Finite Group Databasebeta

← #192740

Subgroups — #192740

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1477C211yesmaximal, characteristic
6#8892C212yesmaximal, complemented
7#8890C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8045C211yesmaximal, characteristic, complemented
11#7901C211yesmaximal, characteristic, complemented
12C2×C4×#41—21no—
13C2×C4×#41C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×#107—21no—
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×#417—21no—
19#2351—21no—
20Q8×C4⋊C4C2211yescharacteristic
21Q8×C4⋊C4C2211yescharacteristic
22#2349—22no—
23#2349—21no—
24D4×C4⋊C4—21no—
25D4×C4⋊C4—21no—
26D4×C4⋊C4C2211yescharacteristic
27D4×C4⋊C4C2211yescharacteristic
28#2296—22no—
29#2296—21no—
30#2285—21no—
31#2285C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1947—21no—
36#1947—21no—
37#1947—22no—
38#1947C2212yes—
39#1947C2212yes—
40#1852—22no—
41#1845C2212yes—
42#1834C2212yes—
43#1803—21no—
44#1803C2212yes—
45#1802—21no—
46#1802C2212yes—
47#1736—21no—
48C2×#340—21no—
49#1477C418yescomplemented
50C2×Q8⋊C8C2212yes—
51C2×D4⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×#41D411yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—44no—
62C4×#41—44no—
63C4×#41C2×C414yes—
64C4×#41Q814yes—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4D411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117—22no—
78C2×#117—21no—
79C2×#117—21no—
80C2×#117D411yescharacteristic
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C4⋊C4D411yescharacteristic
88C22×C42—21no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#431—22no—
93#431C2×C414yescomplemented
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—24no—
97C2×#107—44no—
98#423—22no—
99#423—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2×C414yescomplemented
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107#417—44no—
108C2×#103—22no—
109C2×#102—22no—
110C2×C22⋊C8—21no—
111C2×C4×C8C2312yes—
112#389—22no—
113#389—22no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—22no—
119#389—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#385—22no—
125#385—22no—
126#384—21no—
127#384—21no—
128#384—21no—
129#384—21no—
130#384—21no—
131#384C2312yes—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#382—22no—
135#382—21no—
136#380—22no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380D412yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4D411yescharacteristic
150C4×C4⋊C4D411yescharacteristic
151C4×C4⋊C4D412yes—
152C4×C4⋊C4C2311yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154C2×#95—22no—
155C2×#95—21no—
156#340—22no—
157#340—22no—
158#340—24no—
159#340—22no—
160#340—22no—
161Q8⋊C8C2×C418yescomplemented
162D4⋊C8C2×C418yescomplemented
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41C4⋊C414yes—
167C2×#41D814yes—
168C2×#41Q1614yes—
169C22×Q8C2×D411yescharacteristic
170C22×D4—21no—
171C22×D4C2×D411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—41no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C2×M16—42no—
184C2×M16—42no—
185C2×M16—42no—
186C22×C8—24no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—42no—
191C22×C8—42no—
192C4×Q8—22no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8C4⋊C414yes—
200C4×Q8C22×C412yes—
201C4×Q8C2×Q812yes—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—22no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4C4⋊C414yes—
216C4×D4C22×C412yes—
217C4×D4C2×Q812yes—
218#117—22no—
219#117—24no—
220#117—24no—
221#117—48no—
222#117—42no—
223#117—42no—
224#117C4⋊C414yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—48no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42C22×C412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×Q812yes—
292C2×C42#4112yes—
293C2×C42#4112yes—
294C2×C42C2411yescharacteristic
295#107—44no—
296#107—44no—
297#107—84no—
298C4⋊C8—24no—
299C4⋊C8—24no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—42no—
304C4⋊C8C22×C414yes—
305#103—44no—
306#102—44no—
307C22⋊C8—24no—
308C4×C8—22no—
309C4×C8C22×C414yes—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—21no—
315#95—22no—
316#95—21no—
317#95—22no—
318#95—21no—
319#95—21no—
320C24—21no—
321C24#12011yescharacteristic
322#41—28no—
323#41—28no—
324#41—28no—
325#41#107116yescomplemented
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8C2×C4⋊C412yes—
330C2×Q8C2×D812yes—
331C2×Q8C2×Q1612yes—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4C2×C4⋊C412yes—
339C2×D4C2×D812yes—
340C2×D4C2×Q1612yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4C2×C4⋊C412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4#12011yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4#12011yescharacteristic
390C22×C4C22⋊Q812yes—
391C22×C4C22⋊Q812yes—
392C22×C4#12312yes—
393C22×C4C2×D812yes—
394C22×C4C2×Q1612yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4#13712yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400M16—84no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—48no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—24no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4C2×C4⋊C412yes—
449C4⋊C4C2×C4⋊C412yes—
450C4⋊C4AGL(1,ℤ/8)12yes—
451C4⋊C4#13712yes—
452C22⋊C4—24no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C42—22no—
460C42—22no—
461C42—24no—
462C42—24no—
463C42—44no—
464C42—44no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—42no—
470C42—44no—
471C42—42no—
472C42C2×C4⋊C412yes—
473C42C4×D412yes—
474C42C4×D412yes—
475C42C22⋊Q812yes—
476C42C22⋊Q812yes—
477C42#12312yes—
478C42C23×C411yescharacteristic
479C42C22×Q811yescharacteristic
480C42C2×#4111yescharacteristic
481C42C2×#4111yescharacteristic
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23#38512yes—
494C23#44712yes—
495C23#45112yes—
496C23C2×#12011yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—22no—
500Q8C2×#10714yescomplemented
501D4—216no—
502D4—22no—
503D4—22no—
504D4—22no—
505D4C2×#10714yescomplemented
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—216no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—88no—
554C2×C4—88no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—164no—
559C2×C4#38512yes—
560C2×C4#38512yes—
561C2×C4#38512yes—
562C2×C4C2×#10714yes—
563C2×C4#42814yes—
564C2×C4#44712yes—
565C2×C4#45112yes—
566C2×C4#48014yes—
567C2×C4#48414yes—
568C2×C4C22×C4⋊C411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C2×C22⋊Q811yescharacteristic
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4C2×#12311yescharacteristic
574C2×C4C22×D811yescharacteristic
575C2×C4C22×Q1611yescharacteristic
576C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
577C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
578C2×C4C2×#13711yescharacteristic
579C2×C4C2×#13711yescharacteristic
580C2×C4#57511yescharacteristic
581C2×C4#57511yescharacteristic
582C8—44no—
583C8—44no—
584C8—82no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—24no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22#185214yes—
599C22C2×#38511yescharacteristic
600C22C2×#44711yescharacteristic
601C22C2×#45111yescharacteristic
602C22#300611yescharacteristic
603C22#300711yescharacteristic
604C22#302111yescharacteristic
605C22#302211yescharacteristic
606C4—24no—
607C4—21no—
608C4—22no—
609C4—21no—
610C4—22no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—81no—
616C4—82no—
617C4—81no—
618C4—82no—
619C4—162no—
620C4#185214yes—
621C4C22×#10711yescharacteristic
622C4C2×#42811yescharacteristic
623C4C4×AGL(1,ℤ/8)11yescharacteristic
624C4C4×#13711yescharacteristic
625C4#307911yescharacteristic
626C4#308011yescharacteristic
627C4C2×#48011yescharacteristic
628C4C2×#48411yescharacteristic
629C2—24no—
630C2—44no—
631C2#1878611yescharacteristic
632C2#1878911yescharacteristic
633C2#1879811yescharacteristic
634C2#1880111yescharacteristic
635C2C2×#185211yescharacteristic
636C2#2086511yescharacteristic
637C2#2087511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
3. C2×#1947C2semidirect
6. #8892C2semidirect
7. #8890C2semidirect
8. #8884C2semidirect
9. #8882C2semidirect
10. #8045C2semidirect
11. #7901C2semidirect
49. #1477C4semidirect
93. #431C2×C4semidirect
104. C2×C4⋊C8C2×C4semidirect
161. Q8⋊C8C2×C4semidirect
162. D4⋊C8C2×C4semidirect
325. #41#107semidirect
500. Q8C2×#107semidirect
505. D4C2×#107semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#219822
C2×#19472
#206482
C2×#14772
#88922
#88902
#88842
#88822
#80452
#79012