Finite Group Databasebeta

← #192368

Subgroups — #192368

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#8872C214yesmaximal, complemented
7#8871C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic
9#7907C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#106—21no—
13C22×#103—21no—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2368—22no—
17#2360—21no—
18#2360—21no—
19#2345—22no—
20#2345—21no—
21Q8×C22⋊C4—21no—
22#2329C2212yes—
23#2329C2212yes—
24#2315—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27C2×#389—21no—
28C2×#385—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#2273—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1872—22no—
34#1872C2214yes—
35#1853—22no—
36#1853C2214yes—
37C2×#340—21no—
38C2×#340—21no—
39#1520C418yescomplemented
40C2×#328C2214yes—
41#1444C2212yes—
42#1442C2212yes—
43C4⋊#95—21no—
44C22⋊#95—21no—
45#1294C2214yes—
46#1280C2212yes—
47#1280C2212yes—
48C23×Q8D411yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—21no—
52C23×C8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8C2×C414yes—
58C2×C22⋊Q8D412yes—
59C2×C22⋊Q8Q814yes—
60C2×C22⋊Q8C2312yes—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C2×#106—24no—
84C2×#106—44no—
85C2×#103—24no—
86C2×#103—44no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8C2×C414yes—
93C2×C22⋊C8C2312yes—
94C2×C22⋊C8C2312yes—
95#391—21no—
96#391—21no—
97#391—22no—
98#391—22no—
99#389—21no—
100#389—21no—
101#389—22no—
102#389—22no—
103#389—44no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—44no—
115#380—22no—
116#380—22no—
117#380—21no—
118#380—21no—
119#380—22no—
120#380C2×C414yes—
121#380D412yes—
122#380C2312yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95C2311yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144#340—24no—
145#340—24no—
146#328C2×C4116yescomplemented
147C25C2×D411yescharacteristic
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8C4⋊C414yescomplemented
153C22×Q8C2×D412yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—82no—
168C23×C4C22×C412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×Q812yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C22×C8—24no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8C4⋊C418yescomplemented
198C22⋊Q8C22×C414yes—
199C22⋊Q8C2×Q814yes—
200C4×Q8—82no—
201C4×Q8—82no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4C4⋊C414yescomplemented
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4SD1614yes—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—42no—
267C2×C42—42no—
268#106—48no—
269#103—48no—
270C22⋊C8—24no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—42no—
275C22⋊C8C22×C414yes—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—22no—
291#95—44no—
292#95—44no—
293#95—82no—
294#95C22×C414yes—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24C4×D412yes—
300C24#12011yescharacteristic
301C24#12011yescharacteristic
302C24C22⋊Q812yes—
303C24C2×SD1612yes—
304C24#13512yes—
305C24#13712yes—
306C24#13712yes—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—48no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—84no—
316C2×Q8C2×C4⋊C414yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4C4×D412yes—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4#12011yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4C22⋊Q812yes—
397C22×C4C22⋊Q812yes—
398C22×C4C22⋊Q812yes—
399C22×C4#12312yes—
400C22×C4#12312yes—
401C22×C4C2×SD1612yes—
402C22×C4C2×SD1612yes—
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×Q811yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C2×C8—24no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—48no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—84no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—88no—
438C4⋊C4C2×C4⋊C414yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—28no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—48no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C42—42no—
451C42—42no—
452C42—81no—
453C42—81no—
454C42—82no—
455C42—82no—
456C42—82no—
457C42—82no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23#38512yes—
483C23#38512yes—
484C23C4×SD1614yes—
485C23#44114yes—
486C23#44812yes—
487C23#44812yes—
488C23#45212yes—
489C23#45212yes—
490C23#47514yes—
491C23#47914yes—
492C23C2×C4×D411yescharacteristic
493C23C2×#12011yescharacteristic
494C23C2×C22⋊Q811yescharacteristic
495C23C22×SD1611yescharacteristic
496C23C2×#13511yescharacteristic
497C23C2×#13711yescharacteristic
498C23C2×#13711yescharacteristic
499Q8—44no—
500Q8—48no—
501Q8—82no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—88no—
567C2×C4—88no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—162no—
573C2×C4—162no—
574C2×C4#38512yes—
575C2×C4#38512yes—
576C2×C4B(4,𝔽2)12yes—
577C2×C4#45812yes—
578C2×C4C22×C4⋊C411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C2×C4C2×#12311yescharacteristic
582C8—44no—
583C8—82no—
584C22—21no—
585C22—21no—
586C22—22no—
587C22—22no—
588C22—21no—
589C22—21no—
590C22—22no—
591C22—22no—
592C22—21no—
593C22—21no—
594C22—22no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—21no—
601C22—22no—
602C22—21no—
603C22—22no—
604C22—21no—
605C22—22no—
606C22—21no—
607C22—22no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22#186514yes—
613C22#186514yes—
614C22C2×#38511yescharacteristic
615C22C2×C4×SD1611yescharacteristic
616C22C2×#44111yescharacteristic
617C22C2×#44811yescharacteristic
618C22C2×#45211yescharacteristic
619C22#302111yescharacteristic
620C22#302211yescharacteristic
621C22C2×B(4,𝔽2)11yescharacteristic
622C22C2×#45811yescharacteristic
623C22C2×#47511yescharacteristic
624C22C2×#47911yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—22no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—84no—
635C4—84no—
636C4—81no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4#186714yes—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—24no—
645C2—24no—
646C2—21no—
647C2#1819611yescharacteristic
648C2#1819911yescharacteristic
649C2#1820411yescharacteristic
650C2#1820711yescharacteristic
651C2#2086511yescharacteristic
652C2C2×#186511yescharacteristic
653C2C2×#186711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8872C2semidirect
7. #8871C2semidirect
39. #1520C4semidirect
146. #328C2×C4semidirect
152. C22×Q8C4⋊C4semidirect
197. C22⋊Q8C4⋊C4semidirect
235. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342122
C2×#18722
C2×#18532
C2×#15202
#141302
#88722
#88712
#80232
#79072