Finite Group Databasebeta

← #382606

Subgroups — #382606

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2C2×#1826C212yesmaximal
3#20626C211yesmaximal, characteristic
4C4×#380C211yesmaximal, characteristic
5#11226C214yesmaximal, complemented
6#11209C214yesmaximal, complemented
7#10733C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423C2212yes—
12C2×#407C2212yes—
13#2296—21no—
14#2296—21no—
15#2296—21no—
16#2296—21no—
17#2296C412yescomplemented
18#2296C412yescomplemented
19#2296C2212yes—
20#2296C2212yes—
21C2×#389C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#389C411yescharacteristic, complemented
24C2×#389C411yescharacteristic, complemented
25C2×#385—21no—
26C2×#385—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32#1826—28no—
33#1797—21no—
34#1797—21no—
35#1797—21no—
36#1797—21no—
37C2×#339C2218yes—
38C2×#339C2218yes—
39#1728C2212yes—
40C4⋊#95—21no—
41#1434—21no—
42C22⋊#95—21no—
43C4×#95C2212yes—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16C2314yes—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—44no—
55C2×#117C2×C412yes—
56C2×#117C2×C412yes—
57C2×#117D414yes—
58C2×#117Q814yes—
59C2×#117C2312yes—
60C2×#117C2312yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2314yes—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#423—44no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92#407—24no—
93#407—24no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#389C2×C412yes—
111#389C2×C412yes—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—22no—
120#380—28no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—44no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140#339—216no—
141#339—48no—
142#336—22no—
143#336—22no—
144C25C2×D411yescharacteristic
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q812yes—
177C23×C4#4112yes—
178C23×C4#4112yes—
179C23×C4C2411yescharacteristic
180C2×M16—24no—
181C2×M16—24no—
182C2×M16—28no—
183C2×M16—28no—
184C22×C8—24no—
185#117—24no—
186#117—24no—
187#117—24no—
188#117—24no—
189#117—24no—
190#117—22no—
191#117—22no—
192#117—28no—
193#117—44no—
194#117—44no—
195#117—44no—
196#117C4⋊C418yescomplemented
197#117C2×D414yes—
198#117C2×Q814yes—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4C4⋊C414yescomplemented
223C2×C4⋊C4C4⋊C414yescomplemented
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—28no—
249C2×C42—28no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—44no—
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C4⋊C8—44no—
278C4⋊C8—44no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281#95—28no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—44no—
291#95—44no—
292#95—44no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—41no—
298C24—41no—
299C24C2×C22⋊C411yescharacteristic
300C24C2×C22⋊C411yescharacteristic
301C24#12112yes—
302C24C22⋊Q812yes—
303C24C22×D411yescharacteristic
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
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—41no—
360C22×C4—416no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—416no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4#10414yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4#11712yes—
392C22×C4#11712yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C22⋊Q812yes—
397C22×C4C22⋊Q812yes—
398C22×C4C22⋊Q812yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×Q811yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)14yes—
406C22×C42−1+414yes—
407M16—48no—
408M16—48no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—416no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4C2×C4⋊C414yes—
430C4⋊C4C2×C4⋊C414yes—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—28no—
434C22⋊C4—416no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—24no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—28no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—44no—
457C42—44no—
458C42—48no—
459C42—48no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23#38012yes—
485C23#38012yes—
486C23C2×#10412yes—
487C23#42112yes—
488C23C22×C22⋊C411yescharacteristic
489C23C2×#12111yescharacteristic
490C23C2×C22⋊Q811yescharacteristic
491C23#53412yes—
492C23#53612yes—
493C2×C4—28no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
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—24no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4#38012yes—
563C2×C4#38012yes—
564C2×C4C2×#10412yes—
565C2×C4C2×#10412yes—
566C2×C4#42112yes—
567C2×C4#42112yes—
568C2×C4C22×C4⋊C411yescharacteristic
569C2×C4C2×#11711yescharacteristic
570C2×C4#51812yes—
571C2×C4#51912yes—
572C2×C4C2×#12111yescharacteristic
573C2×C4C2×C22⋊Q811yescharacteristic
574C2×C4#53412yes—
575C2×C4#53612yes—
576C2×C4#53714yes—
577C2×C4#53914yes—
578C2×C4#53914yes—
579C2×C4#54414yes—
580C8—84no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—48no—
599C22—48no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22#180614yes—
605C22C2×#38011yescharacteristic
606C22#230811yescharacteristic
607C22#231011yescharacteristic
608C22#261812yes—
609C22#262712yes—
610C22C22×#10411yescharacteristic
611C22C2×#42111yescharacteristic
612C22C2×#42111yescharacteristic
613C22C2×#42111yescharacteristic
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—44no—
625C4—44no—
626C4—88no—
627C4—82no—
628C4—82no—
629C4#231312yes—
630C4#231612yes—
631C4#261812yes—
632C4#262712yes—
633C2—22no—
634C2—24no—
635C2—22no—
636C2—42no—
637C2#1433911yescharacteristic
638C2C2×#180611yescharacteristic
639C2#2065411yescharacteristic
640C2#2177311yescharacteristic
641C2#2177411yescharacteristic
642C2#2184411yescharacteristic
643C2#2184511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11226C2semidirect
6. #11209C2semidirect
17. #2296C4semidirect
18. #2296C4semidirect
21. C2×#389C4semidirect
22. C2×#389C4semidirect
23. C2×#389C4semidirect
24. C2×#389C4semidirect
196. #117C4⋊C4semidirect
222. C2×C4⋊C4C4⋊C4semidirect
223. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#18262
#206262
C4×#3802
#112262
#112092
#107332