Finite Group Databasebeta

← #382784

Subgroups — #382784

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#11204C212yesmaximal, complemented
5#11198C212yesmaximal
6#11045C212yesmaximal, complemented
7#10968C212yesmaximal
8#10802C212yesmaximal, complemented
9#10763C212yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#423C2212yes—
14C22×#101C2214yes—
15C2×#407C2212yes—
16C2×#407C2212yes—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C4×C22⋊Q8—21no—
20C4×C22⋊Q8—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#387C412yes—
28C2×#385C412yes—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4C412yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C412yes—
33#1817—21no—
34#1817—21no—
35#1817—21no—
36#1817—21no—
37#1797—21no—
38#1797—21no—
39#1797—21no—
40#1797—21no—
41C2×#343C2214yes—
42C2×#343C2214yes—
43C2×#343C2214yes—
44#1728C2212yes—
45#1728C2212yes—
46#1728C2212yes—
47C23×Q8C2311yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16C2314yes—
52C22×M16C2314yes—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—24no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—44no—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C42—44no—
73C22×C42C2×C412yes—
74C22×C42C2×C412yes—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#423—44no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×#101—28no—
91C2×#101—44no—
92C2×#101—44no—
93#407—44no—
94#407—44no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136#343—28no—
137#343—44no—
138#343—44no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145C25C22×C411yescharacteristic
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8#4114yes—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
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×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2411yescharacteristic
186C2×M16—24no—
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C22×C8—24no—
199C22×C8—24no—
200C22⋊Q8—28no—
201C22⋊Q8—28no—
202C22⋊Q8—28no—
203C22⋊Q8—28no—
204C4×Q8—44no—
205C4×Q8—44no—
206C4×Q8—44no—
207C4×Q8—44no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4#4114yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—44no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—44no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—44no—
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C4⋊C8—44no—
271C4⋊C8—44no—
272#101—416no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C22⋊C8—44no—
276C22⋊C8—44no—
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—22no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—41no—
294C24—41no—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C23×C411yescharacteristic
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—28no—
301C2×Q8—28no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—44no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
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—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4#10114yescomplemented
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4#11714yes—
396C22×C4#11714yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4H(5,𝔽2)14yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404M16—44no—
405M16—44no—
406M16—44no—
407M16—44no—
408M16—44no—
409M16—44no—
410M16—44no—
411M16—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C22⋊C4—28no—
432C22⋊C4—28no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42—44no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—24no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C2×#10112yes—
482C23#41312yes—
483C23C22×C22⋊C411yescharacteristic
484C23C2×#11711yescharacteristic
485C23C2×#11711yescharacteristic
486C23#51812yes—
487C23#52012yes—
488Q8—48no—
489Q8—44no—
490Q8—44no—
491C2×C4—28no—
492C2×C4—21no—
493C2×C4—21no—
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—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—28no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—24no—
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—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4C2×#10112yes—
556C2×C4C2×#10112yes—
557C2×C4#41312yes—
558C2×C4#41312yes—
559C2×C4#52512yes—
560C2×C4#52512yes—
561C2×C4#53412yes—
562C2×C4#53412yes—
563C2×C4#53511yescharacteristic
564C2×C4#53611yescharacteristic
565C2×C4#53712yes—
566C2×C4#53814yes—
567C2×C4#53914yes—
568C2×C4#54014yes—
569C2×C4#54214yes—
570C8—84no—
571C8—84no—
572C22—22no—
573C22—22no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—48no—
587C22—48no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22#228011yescharacteristic
597C22#230811yescharacteristic
598C22#231111yescharacteristic
599C22#231812yes—
600C22#231912yes—
601C22C22×#10111yescharacteristic
602C22C2×#41311yescharacteristic
603C22C2×#41311yescharacteristic
604C22C2×#41311yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—44no—
612C4—44no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—82no—
618C4—82no—
619C4#263014yes—
620C4#263512yes—
621C4#263812yes—
622C2—22no—
623C2—22no—
624C2—42no—
625C2—42no—
626C2#1434411yescharacteristic
627C2#2072711yescharacteristic
628C2#2072811yescharacteristic
629C2#2175311yescharacteristic
630C2#2175411yescharacteristic
631C2#2187911yescharacteristic
632C2#2188011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11204C2semidirect
6. #11045C2semidirect
8. #10802C2semidirect
9. #10763C2semidirect
384. C22×C4#101semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#206822
#206262
#112042
#111982
#110452
#109682
#108022
#107632