Finite Group Databasebeta

← #393310

Subgroups — #393310

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1987C211yesmaximal, characteristic, complemented
5C2×#1798C211yesmaximal, characteristic
6#10885C212yesmaximal, complemented
7#10884C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19#2628—24no—
20#2628C418yescomplemented
21C2×#400—21no—
22C2×#399—21no—
23C2×#395—21no—
24C2×#393—21no—
25C2×#389C2212yes—
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28#2274—24no—
29#2274—24no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4—21no—
38#1988—24no—
39#1987—24no—
40#1798—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C23×Q8—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yescomplemented
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117C2×C414yes—
71C2×#117D414yes—
72C2×#117Q814yescomplemented
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4Q814yescomplemented
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42—21no—
96C22×C42—21no—
97C22×C42—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—24no—
110#400—28no—
111#399—28no—
112#395—28no—
113#393—28no—
114#389—24no—
115#389C2×C418yescomplemented
116#382—22no—
117#382C2×C414yescomplemented
118#380—24no—
119#380C2×C418yes—
120C4×C4⋊C4—28no—
121C4×C4⋊C4—28no—
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⋊C4D414yes—
129C4×C4⋊C4D414yes—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—28no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—24no—
146#340—24no—
147#340—24no—
148#336—24no—
149C25C2×D411yescharacteristic
150C22×Q8—24no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—82no—
154C22×Q8—82no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4C22×C412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×Q812yes—
182C23×C4C2×Q812yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C4⋊Q8—28no—
201C4⋊Q8—28no—
202C4⋊Q8—44no—
203C4⋊Q8—44no—
204#117—28no—
205#117—28no—
206#117—28no—
207#117—28no—
208#117—28no—
209#117—28no—
210#117—28no—
211#117—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4C22⋊C414yes—
252C2×C4⋊C4C22⋊C414yes—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×Q812yes—
260C2×C4⋊C4C2×Q812yes—
261C2×C4⋊C4C2×Q812yes—
262C2×C4⋊C4C2×Q812yes—
263C2×C4⋊C4#4112yes—
264C2×C4⋊C4#4112yes—
265C2×C4⋊C4#4112yes—
266C2×C4⋊C4#4112yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4—82no—
277C2×C22⋊C4C22⋊C414yes—
278C2×C22⋊C4C22⋊C414yes—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—82no—
291C2×C42—82no—
292C2×C42—82no—
293C2×C42—82no—
294C2×C42C22×C412yes—
295C2×C42C22×C412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42#4112yes—
301C2×C42#4112yes—
302C4⋊C8—44no—
303C4⋊C8—44no—
304C22⋊C8—48no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—24no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24C2×C22⋊C412yes—
322C24C22⋊Q812yes—
323C24C22⋊Q812yes—
324C24C22×D411yescharacteristic
325C2×Q8—48no—
326C2×Q8—88no—
327C2×Q8—88no—
328C2×Q8—84no—
329C2×Q8—84no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
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—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C4×Q814yes—
415C22×C4C4×Q814yes—
416C22×C4#12012yes—
417C22×C4#12012yes—
418C22×C4C22⋊Q812yes—
419C22×C4C22⋊Q812yes—
420C22×C4#12312yes—
421C22×C4#12312yes—
422C22×C4#12414yes—
423C22×C4C4⋊Q814yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×Q811yescharacteristic
429C22×C4C22×Q811yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C4⋊C4—216no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—48no—
476C4⋊C4—88no—
477C4⋊C4C22⋊Q814yes—
478C4⋊C4C22⋊Q814yes—
479C4⋊C4C22⋊Q814yes—
480C4⋊C4C22⋊Q814yes—
481C22⋊C4—216no—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—88no—
485C42—216no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—84no—
491C42—84no—
492C42—84no—
493C42—84no—
494C42C4×D414yes—
495C42C4×D414yes—
496C42#12114yes—
497C42#12114yes—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23#39114yes—
521C23C22×C22⋊C411yescharacteristic
522C23C2×#12011yescharacteristic
523C23C2×C22⋊Q811yescharacteristic
524C23C2×C22⋊Q811yescharacteristic
525C23C2×#12311yescharacteristic
526C23#52511yescharacteristic
527C23#52511yescharacteristic
528C23#57612yes—
529C23#57712yes—
530C23#57712yes—
531C23#57812yes—
532Q8—816no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—816no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4#38614yes—
601C2×C4#38614yes—
602C2×C4#39114yes—
603C2×C4C2×C4×D411yescharacteristic
604C2×C4C2×C4×D411yescharacteristic
605C2×C4C2×C4×Q811yescharacteristic
606C2×C4C2×C4×Q811yescharacteristic
607C2×C4C4×#4111yescharacteristic
608C2×C4C4×#4111yescharacteristic
609C2×C4C2×#12111yescharacteristic
610C2×C4C2×#12111yescharacteristic
611C2×C4C2×C22⋊Q811yescharacteristic
612C2×C4C2×C22⋊Q811yescharacteristic
613C2×C4C2×C22⋊Q811yescharacteristic
614C2×C4C2×C22⋊Q811yescharacteristic
615C2×C4#52511yescharacteristic
616C2×C4#52511yescharacteristic
617C2×C4#52511yescharacteristic
618C2×C4#52511yescharacteristic
619C2×C4C2×#12411yescharacteristic
620C2×C4#52911yescharacteristic
621C2×C4#52911yescharacteristic
622C2×C4#52911yescharacteristic
623C2×C4#52911yescharacteristic
624C2×C4C2×C4⋊Q811yescharacteristic
625C2×C4#53211yescharacteristic
626C2×C4#53211yescharacteristic
627C2×C4#53311yescharacteristic
628C2×C4#53311yescharacteristic
629C2×C4#53311yescharacteristic
630C2×C4#53311yescharacteristic
631C8—82no—
632C8—82no—
633C22—28no—
634C22—24no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C2×#38611yescharacteristic
645C22C2×#39111yescharacteristic
646C22#229711yescharacteristic
647C22#284811yescharacteristic
648C22#284811yescharacteristic
649C22#285511yescharacteristic
650C22#285511yescharacteristic
651C22#289712yes—
652C22#289812yes—
653C22#300811yescharacteristic
654C22#300911yescharacteristic
655C22#301011yescharacteristic
656C22#301111yescharacteristic
657C22#308112yes—
658C22#308212yes—
659C22#308312yes—
660C22#308412yes—
661C22#309612yes—
662C22#309712yes—
663C22#311812yes—
664C22#312112yes—
665C22#312412yes—
666C22#312512yes—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—84no—
677C4—84no—
678C4C4×C22⋊Q811yescharacteristic
679C4C4×C22⋊Q811yescharacteristic
680C4C4×C22⋊Q811yescharacteristic
681C4C4×C22⋊Q811yescharacteristic
682C4#285011yescharacteristic
683C4#285011yescharacteristic
684C4#285711yescharacteristic
685C4#285711yescharacteristic
686C2—44no—
687C2#1470211yescharacteristic
688C2#2091311yescharacteristic
689C2#2091411yescharacteristic
690C2#2131512yes—
691C2#2139712yes—
692C2#3296211yescharacteristic
693C2#3296311yescharacteristic
694C2#3296411yescharacteristic
695C2#3296511yescharacteristic
696C2#3315512yes—
697C2#3315912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1988C2semidirect
4. C2×#1987C2semidirect
6. #10885C2semidirect
7. #10884C2semidirect
8. #10850C2semidirect
9. #10850C2semidirect
20. #2628C4semidirect
60. C2×C4⋊Q8C2×C4semidirect
72. C2×#117Q8semidirect
83. C22×C4⋊C4Q8semidirect
115. #389C2×C4semidirect
117. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#22742
C2×#19882
C2×#19872
C2×#17982
#108852
#108842
#108502
#107312
#107302