C2×D411yescharacteristic
343C4⋊C8C2×D412yes—
344#104—44no—
345#104—44no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348C22⋊C8—22no—
349C22⋊C8—22no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—22no—
353C22⋊C8—22no—
354C22⋊C8—21no—
355C22⋊C8—22no—
356C22⋊C8—21no—
357C22⋊C8—21no—
358C22⋊C8—21no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361#97—22no—
362#97—22no—
363#97—22no—
364#97—22no—
365C4×C8—22no—
366C4×C8—22no—
367C4×C8—21no—
368C4×C8—22no—
369C4×C8—21no—
370C4×C8—21no—
371C4×C8—21no—
372C4×C8—22no—
373C4×C8—22no—
374C4×C8C22×C412yes—
375C4×C8C22×C412yes—
376#95—22no—
377#95—41no—
378C24—21no—
379C24#12311yescharacteristic
380#41—21no—
381#41—21no—
382#41—22no—
383#41—22no—
384#41—21no—
385#41—21no—
386#41—21no—
387#41—21no—
388#41—42no—
389#41—42no—
390#41—41no—
391#41—41no—
392#41—42no—
393#41—42no—
394#41AGL(1,ℤ/8)12yes—
395#41#13712yes—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—41no—
399C2×Q8C2×C4⋊C411yescharacteristic
400C2×Q8C2×C4⋊C411yescharacteristic
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—41no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×C4⋊C411yescharacteristic
411C2×D4C2×C4⋊C411yescharacteristic
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C2×C4⋊C411yescharacteristic
444C22×C4C2×C4⋊C411yescharacteristic
445C22×C4C4×D411yescharacteristic
446C22×C4C4×D411yescharacteristic
447C22×C4C4×D411yescharacteristic
448C22×C4C4×D411yescharacteristic
449C22×C4#12011yescharacteristic
450C22×C4#12011yescharacteristic
451C22×C4C22⋊Q811yescharacteristic
452C22×C4C22⋊Q811yescharacteristic
453C22×C4C22⋊Q811yescharacteristic
454C22×C4C22⋊Q811yescharacteristic
455C22×C4#12311yescharacteristic
456C22×C4#12311yescharacteristic
457C22×C4#12311yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461M16—44no—
462M16—44no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—42no—
468M16—42no—
469M16—42no—
470M16—42no—
471M16—82no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—22no—
481C2×C8—21no—
482C2×C8—44no—
483C2×C8—44no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—42no—
489C2×C8—82no—
490C2×C8—82no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4C2×C4⋊C411yescharacteristic
505C4⋊C4C2×C4⋊C411yescharacteristic
506C4⋊C4C2×C4⋊C411yescharacteristic
507C4⋊C4C2×C4⋊C411yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—42no—
534C42—82no—
535C42C2×C4⋊C411yescharacteristic
536C42C2×C4⋊C411yescharacteristic
537C42C4×D411yescharacteristic
538C42C4×D411yescharacteristic
539C42C4×D411yescharacteristic
540C42C4×D411yescharacteristic
541C42#12011yescharacteristic
542C42#12011yescharacteristic
543C42C22⋊Q811yescharacteristic
544C42C22⋊Q811yescharacteristic
545C42C22⋊Q811yescharacteristic
546C42C22⋊Q811yescharacteristic
547C42C23×C411yescharacteristic
548C42C22×D411yescharacteristic
549C42C22×D411yescharacteristic
550C42C22×Q811yescharacteristic
551C23—21no—
552C23—21no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—81no—
560C23#38511yescharacteristic
561C23#38511yescharacteristic
562C23C2×#12311yescharacteristic
563Q8—21no—
564Q8—21no—
565Q8—21no—
566Q8—42no—
567Q8#42812yes—
568D4—22no—
569D4—22no—
570D4—21no—
571D4—21no—
572D4—21no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—41no—
577D4—41no—
578D4#42812yes—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—82no—
629C2×C4—88no—
630C2×C4—82no—
631C2×C4#38511yescharacteristic
632C2×C4#38511yescharacteristic
633C2×C4#38511yescharacteristic
634C2×C4#38511yescharacteristic
635C2×C4#38511yescharacteristic
636C2×C4#38511yescharacteristic
637C2×C4#44312yes—
638C2×C4#44412yes—
639C2×C4C22×C4⋊C411yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×C4×D411yescharacteristic
642C2×C4C2×#12011yescharacteristic
643C2×C4C2×C22⋊Q811yescharacteristic
644C2×C4C2×C22⋊Q811yescharacteristic
645C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
646C2×C4C2×#13711yescharacteristic
647C2×C4#57511yescharacteristic
648C2×C4#57511yescharacteristic
649C8—42no—
650C8—42no—
651C8—42no—
652C8—42no—
653C8—41no—
654C8—42no—
655C8—41no—
656C8—41no—
657C8—42no—
658C8—41no—
659C8—82no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—22no—
666C22—21no—
667C22—21no—
668C22—42no—
669C22—42no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—84no—
676C22#186612yes—
677C22C2×#38511yescharacteristic
678C22#309411yescharacteristic
679C22#309511yescharacteristic
680C4—21no—
681C4—21no—
682C4—22no—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—84no—
695C4#186612yes—
696C4C2×#42811yescharacteristic
697C4#291111yescharacteristic
698C4C2×#44311yescharacteristic
699C4C2×#44411yescharacteristic
700C4#300611yescharacteristic
701C4#300711yescharacteristic
702C2—21no—
703C2—22no—
704C2—21no—
705C2—41no—
706C2—42no—
707C2—81no—
708C2#2084611yescharacteristic
709C2#2086711yescharacteristic
710C2C2×#186611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17817C2semidirect
4. #17809C2semidirect
6. #8905C2semidirect
7. #8904C2semidirect
8. #8903C2semidirect
9. #8902C2semidirect
10. #8098C2semidirect
11. #7892C2semidirect
43. #1571C22semidirect
44. #1570C22semidirect
45. #1567C22semidirect
46. #1566C22semidirect
49. #1488C4semidirect
50. #1488C4semidirect
51. #1487C4semidirect
52. #1487C4semidirect
53. #1486C4semidirect
54. #1486C4semidirect
57. #1303C22semidirect
58. #1303C22semidirect
59. #1279C22semidirect
60. #1279C22semidirect
179. Q8⋊C8C2×C4semidirect
180. Q8⋊C8C2×C4semidirect
181. Q8⋊C8C2×C4semidirect
182. Q8⋊C8C2×C4semidirect
183. D4⋊C8C2×C4semidirect
184. D4⋊C8C2×C4semidirect
185. D4⋊C8C2×C4semidirect
186. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#601272
C2×#18062
#178172
#178092
#174432
#89052
#89042
#89032
#89022
#80982
#78922