Finite Group Databasebeta

← #359933

Subgroups — #359933

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3C2×#1826C212yesmaximal
4C4×#381C211yesmaximal, characteristic
5#11209C214yesmaximal, complemented
6#10886C214yesmaximal, complemented
7#10733C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432—21no—
14C2×#431—21no—
15C2×#431—21no—
16C2×#423C2212yes—
17C2×#407C2212yes—
18C2×C4×M16C2212yes—
19#2296—21no—
20#2296—21no—
21#2296—21no—
22#2296—21no—
23#2296C412yescomplemented
24#2296C412yescomplemented
25#2296C2212yes—
26#2296C2212yes—
27C2×#389C411yescharacteristic, complemented
28C2×#389C411yescharacteristic, complemented
29C2×#389C411yescharacteristic, complemented
30C2×#389C411yescharacteristic, complemented
31C2×#385—21no—
32C2×#385—21no—
33C2×#381—21no—
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4—21no—
36C2×C4×C22⋊C4C2212yes—
37#1826—28no—
38C2×#339C2218yes—
39C2×#339C2218yes—
40#1434—21no—
41#1434—21no—
42C22⋊#95—21no—
43C4×#95C2212yes—
44C4×#95C2212yes—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16C2314yes—
49C2×#117—21no—
50C2×#117—21no—
51C2×#117—28no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117C2×C412yes—
57C2×#117C2×C412yes—
58C2×#117C2312yes—
59C2×#117C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C42—24no—
72C22×C42—21no—
73C22×C42—24no—
74C22×C42—21no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2314yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—28no—
89#432—28no—
90#431—28no—
91#431—28no—
92#423—28no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95#407—24no—
96#407D418yes—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C4×M16—28no—
100C2×#97—22no—
101C2×C4×C8—22no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389C2×C412yes—
111#389C2×C412yes—
112#389C2×C412yes—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2×C412yes—
116#389C2×C412yes—
117#389C2×C412yes—
118#385—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#385—22no—
125#385—22no—
126#381—28no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146#339—216no—
147#339—216no—
148C25C2×D411yescharacteristic
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D414yes—
180C23×C4#4114yes—
181C23×C4C2411yescharacteristic
182C2×M16—24no—
183C2×M16—28no—
184C2×M16—28no—
185C2×M16C2×D418yes—
186C22×C8—24no—
187#117—28no—
188#117—28no—
189#117—24no—
190#117—24no—
191#117—24no—
192#117—28no—
193#117—24no—
194#117—24no—
195#117—42no—
196#117—44no—
197#117—42no—
198#117#4118yes—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
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—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C42—28no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—42no—
260C2×C42—42no—
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—28no—
278C4⋊C8—28no—
279C22⋊C8—28no—
280C22⋊C8—28no—
281#97—28no—
282C4×C8—28no—
283#95—28no—
284#95—28no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—44no—
294#95—44no—
295#95—44no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—41no—
301C24—41no—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304C24#12114yes—
305C24C22×D411yescharacteristic
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—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—216no—
320C22×C4—216no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
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—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
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—416no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
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×C4C4×D414yes—
390C22×C4C4×D414yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4#12114yes—
394C22×C4#12312yes—
395C22×C4#12312yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4114yes—
400C22×C4C2×#4112yes—
401C22×C42−1+414yes—
402M16—216no—
403M16—48no—
404C2×C8—28no—
405C2×C8—44no—
406C2×C8—44no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4#11714yes—
424C4⋊C4#11714yes—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—416no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C42—28no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—28no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—42no—
453C42—42no—
454C42—44no—
455C42—48no—
456C42—48no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—28no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—28no—
472C23—21no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23#38612yes—
479C23#38612yes—
480C23C2×#10412yes—
481C23#42112yes—
482C23C22×C22⋊C411yescharacteristic
483C23C2×#12112yes—
484C23#52512yes—
485C23#53612yes—
486C2×C4—28no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—28no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—28no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4#38612yes—
553C2×C4#38612yes—
554C2×C4C2×#10412yes—
555C2×C4C2×#10412yes—
556C2×C4#42112yes—
557C2×C4#42112yes—
558C2×C4C2×#11712yes—
559C2×C4C2×C4×D412yes—
560C2×C4#51912yes—
561C2×C4C2×#12011yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4#52512yes—
564C2×C4#53612yes—
565C2×C4#54718yes—
566C2×C4#55218yes—
567C8—48no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—416no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22#187114yes—
589C22#228011yescharacteristic
590C22C2×#38611yescharacteristic
591C22#231011yescharacteristic
592C22#241314yes—
593C22C22×#10411yescharacteristic
594C22C2×#42111yescharacteristic
595C22C2×#42111yescharacteristic
596C22C2×#42111yescharacteristic
597C4—22no—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—44no—
604C4—44no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—88no—
610C4—82no—
611C4—82no—
612C4#234414yes—
613C4#240612yes—
614C4#248412yes—
615C2—24no—
616C2—24no—
617C2—42no—
618C2#1387011yescharacteristic
619C2#2074711yescharacteristic
620C2#2074811yescharacteristic
621C2C2×#187111yescharacteristic
622C2#2090211yescharacteristic
623C2#2184411yescharacteristic
624C2#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. #11209C2semidirect
6. #10886C2semidirect
23. #2296C4semidirect
24. #2296C4semidirect
27. C2×#389C4semidirect
28. C2×#389C4semidirect
29. C2×#389C4semidirect
30. C2×#389C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#22962
C2×#18262
C4×#3812
#112092
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