Finite Group Databasebeta

← #6447190

Subgroups — #6447190

Order 512.

2316 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic, complemented
2C2×D4×M16C211yesmaximal, characteristic
3#34946C211yesmaximal, characteristic
4C2×#2536C211yesmaximal, characteristic
5#25356C214yesmaximal, complemented
6#25352C214yesmaximal, complemented
7#25345C214yesmaximal, complemented
8#25343C214yesmaximal, complemented
9#23642C212yesmaximal, complemented
10#23481C212yesmaximal, complemented
11#21323C211yesmaximal, characteristic, complemented
12#21172C211yesmaximal, characteristic, complemented
13#20972C212yesmaximal, complemented
14#20661C212yesmaximal, complemented
15C4×#419C211yesmaximal, characteristic, complemented
16C23×M16C2212yes—
17C2×#533C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3078—28no—
22C2×#479C2214yescomplemented
23C2×#477C2212yescomplemented
24C2×#475C2212yescomplemented
25D4×M16—28no—
26C2×#436C2212yes—
27C2×#435C2214yes—
28C2×C8×D4C2212yes—
29C2×#428C2212yes—
30C2×#428C2212yes—
31C2×#423C2211yescharacteristic
32C2×#419C2212yes—
33C2×#419C2211yescharacteristic
34C2×#419C2211yescharacteristic
35C2×#407C2212yes—
36C2×C4×M16C2211yescharacteristic
37#2563C2212yes—
38#2562C2212yes—
39#2562C2212yes—
40#2561—22no—
41#2547C2212yes—
42#2536—21no—
43#2536—21no—
44#2536—22no—
45#2536C2212yes—
46#2536C2214yes—
47#2536C2212yes—
48#2503—21no—
49#2435—21no—
50#2398—21no—
51#2398—21no—
52C2×#398—21no—
53C2×#394—21no—
54C4×#124—22no—
55C2×#388—22no—
56C2×#382C2211yescharacteristic
57#2280C2212yes—
58#2280C2212yes—
59C2×#380C2212yes—
60C4×#117C2211yescharacteristic
61#2077C2214yescomplemented
62#2077C2214yescomplemented
63#2076C2214yescomplemented
64#2075C2214yescomplemented
65#2036C2214yescomplemented
66#2036C2214yescomplemented
67#2035C2214yescomplemented
68#2035C2214yescomplemented
69#1993C2212yescomplemented
70#1993C2212yescomplemented
71#1991C2214yescomplemented
72#1949C2214yescomplemented
73#1949C2214yescomplemented
74#1894C2214yescomplemented
75#1894C2214yescomplemented
76#1892C2214yescomplemented
77#1892C2214yescomplemented
78#1809C2214yescomplemented
79#1809C2214yescomplemented
80#1808C2214yescomplemented
81#1807C2214yescomplemented
82#1763C2214yescomplemented
83C4×#103C2214yescomplemented
84#1736C2212yes—
85#1736C2212yes—
86#1736C2211yescharacteristic
87#1736C2212yes—
88#1736C2211yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×D4D411yescharacteristic
91C24×C4C2312yes—
92C22×M16—28no—
93C22×M16—28no—
94C22×M16—44no—
95C22×M16C2311yescharacteristic
96C22×M16C2314yes—
97C22×M16C2312yes—
98C22×M16C2311yescharacteristic
99C23×C8C2314yes—
100#533—24no—
101#533—24no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125C2312yes—
105C2×#124—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—28no—
111C2×C22⋊Q8—44no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2314yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—24no—
123C2×C4×D4—24no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D412yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2312yes—
132C2×#117C2311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4D411yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2312yes—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C22⋊C4D412yes—
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#479—216no—
144#477—28no—
145#475—28no—
146#436—28no—
147#435—216no—
148C8×D4—28no—
149#428—28no—
150#428—44no—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155#423D418yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158#419—24no—
159#419—28no—
160#419—24no—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165#407—28no—
166C2×C22⋊C8C2314yes—
167C2×C22⋊C8C2314yes—
168C4×M16—24no—
169C2×#97C2312yes—
170C2×C4×C8C2312yes—
171#400—22no—
172#400—22no—
173#398—24no—
174#398—22no—
175#398—22no—
176#397—22no—
177#397—22no—
178#394—22no—
179#394—22no—
180#394—22no—
181#394—24no—
182#394—22no—
183#393—22no—
184#393—22no—
185#391—22no—
186#390—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#382—22no—
196#382—21no—
197#382—21no—
198#382C2312yes—
199#382C2312yes—
200#381—22no—
201#381—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211#380C2312yes—
212#380C2314yes—
213#380C2314yes—
214#379—22no—
215#379—22no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224#376—21no—
225C2×#95—21no—
226C2×#95—22no—
227C2×#95—21no—
228C43—21no—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237C25C2×D412yes—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8#4114yes—
247C22×Q8C2412yes—
248C22×D4—24no—
249C22×D4—44no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D412yes—
252C22×D4C2×D411yescharacteristic
253C23×C4—21no—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C2×M16—22no—
271C2×M16—24no—
272C2×M16—22no—
273C2×M16—24no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—28no—
278C2×M16—28no—
279C2×M16—44no—
280C2×M16—432no—
281C2×M16C2×D418yes—
282C2×M16#4114yes—
283C22×C8—28no—
284C22×C8—28no—
285C22×C8—44no—
286C22×C8C2412yes—
287C22×C8C2414yes—
288C22×C8C2412yes—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293#125—24no—
294#125—24no—
295#124—48no—
296C22⋊Q8—28no—
297C22⋊Q8—28no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300C22⋊Q8—48no—
301C22⋊Q8—48no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—28no—
315C4×D4—22no—
316C4×D4—24no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—24no—
321#117—24no—
322#117—24no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—48no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—28no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—48no—
370C2×C22⋊C4—48no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—44no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D414yes—
386C2×C42#4114yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—28no—
390#107—44no—
391#106—28no—
392#106—44no—
393C4⋊C8C2×D418yes—
394C4⋊C8C2×D418yes—
395#103—28no—
396#103—28no—
397#103—28no—
398#103—28no—
399C22⋊C8—28no—
400C22⋊C8—28no—
401#97—24no—
402C4×C8—24no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419C24—24no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24C22×D412yes—
427C24C22×D412yes—
428C24C22×D411yescharacteristic
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—28no—
432C2×Q8—28no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—42no—
444C2×Q8—82no—
445C2×Q8C2×#4114yes—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—24no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—416no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4#12114yes—
497C22×C4#12114yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4H(5,𝔽2)12yes—
510C22×C42−1+412yes—
511C22×C4C2511yescharacteristic
512M16—28no—
513M16—216no—
514M16—416no—
515M16—44no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—416no—
524C2×C8C22×D418yes—
525C2×C8C2×#4114yes—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—24no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—48no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C42—44no—
571C42—44no—
572C42—44no—
573C42—44no—
574C42—42no—
575C42—42no—
576C42—82no—
577C42#12114yes—
578C42#12114yes—
579C42C22×D412yes—
580C42C22×D412yes—
581C42C2×#4112yes—
582C42C2×#4112yes—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—48no—
599C23—48no—
600C23—48no—
601C23—48no—
602C23—84no—
603C23—84no—
604C23—84no—
605C23C2×#12112yes—
606C23#54612yes—
607C23#54712yes—
608C23#57514yes—
609C23C23×D411yescharacteristic
610Q8—24no—
611Q8—24no—
612Q8—48no—
613Q8—42no—
614Q8—42no—
615Q8—84no—
616Q8—84no—
617D4—28no—
618D4—48no—
619D4—44no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—84no—
665C2×C4—88no—
666C2×C4C2×#12112yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#12112yes—
669C2×C4C2×#12112yes—
670C2×C4C2×#12112yes—
671C2×C4#52914yes—
672C2×C4#54214yes—
673C2×C4#54612yes—
674C2×C4#54614yes—
675C2×C4#54712yes—
676C2×C4#54814yes—
677C2×C4C23×D411yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C22×#4111yescharacteristic
680C2×C4C22×#4111yescharacteristic
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4C2×2−1+411yescharacteristic
683C8—24no—
684C8—28no—
685C8—42no—
686C22—22no—
687C22—22no—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—816no—
705C22#242714yes—
706C22C22×#12111yescharacteristic
707C22C2×#54611yescharacteristic
708C22C2×#54711yescharacteristic
709C22C2×#57512yes—
710C22#358612yes—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—42no—
716C4—42no—
717C4—44no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—84no—
724C4#242714yes—
725C4C2×#52911yescharacteristic
726C4C2×#54211yescharacteristic
727C4C2×#54611yescharacteristic
728C4C2×#54811yescharacteristic
729C2—24no—
730C2—44no—
731C2—82no—
732C2C2×#242711yescharacteristic
733C2#6044411yescharacteristic
734C2#6045511yescharacteristic
735C2#6115812yes—
736C2#6125312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3078C2semidirect
5. #25356C2semidirect
6. #25352C2semidirect
7. #25345C2semidirect
8. #25343C2semidirect
9. #23642C2semidirect
10. #23481C2semidirect
11. #21323C2semidirect
12. #21172C2semidirect
13. #20972C2semidirect
14. #20661C2semidirect
15. C4×#419C2semidirect
22. C2×#479C22semidirect
23. C2×#477C22semidirect
24. C2×#475C22semidirect
61. #2077C22semidirect
62. #2077C22semidirect
63. #2076C22semidirect
64. #2075C22semidirect
65. #2036C22semidirect
66. #2036C22semidirect
67. #2035C22semidirect
68. #2035C22semidirect
69. #1993C22semidirect
70. #1993C22semidirect
71. #1991C22semidirect
72. #1949C22semidirect
73. #1949C22semidirect
74. #1894C22semidirect
75. #1894C22semidirect
76. #1892C22semidirect
77. #1892C22semidirect
78. #1809C22semidirect
79. #1809C22semidirect
80. #1808C22semidirect
81. #1807C22semidirect
82. #1763C22semidirect
83. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×D4×M162
#349462
C2×#25362
#253562
#253522
#253452
#253432
#236422
#234812
#213232
#211722
#209722
#206612
C4×#4192