Finite Group Databasebeta

← #360095

Subgroups — #360095

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#1901C212yesmaximal
4C2×#1801C212yesmaximal, complemented
5#20626C211yesmaximal, characteristic
6#11204C212yesmaximal, complemented
7#11045C212yesmaximal, complemented
8#10847C212yesmaximal, complemented
9#10739C212yesmaximal
10C23×M16C2212yes—
11C22×C22⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C2×#423—21no—
15C2×#423—21no—
16C2×#423—28no—
17C2×#423C2212yes—
18C2×#423C2212yes—
19C2×#423C2212yes—
20C22×C4⋊C8—21no—
21C22×C4⋊C8—21no—
22C22×#101C2214yes—
23C2×#407C2212yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28C4×C22⋊Q8—21no—
29C4×C22⋊Q8—21no—
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8—21no—
32C2×#391C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#387C412yes—
35C2×#385C412yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C412yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C412yes—
40#1901—28no—
41#1801—28no—
42#1797—21no—
43#1797—21no—
44#1797—21no—
45#1797—21no—
46C2×#343C2214yes—
47C2×#343C2214yes—
48C2×#343C2214yes—
49#1728C2212yes—
50C23×Q8C2311yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—28no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16—44no—
59C22×M16C2314yes—
60C22×M16C2314yes—
61C23×C8—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8D414yes—
69C2×C22⋊Q8Q814yes—
70C2×C4×Q8—24no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—24no—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4C2×C412yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4C2×C412yes—
83C22×C22⋊C4C2×C412yes—
84C22×C42—24no—
85C22×C42—24no—
86C22×C42C2×C412yes—
87C22×C42C2×C411yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C412yes—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42D411yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#423—28no—
101#423—28no—
102#423—28no—
103#423—24no—
104#423—24no—
105#423—24no—
106#423D418yes—
107C2×C4⋊C8—28no—
108C2×C4⋊C8—28no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—22no—
115C2×#101—28no—
116C2×#101—28no—
117C2×#101—44no—
118#407—24no—
119#407—24no—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—22no—
122#391—21no—
123#391—21no—
124#391—21no—
125#391—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#385—22no—
135#385—22no—
136#385—22no—
137#385—22no—
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C2×#95C2×C412yes—
155C2×#95C2×C412yes—
156C2×#95C2×C412yes—
157C2×#95C2×C412yes—
158C2×#95C2×C412yes—
159#343—28no—
160#343—28no—
161#343—44no—
162#336—22no—
163#336—22no—
164C25C22×C411yescharacteristic
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8C2×D412yes—
174C22×Q8C2×Q812yes—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22×C412yes—
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C412yes—
199C23×C4C22×C412yes—
200C23×C4C22×C412yes—
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D412yes—
207C23×C4C2×Q812yes—
208C23×C4#4112yes—
209C23×C4#4112yes—
210C23×C4C2411yescharacteristic
211C2×M16—28no—
212C2×M16—24no—
213C2×M16—24no—
214C2×M16—24no—
215C2×M16—24no—
216C2×M16—24no—
217C2×M16—24no—
218C2×M16—24no—
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—24no—
222C2×M16—24no—
223C2×M16—416no—
224C2×M16—416no—
225C2×M16—44no—
226C2×M16—44no—
227C2×M16C2×D418yes—
228C22×C8—28no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—44no—
232C22×C8—44no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C4×Q8—42no—
242C4×Q8—42no—
243C4×Q8—44no—
244C4×Q8—44no—
245C4×Q8—42no—
246C4×Q8—42no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C4⋊C418yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×Q812yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C4⋊C418yes—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—28no—
299C2×C42—44no—
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22⋊C411yescharacteristic
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22⋊C411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D414yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42#4114yes—
318C4⋊C8—28no—
319C4⋊C8—28no—
320C4⋊C8—28no—
321C4⋊C8—28no—
322C4⋊C8—44no—
323C4⋊C8—44no—
324#101—416no—
325C22⋊C8—44no—
326C22⋊C8—44no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—41no—
342C24—41no—
343C24C2×C22⋊C411yescharacteristic
344C24C2×C22⋊C411yescharacteristic
345C24C2×C4⋊C412yes—
346C24#11712yes—
347C24C23×C411yescharacteristic
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—41no—
357C2×Q8—42no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4#10114yescomplemented
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C4⋊C414yes—
451C22×C4C2×C4⋊C412yes—
452C22×C4C2×C4⋊C414yes—
453C22×C4#11712yes—
454C22×C4C4×D414yes—
455C22×C4C4×D414yes—
456C22×C4#12112yes—
457C22×C4#12112yes—
458C22×C4#12112yes—
459C22×C4#12112yes—
460C22×C4C22⋊Q812yes—
461C22×C4C22⋊Q812yes—
462C22×C4C22⋊Q812yes—
463C22×C4C22⋊Q812yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×Q811yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474M16—28no—
475M16—28no—
476M16—416no—
477M16—44no—
478M16—44no—
479M16—44no—
480M16—44no—
481M16—44no—
482M16—44no—
483C2×C8—28no—
484C2×C8—416no—
485C2×C8—44no—
486C2×C8—44no—
487C2×C8—44no—
488C2×C8—44no—
489C2×C8—44no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—44no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C42—21no—
511C42—21no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—44no—
525C42—44no—
526C42—44no—
527C42—44no—
528C42#12012yes—
529C42#12012yes—
530C42#12114yes—
531C42#12114yes—
532C42#12312yes—
533C42#12312yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23#38012yes—
559C23#38012yes—
560C23C2×#10112yes—
561C23#41312yes—
562C23C22×C22⋊C411yescharacteristic
563C23C22×C4⋊C411yescharacteristic
564C23C2×#11711yescharacteristic
565C23C2×C4×D412yes—
566C23#52012yes—
567Q8—42no—
568Q8—42no—
569Q8—44no—
570Q8—44no—
571Q8—42no—
572Q8—42no—
573C2×C4—28no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4#38012yes—
638C2×C4#38012yes—
639C2×C4#38612yes—
640C2×C4#38612yes—
641C2×C4#38612yes—
642C2×C4#38612yes—
643C2×C4C2×#10112yes—
644C2×C4C2×#10112yes—
645C2×C4#41312yes—
646C2×C4#41312yes—
647C2×C4C2×#12011yescharacteristic
648C2×C4C2×#12111yescharacteristic
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×#12112yes—
651C2×C4C2×C22⋊Q811yescharacteristic
652C2×C4C2×C22⋊Q811yescharacteristic
653C2×C4C2×#12311yescharacteristic
654C2×C4#53511yescharacteristic
655C2×C4#53611yescharacteristic
656C2×C4#53712yes—
657C2×C4D4212yes—
658C2×C4#54612yes—
659C2×C4#54712yes—
660C2×C4#54812yes—
661C2×C4D4×Q812yes—
662C2×C4#55212yes—
663C2×C4#55312yes—
664C2×C4#55412yes—
665C8—48no—
666C8—84no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—48no—
685C22—48no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22#180114yescomplemented
691C22C2×#38011yescharacteristic
692C22C2×#38611yescharacteristic
693C22#231111yescharacteristic
694C22D4×C4⋊C412yes—
695C22#235212yes—
696C22C22×#10111yescharacteristic
697C22C2×#41311yescharacteristic
698C22C2×#41311yescharacteristic
699C22C2×#41311yescharacteristic
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—44no—
708C4—44no—
709C4—41no—
710C4—41no—
711C4—82no—
712C4—82no—
713C4#190114yescomplemented
714C4#240811yescharacteristic
715C4#240911yescharacteristic
716C4#241612yes—
717C4#241712yes—
718C4#248611yescharacteristic
719C4#248711yescharacteristic
720C2—24no—
721C2—22no—
722C2—22no—
723C2—42no—
724C2#1387211yescharacteristic
725C2C2×#180111yescharacteristic
726C2#2063711yescharacteristic
727C2C2×#190111yescharacteristic
728C2#2098911yescharacteristic
729C2#2187911yescharacteristic
730C2#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. C2×#1801C2semidirect
6. #11204C2semidirect
7. #11045C2semidirect
8. #10847C2semidirect
439. C22×C4#101semidirect
690. C22#1801semidirect
713. C4#1901semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
C2×C4×C22⋊Q82
C2×#19012
C2×#18012
#206262
#112042
#110452
#108472
#107392