Finite Group Databasebeta

← #6477447

Subgroups — #6477447

Order 512.

3304 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#3189C211yesmaximal, characteristic, complemented
3C22×#484C211yesmaximal, characteristic, complemented
4#35010C211yesmaximal, characteristic
5#33010C212yesmaximal, complemented
6#32983C212yesmaximal, complemented
7#29642C212yesmaximal, complemented
8#29597C212yesmaximal, complemented
9#25545C218yesmaximal, complemented
10#24098C214yesmaximal, complemented
11#23970C214yesmaximal, complemented
12#21759C211yesmaximal, characteristic, complemented
13#21738C211yesmaximal, characteristic
14C2×#1824C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—22no—
17C2×#554—22no—
18C2×#554—28no—
19C2×#554C2214yes—
20C2×#554C2214yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125—21no—
23C22×C22⋊Q8—21no—
24C22×C22⋊Q8—21no—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4—21no—
30C22×C4×D4C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3189—28no—
34C2×#484—28no—
35C2×#484C2212yes—
36C2×#484C2212yes—
37C2×#484C2212yes—
38C2×#484C2212yescomplemented
39C2×#484C2214yescomplemented
40C2×#467C2214yescomplemented
41C2×#451C2212yes—
42C2×#451C2212yescomplemented
43C22×#107C2212yescomplemented
44C22×#103C2212yescomplemented
45C22×C22⋊C8C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2627—21no—
50#2626—21no—
51#2579—22no—
52#2564—22no—
53#2487C2212yes—
54#2417C2214yes—
55#2409C2212yes—
56#2315—21no—
57#2308—21no—
58C2×#380C2211yescharacteristic
59#2057C2218yescomplemented
60#2036C2218yescomplemented
61#2024C2218yescomplemented
62#2001C2218yescomplemented
63#1892C2214yescomplemented
64#1878C2218yescomplemented
65#1872C2218yescomplemented
66#1872C2214yescomplemented
67#1869C2214yes—
68#1869C2214yes—
69#1863C2218yescomplemented
70#1852C2218yescomplemented
71#1824C2212yes—
72#1824C2212yes—
73#1824C2212yes—
74#1824C2214yescomplemented
75#1817C2212yes—
76#1787C2212yes—
77#1779C2212yes—
78C2×#340C2214yescomplemented
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4D411yescharacteristic
83C24×C4—21no—
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C22×Q16C2314yes—
88C23×C8C2312yes—
89C23×C8C2312yes—
90#554—232no—
91#554—232no—
92#554—216no—
93#554—216no—
94#554—216no—
95#554—216no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8C2314yes—
100C2×#125—216no—
101C2×#125—24no—
102C2×#125—24no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—216no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8D418yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2314yes—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125C2×C22⋊Q8C2314yes—
126C2×C4×Q8—24no—
127C2×C4×Q8—24no—
128C2×C4×Q8—22no—
129C2×C4×Q8C2314yes—
130C2×C4×D4—216no—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4D412yes—
136C2×C4×D4D412yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2312yes—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—24no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—24no—
150C22×C4⋊C4—24no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4D411yescharacteristic
153C22×C4⋊C4D418yes—
154C22×C4⋊C4C2311yescharacteristic
155C22×C4⋊C4C2314yes—
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2312yes—
161C22×C4⋊C4C2312yes—
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4D411yescharacteristic
167C22×C22⋊C4D411yescharacteristic
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170#484—28no—
171#484—28no—
172#484—28no—
173#484—216no—
174#484—28no—
175#467—216no—
176#451—28no—
177#451—28no—
178C2×#107—28no—
179C2×#107C2314yes—
180C2×#107C2314yes—
181C2×#107C2314yes—
182C2×#103—28no—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2318yes—
188C2×C22⋊C8—24no—
189C2×C22⋊C8—24no—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2318yes—
195#400—22no—
196#399—22no—
197#398—24no—
198#396—24no—
199#395—22no—
200#393—22no—
201#391C2314yes—
202#389C2314yes—
203#387—24no—
204#386—24no—
205#386—22no—
206#386—22no—
207#385—24no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#381—22no—
213#380—21no—
214#380—21no—
215#380—24no—
216#380C2312yes—
217#380C2312yes—
218#380C2314yes—
219#379—22no—
220C2×#95—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C25C2×D411yescharacteristic
227C22×Q8—28no—
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8—84no—
237C22×Q8C2×D414yes—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×D4—24no—
241C22×D4—24no—
242C22×D4Q1618yescomplemented
243C22×D4C2×D412yes—
244C22×D4C2×D412yes—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—28no—
250C23×C4—28no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—84no—
262C23×C4—84no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D414yes—
271C23×C4C2×D414yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C2×Q16—28no—
277C2×Q16—48no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8—44no—
281C22×C8—44no—
282C22×C8C2414yes—
283C22×C8C2414yes—
284C4⋊Q8—216no—
285C4⋊Q8—28no—
286C4⋊Q8—28no—
287#125—232no—
288#125—232no—
289C22⋊Q8—216no—
290C22⋊Q8—232no—
291C22⋊Q8—232no—
292C22⋊Q8—232no—
293C22⋊Q8—232no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C22⋊Q8—48no—
307C22⋊Q8C2×D418yes—
308C22⋊Q8C2×D418yes—
309C4×Q8—216no—
310C4×Q8—28no—
311C4×Q8—28no—
312C4×D4—232no—
313C4×D4—232no—
314C4×D4—28no—
315C4×D4—28no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—24no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—216no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D418yes—
364C2×C4⋊C4C2×D418yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—44no—
384C2×C22⋊C4—44no—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C22⋊C4C2×D412yes—
388C2×C22⋊C4C2×D412yes—
389C2×C42—28no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—24no—
393C2×C42C2×D412yes—
394C2×C42C2412yes—
395#107—28no—
396#107—28no—
397#107—28no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—28no—
402#103—48no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—48no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414C24—22no—
415C24—22no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24#12014yes—
421C24C2×Q1614yes—
422C24C2×Q1614yes—
423C24#13714yes—
424C24C22×D412yes—
425C24C22×D411yescharacteristic
426C24C22×D411yescharacteristic
427C24C22×D411yescharacteristic
428C2×Q8—216no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—28no—
434C2×Q8—28no—
435C2×Q8—48no—
436C2×Q8—44no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8—42no—
440C2×Q8—82no—
441C2×Q8—88no—
442C2×Q8—84no—
443C2×Q8—816no—
444C2×Q8C22×D414yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—232no—
450C2×D4C2×Q1618yes—
451C2×D4C2×Q1618yes—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—216no—
468C22×C4—28no—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—28no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—82no—
495C22×C4—84no—
496C22×C4—84no—
497C22×C4—88no—
498C22×C4—82no—
499C22×C4—816no—
500C22×C4—816no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4#12014yes—
504C22×C4#12014yes—
505C22×C4#12014yes—
506C22×C4C2×Q1614yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C2511yescharacteristic
520Q16—88no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—28no—
524C2×C8—28no—
525C2×C8—48no—
526C2×C8—84no—
527C4⋊C4—216no—
528C4⋊C4—216no—
529C4⋊C4—216no—
530C4⋊C4—216no—
531C4⋊C4—24no—
532C4⋊C4—28no—
533C4⋊C4—28no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—28no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—84no—
550C4⋊C4—82no—
551C4⋊C4—84no—
552C4⋊C4C22×D414yes—
553C4⋊C4C22×D414yes—
554C4⋊C4C22×D414yes—
555C22⋊C4—216no—
556C22⋊C4—28no—
557C22⋊C4—28no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—84no—
567C22⋊C4—84no—
568C42—216no—
569C42—24no—
570C42—22no—
571C42—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—22no—
580C23—24no—
581C23—28no—
582C23—22no—
583C23—44no—
584C23—44no—
585C23—48no—
586C23—48no—
587C23—48no—
588C23—44no—
589C23—44no—
590C23—48no—
591C23#451116yes—
592C23C2×#12012yes—
593C23C2×#12012yes—
594C23C2×#12012yes—
595C23C22×Q1614yes—
596C23C22×Q1612yes—
597C23C22×Q1612yes—
598C23C22×Q1612yes—
599C23C2×#13712yes—
600C23C2×#13712yes—
601C23C2×#13712yes—
602C23C23×D411yescharacteristic
603Q8—28no—
604Q8—48no—
605Q8—44no—
606Q8—44no—
607Q8—44no—
608Q8—84no—
609Q8—816no—
610D4—216no—
611D4—216no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—48no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—88no—
656C2×C4—88no—
657C2×C4—816no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—164no—
661C2×C4C2×#12012yes—
662C2×C4C2×#12012yes—
663C2×C4C2×#12012yes—
664C2×C4C22×Q1614yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C23×D411yescharacteristic
667C2×C4#58514yes—
668C8—44no—
669C8—44no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—28no—
683C22—24no—
684C22—416no—
685C22—416no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22C2×#45114yes—
690C22C2×#45114yes—
691C22C2×#45114yes—
692C22C22×#12011yescharacteristic
693C22#348512yes—
694C22C23×Q1611yescharacteristic
695C22C22×#13711yescharacteristic
696C22#358512yes—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—24no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—84no—
708C4—82no—
709C4—82no—
710C4#344912yes—
711C4#348512yes—
712C2—24no—
713C2—24no—
714C2—24no—
715C2—44no—
716C2#3508811yescharacteristic
717C2C22×#45111yescharacteristic
718C2#6036711yescharacteristic
719C2#6169812yes—
720C2#6170912yes—
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
2. C2×#3189C2semidirect
3. C22×#484C2semidirect
5. #33010C2semidirect
6. #32983C2semidirect
7. #29642C2semidirect
8. #29597C2semidirect
9. #25545C2semidirect
10. #24098C2semidirect
11. #23970C2semidirect
12. #21759C2semidirect
14. C2×#1824C2semidirect
38. C2×#484C22semidirect
39. C2×#484C22semidirect
40. C2×#467C22semidirect
42. C2×#451C22semidirect
43. C22×#107C22semidirect
44. C22×#103C22semidirect
59. #2057C22semidirect
60. #2036C22semidirect
61. #2024C22semidirect
62. #2001C22semidirect
63. #1892C22semidirect
64. #1878C22semidirect
65. #1872C22semidirect
66. #1872C22semidirect
69. #1863C22semidirect
70. #1852C22semidirect
74. #1824C22semidirect
78. C2×#340C22semidirect
242. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#31892
C22×#4842
#350102
#330102
#329832
#296422
#295972
#255452
#240982
#239702
#217592
#217382
C2×#18242