Finite Group Databasebeta

← #519093

Subgroups — #519093

Order 512.

4922 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×C42×D4C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic, complemented
5#14702C212yesmaximal, complemented
6#14161C211yesmaximal, characteristic, complemented
7#14139C212yesmaximal, complemented
8#14054C214yesmaximal, complemented
9#14052C218yesmaximal, complemented
10#13926C211yesmaximal, characteristic, complemented
11#13924C212yesmaximal, complemented
12D4×#95C214yesmaximal
13C4×#391C211yesmaximal, characteristic, complemented
14C2×C4⋊#95C212yesmaximal, complemented
15C2×#554—28no—
16C2×#554—28no—
17C2×#554C4116yescomplemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2214yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2212yes—
24C22×C4×D4C2212yes—
25C22×C4×D4C2212yes—
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2212yes—
29C23×C42C2212yes—
30D4×C4⋊C4—28no—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#391C2214yes—
35C2×#389C2212yes—
36C2×#385C2214yes—
37C2×#384C2211yescharacteristic
38C2×#380C2212yes—
39C42×D4—28no—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2214yes—
45C22×#95C2218yes—
46C2×C43C2211yescharacteristic
47#1446C2214yes—
48#1445C2218yes—
49#1441C2218yes—
50#1441C2214yes—
51#1436C2218yes—
52#1436C2214yes—
53C4⋊#95C2212yes—
54C4⋊#95C2212yes—
55C4⋊#95C2212yes—
56C4⋊#95C2214yes—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2212yes—
59C4⋊#95C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2214yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2218yes—
64C22⋊#95C2218yes—
65C22⋊#95C2218yes—
66C22⋊#95C2218yes—
67C4×#95C2214yes—
68C23×Q8C2311yescharacteristic
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2314yes—
72C24×C4C2314yes—
73C24×C4C2312yes—
74C24×C4C2314yes—
75#554—2128no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8C2×C418yescomplemented
79C2×#125—28no—
80C2×#125—28no—
81C2×#125C2×C4116yescomplemented
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8C2×C4116yescomplemented
87C2×C22⋊Q8C2×C4132yescomplemented
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8C2×C418yescomplemented
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—216no—
97C2×C4×D4C2×C4116yes—
98C2×C4×D4C2×C418yes—
99C2×C4×D4D418yes—
100C2×C4×D4D418yes—
101C2×C4×D4Q8116yescomplemented
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4C2×C4116yescomplemented
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42—28no—
127C22×C42—28no—
128C22×C42—44no—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#391—24no—
139#391—24no—
140#391—48no—
141#391—44no—
142#389—28no—
143#385—216no—
144#384—24no—
145#380—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—216no—
151C2×#95—416no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C43—24no—
167C25C2412yes—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8C22×C414yes—
179C22×Q8C22×C418yes—
180C22×D4C22×C414yes—
181C22×D4C2×D414yes—
182C22×D4C2×D414yes—
183C22×D4C2×Q818yes—
184C22×D4#4118yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—216no—
197C23×C4—28no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—48no—
202C23×C4—48no—
203C23×C4—48no—
204C23×C4C22×C418yes—
205C23×C4C22×C4116yes—
206C23×C4C2×D418yes—
207C23×C4C2×D418yes—
208C23×C4C2×Q8116yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4⋊Q8—232no—
223#125—264no—
224C22⋊Q8—264no—
225C22⋊Q8—2128no—
226C4×Q8—232no—
227C4×D4—264no—
228C4×D4—264no—
229C4×D4—432no—
230C4×D4C22⋊C4164yes—
231C2×C4⋊C4—264no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C4116yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C418yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C4116yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×Q818yes—
272C2×C4⋊C4#4114yes—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4C22×C4116yes—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C2×D418yes—
282C2×C22⋊C4C2×D418yes—
283C2×C22⋊C4C2×Q8116yes—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—432no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42C22×C418yes—
306C2×C42C22×C414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×Q818yes—
310C2×C42#4114yes—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—216no—
319#95—48no—
320#95—48no—
321#95—44no—
322#95—48no—
323#95—48no—
324#95—48no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24C23×C414yes—
332C24C22×D414yes—
333C24C22×D414yes—
334C24C22×Q818yes—
335C24C2×#4118yes—
336C24C2511yescharacteristic
337C2×Q8—28no—
338C2×Q8—216no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—416no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—88no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×D4C2×C22⋊C4116yes—
350C2×D4C4×Q8132yes—
351C2×D4C22⋊Q8132yes—
352C2×D4#124116yes—
353C2×D4C4⋊Q8116yes—
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—22no—
363C22×C4—22no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—232no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—232no—
402C22×C4—232no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—416no—
414C22×C4—432no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—432no—
421C22×C4—416no—
422C22×C4—816no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—816no—
427C22×C4C2×C22⋊C4132yes—
428C22×C4C23×C414yes—
429C22×C4C23×C414yes—
430C22×C4C23×C418yes—
431C22×C4C23×C412yes—
432C22×C4C23×C414yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q818yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q814yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C4⋊C4—28no—
456C4⋊C4—232no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—232no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—84no—
468C4⋊C4C2×C22⋊C4116yes—
469C22⋊C4—232no—
470C22⋊C4—232no—
471C22⋊C4—416no—
472C22⋊C4C2×C22⋊C4132yes—
473C42—216no—
474C42—216no—
475C42—28no—
476C42—48no—
477C42—416no—
478C42—48no—
479C42—48no—
480C42—44no—
481C42C2×C22⋊C4116yes—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23C22×C22⋊C418yes—
501C23C2×C4×Q8116yes—
502C23C2×C22⋊Q8116yes—
503C23C2×#12418yes—
504C23C2×C4⋊Q818yes—
505C23C24×C411yescharacteristic
506C23C23×D411yescharacteristic
507C23C23×D411yescharacteristic
508C23C23×Q812yes—
509C23C22×#4112yes—
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+411yescharacteristic
515C23#58512yes—
516Q8—416no—
517Q8—816no—
518D4#391164yescomplemented
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—264no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—216no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—464no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—832no—
584C2×C4—832no—
585C2×C4C22×C22⋊C414yes—
586C2×C4C22×C22⋊C418yes—
587C2×C4C22×C22⋊C414yes—
588C2×C4C22×C22⋊C414yes—
589C2×C4C2×C4×Q818yes—
590C2×C4C4×#4114yes—
591C2×C4#52014yes—
592C2×C4C2×C22⋊Q818yes—
593C2×C4C2×#12414yes—
594C2×C4C2×C4⋊Q814yes—
595C2×C4#54614yes—
596C2×C4#54614yes—
597C2×C4#54714yes—
598C2×C4#54714yes—
599C2×C4#55418yescomplemented
600C22—216no—
601C22—28no—
602C22—216no—
603C22—28no—
604C22—28no—
605C22—216no—
606C22—216no—
607C22—28no—
608C22—28no—
609C22—28no—
610C22C2×#391116yes—
611C22C23×C22⋊C411yescharacteristic
612C22C22×C4×Q812yes—
613C22C2×C4×#4111yescharacteristic
614C22C2×#52011yescharacteristic
615C22C4×H(5,𝔽2)12yes—
616C22C22×C22⋊Q812yes—
617C22C22×#12411yescharacteristic
618C22C22×C4⋊Q811yescharacteristic
619C22C2×#54611yescharacteristic
620C22C2×#54611yescharacteristic
621C22C2×#54711yescharacteristic
622C22C2×#54711yescharacteristic
623C22C2×#55412yes—
624C22#347512yes—
625C22Q8×#4112yes—
626C22#348112yes—
627C22#348812yes—
628C22#349811yescharacteristic
629C22#350011yescharacteristic
630C22#350112yes—
631C22#350312yes—
632C22#351512yes—
633C22#351912yes—
634C4—28no—
635C4—28no—
636C4—24no—
637C4—44no—
638C4—48no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—48no—
643C4—82no—
644C4—84no—
645C4C2×#39118yes—
646C4#233918yes—
647C2—216no—
648C2C22×#39111yescharacteristic
649C2C2×#233911yescharacteristic
650C2C4×#55412yes—
651C2#3425811yescharacteristic
652C2#3433212yes—
653C2#3615412yes—
654C2#3616411yescharacteristic
655C2#3618011yescharacteristic
656C2#3627512yes—
657C2#3627611yescharacteristic
658C2#3628911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14714C2semidirect
5. #14702C2semidirect
6. #14161C2semidirect
7. #14139C2semidirect
8. #14054C2semidirect
9. #14052C2semidirect
10. #13926C2semidirect
11. #13924C2semidirect
13. C4×#391C2semidirect
14. C2×C4⋊#95C2semidirect
17. C2×#554C4semidirect
78. C2×C4⋊Q8C2×C4semidirect
81. C2×#125C2×C4semidirect
86. C2×C22⋊Q8C2×C4semidirect
87. C2×C22⋊Q8C2×C4semidirect
90. C2×C4×Q8C2×C4semidirect
101. C2×C4×D4Q8semidirect
107. C22×C4⋊C4C2×C4semidirect
518. D4#391semidirect
599. C2×C4#554semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×D4×C4⋊C42
C2×C42×D42
#147142
#147022
#141612
#141392
#140542
#140522
#139262
#139242
D4×#952
C4×#3912
C2×C4⋊#952