Finite Group Databasebeta

← #1393182

Subgroups — #1393182

Order 512.

5237 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#2443C213yesmaximal
3C2×#2614C211yesmaximal, characteristic, complemented
4C22×#395C211yesmaximal, characteristic, complemented
5#16511C213yesmaximal, complemented
6#16507C213yesmaximal, complemented
7#16262C213yesmaximal, complemented
8#16251C213yesmaximal, complemented
9#15176C211yesmaximal, characteristic, complemented
10#14874C211yesmaximal, characteristic, complemented
11#14873C211yesmaximal, characteristic, complemented
12#14759C213yesmaximal, complemented
13#14720C213yesmaximal, complemented
14#14302C211yesmaximal, characteristic, complemented
15#13894C213yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18C2×D4×Q8—224no—
19C22×C4⋊Q8C2213yes—
20C22×#123C2213yes—
21C22×C22⋊Q8C2213yes—
22C22×C22⋊Q8C2213yes—
23C22×#121C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2213yes—
26C23×C4⋊C4C2213yes—
27#2614—28no—
28#2443—224no—
29C2×#399C2213yes—
30C2×#398C2213yes—
31C2×#397C2213yes—
32C2×#395—28no—
33C2×#395C2211yescharacteristic
34C2×#395C2213yes—
35C2×#395C2213yes—
36C2×#395C2213yes—
37C2×#395C2213yes—
38C2×#395C2211yescharacteristic
39C2×#394C2213yes—
40C2×#393C2211yescharacteristic
41C2×#393C2213yes—
42C2×#393C2213yes—
43C2×#388C2213yes—
44C2×#387C2213yes—
45C2×#386C2213yes—
46C2×#386C2213yes—
47C2×#386C2213yes—
48C2×#385C2213yes—
49C2×#385C2213yes—
50C2×#381C2213yes—
51C22×#95C2213yes—
52#1446C2211yescharacteristic
53#1446C2213yes—
54#1446C2213yes—
55#1445C2216yes—
56#1444C2216yes—
57#1444C2216yes—
58#1442C2216yes—
59#1442C2213yes—
60#1441C2213yes—
61#1441C2213yes—
62#1440C2216yes—
63#1439C2213yes—
64#1439C2216yes—
65#1434C2213yes—
66#1434C2213yes—
67C22⋊#95C2213yes—
68C22⋊#95C2213yes—
69C22⋊#95C2216yes—
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—212no—
73C23×Q8—212no—
74C23×Q8—46no—
75C23×Q8—46no—
76C23×Q8C2311yescharacteristic
77C23×Q8C2313yes—
78C23×Q8C2313yes—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2313yes—
83C24×C4C2311yescharacteristic
84C24×C4C2313yes—
85C24×C4C2313yes—
86D4×Q8—2128no—
87C2×C4⋊Q8—224no—
88C2×C4⋊Q8—212no—
89C2×#123—212no—
90C2×#123—212no—
91C2×#123—46no—
92C2×C22⋊Q8—212no—
93C2×C22⋊Q8—212no—
94C2×C22⋊Q8—224no—
95C2×C22⋊Q8—224no—
96C2×#121—212no—
97C2×C4×Q8—212no—
98C2×C4×D4—224no—
99C2×C4×D4D4124yes—
100C22×C4⋊C4—212no—
101C22×C4⋊C4—212no—
102C22×C4⋊C4—46no—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2313yes—
106C22×C4⋊C4C2316yes—
107C22×C4⋊C4C2313yes—
108C22×C4⋊C4C2313yes—
109C22×C4⋊C4C2313yes—
110C22×C4⋊C4C2313yes—
111C22×C4⋊C4C2313yes—
112C22×C4⋊C4C2313yes—
113C22×C22⋊C4C2313yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2313yes—
117C22×C22⋊C4C2313yes—
118C22×C22⋊C4C2313yes—
119C22×C22⋊C4C2313yes—
120C22×C22⋊C4C2313yes—
121C22×C22⋊C4C2313yes—
122C22×C22⋊C4C2313yes—
123C22×C22⋊C4C2313yes—
124C22×C42C2313yes—
125#399—212no—
126#398—212no—
127#397—212no—
128#395—24no—
129#395—212no—
130#395—212no—
131#395—212no—
132#395—46no—
133#395D418yescomplemented
134#394—212no—
135#393—24no—
136#393—212no—
137#393—46no—
138#388—212no—
139#387—212no—
140#386—212no—
141#386—212no—
142#386—212no—
143#385—212no—
144#385—212no—
145#381—212no—
146C2×#95—212no—
147C2×#95C2313yes—
148C2×#95C2313yes—
149C2×#95C2316yes—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2316yes—
153C2×#95C2313yes—
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2313yes—
159C2×#95C2316yes—
160C2×#95C2313yes—
161C2×#95C2316yes—
162C2×#95C2316yes—
163C2×#95C2316yes—
164C2×#95C2313yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C22×Q8—232no—
169C22×Q8—232no—
170C22×Q8—26no—
171C22×Q8—212no—
172C22×Q8—212no—
173C22×Q8—26no—
174C22×Q8—26no—
175C22×Q8—26no—
176C22×Q8—26no—
177C22×Q8—26no—
178C22×Q8—26no—
179C22×Q8—26no—
180C22×Q8—46no—
181C22×Q8—46no—
182C22×Q8—448no—
183C22×Q8—448no—
184C22×Q8—448no—
185C22×Q8—448no—
186C22×Q8C2×D414yes—
187C22×D4—26no—
188C22×D4—22no—
189C22×D4—26no—
190C22×D4—26no—
191C22×D4—22no—
192C22×D4—26no—
193C22×D4—22no—
194C22×D4—43no—
195C22×D4C2×D4112yes—
196C23×C4—212no—
197C23×C4—212no—
198C23×C4—26no—
199C23×C4—26no—
200C23×C4—26no—
201C23×C4—26no—
202C23×C4—26no—
203C23×C4—26no—
204C23×C4—26no—
205C23×C4—26no—
206C23×C4—26no—
207C23×C4—26no—
208C23×C4—22no—
209C23×C4—43no—
210C23×C4—43no—
211C23×C4—46no—
212C23×C4—46no—
213C23×C4—43no—
214C23×C4C2×D4112yes—
215C23×C4C2×D4112yes—
216C23×C4C2413yes—
217C23×C4C2413yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2413yes—
221C23×C4C2413yes—
222C23×C4C2413yes—
223C23×C4C2413yes—
224C23×C4C2411yescharacteristic
225C23×C4C2413yes—
226C23×C4C2411yescharacteristic
227C23×C4C2413yes—
228C4⋊Q8—296no—
229#123—448no—
230C22⋊Q8—296no—
231C22⋊Q8—296no—
232#121—416no—
233C4×Q8—232no—
234C4×D4—296no—
235C2×C4⋊C4—26no—
236C2×C4⋊C4—212no—
237C2×C4⋊C4—212no—
238C2×C4⋊C4—26no—
239C2×C4⋊C4—212no—
240C2×C4⋊C4—212no—
241C2×C4⋊C4—212no—
242C2×C4⋊C4—26no—
243C2×C4⋊C4—26no—
244C2×C4⋊C4—26no—
245C2×C4⋊C4—26no—
246C2×C4⋊C4—26no—
247C2×C4⋊C4—26no—
248C2×C4⋊C4—26no—
249C2×C4⋊C4—26no—
250C2×C4⋊C4—26no—
251C2×C4⋊C4—26no—
252C2×C4⋊C4—26no—
253C2×C4⋊C4—26no—
254C2×C4⋊C4—448no—
255C2×C4⋊C4—43no—
256C2×C4⋊C4—43no—
257C2×C4⋊C4—43no—
258C2×C4⋊C4—46no—
259C2×C4⋊C4C2×D4112yes—
260C2×C4⋊C4C2×D4112yes—
261C2×C22⋊C4—212no—
262C2×C22⋊C4—212no—
263C2×C22⋊C4—26no—
264C2×C22⋊C4—26no—
265C2×C22⋊C4—26no—
266C2×C22⋊C4—26no—
267C2×C22⋊C4—212no—
268C2×C22⋊C4—26no—
269C2×C22⋊C4—26no—
270C2×C22⋊C4—26no—
271C2×C22⋊C4—26no—
272C2×C22⋊C4—26no—
273C2×C22⋊C4—26no—
274C2×C22⋊C4—26no—
275C2×C22⋊C4—26no—
276C2×C22⋊C4—26no—
277C2×C22⋊C4—26no—
278C2×C22⋊C4—26no—
279C2×C22⋊C4—43no—
280C2×C22⋊C4—43no—
281C2×C22⋊C4—43no—
282C2×C22⋊C4—43no—
283C2×C22⋊C4—46no—
284C2×C22⋊C4C2×D4112yes—
285C2×C22⋊C4C2×D4112yes—
286C2×C42—212no—
287C2×C42C2×D4112yes—
288#95—26no—
289#95—212no—
290#95—212no—
291#95—26no—
292#95—26no—
293#95—26no—
294#95—26no—
295#95—26no—
296#95—26no—
297#95—26no—
298#95—26no—
299#95—26no—
300#95—26no—
301#95—212no—
302#95—46no—
303#95—46no—
304#95—46no—
305#95C2×D4112yes—
306C24—21no—
307C24—23no—
308C24—23no—
309C24—21no—
310C24—21no—
311C24—23no—
312C24—23no—
313C24—23no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—23no—
318C24—23no—
319C24—23no—
320C24—23no—
321C24—21no—
322C24—21no—
323C24C22×D416yes—
324C24C22×D416yes—
325C242−1+416yes—
326C242−1+412yes—
327C24C2511yescharacteristic
328C2×Q8—28no—
329C2×Q8—224no—
330C2×Q8—224no—
331C2×Q8—224no—
332C2×Q8—224no—
333C2×Q8—424no—
334C2×Q8—424no—
335C2×Q8—424no—
336C2×Q8—424no—
337C2×Q8—412no—
338C2×Q8—412no—
339C2×Q8—4128no—
340C2×Q8—4128no—
341C2×Q8—412no—
342C2×Q8—412no—
343C2×D4—224no—
344C2×D4—224no—
345C2×D4—412no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—412no—
349C2×D4—412no—
350C2×D4—86no—
351C2×D4#120116yes—
352C22×C4—26no—
353C22×C4—26no—
354C22×C4—23no—
355C22×C4—21no—
356C22×C4—224no—
357C22×C4—224no—
358C22×C4—224no—
359C22×C4—23no—
360C22×C4—23no—
361C22×C4—23no—
362C22×C4—23no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—26no—
366C22×C4—26no—
367C22×C4—26no—
368C22×C4—23no—
369C22×C4—26no—
370C22×C4—26no—
371C22×C4—23no—
372C22×C4—23no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—26no—
376C22×C4—23no—
377C22×C4—23no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4—23no—
385C22×C4—224no—
386C22×C4—224no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—21no—
393C22×C4—26no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—26no—
397C22×C4—26no—
398C22×C4—26no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—21no—
402C22×C4—23no—
403C22×C4—412no—
404C22×C4—412no—
405C22×C4—412no—
406C22×C4—412no—
407C22×C4—412no—
408C22×C4—412no—
409C22×C4—424no—
410C22×C4—424no—
411C22×C4—424no—
412C22×C4—424no—
413C22×C4—412no—
414C22×C4—412no—
415C22×C4—412no—
416C22×C4—44no—
417C22×C4—86no—
418C22×C4C22×D416yes—
419C22×C4C22×D416yes—
420C22×C4C22×D416yes—
421C22×C4C22×D416yes—
422C22×C4C22×D412yes—
423C22×C4C22×D416yes—
424C22×C4C22×D416yes—
425C22×C4C22×D416yes—
426C22×C4H(5,𝔽2)16yes—
427C22×C42−1+416yes—
428C4⋊C4—224no—
429C4⋊C4—248no—
430C4⋊C4—224no—
431C4⋊C4—224no—
432C4⋊C4—412no—
433C4⋊C4—412no—
434C4⋊C4—412no—
435C4⋊C4—44no—
436C4⋊C4—412no—
437C4⋊C4—86no—
438C22⋊C4—224no—
439C22⋊C4—224no—
440C22⋊C4—412no—
441C22⋊C4—412no—
442C22⋊C4—412no—
443C22⋊C4—412no—
444C22⋊C4—44no—
445C22⋊C4—412no—
446C22⋊C4—44no—
447C22⋊C4—412no—
448C22⋊C4—86no—
449C42—224no—
450C23—26no—
451C23—26no—
452C23—26no—
453C23—22no—
454C23—26no—
455C23—26no—
456C23—26no—
457C23—26no—
458C23—212no—
459C23—26no—
460C23—26no—
461C23—26no—
462C23—26no—
463C23—26no—
464C23—26no—
465C23—43no—
466C23—41no—
467C23—43no—
468C23—43no—
469C23—46no—
470C23—41no—
471C23—43no—
472C23—43no—
473C23—43no—
474C23—43no—
475C23—43no—
476C23—43no—
477C23—41no—
478C23—43no—
479C23—43no—
480C23—43no—
481C23—46no—
482C23C2×#12014yes—
483C23C2×#12014yes—
484C23#536112yes—
485C23#536112yes—
486C23C23×D413yes—
487C23C23×D411yescharacteristic
488C23C2×H(5,𝔽2)13yes—
489C23C2×2−1+413yes—
490C23C2×2−1+411yescharacteristic
491C23C2×2−1+413yes—
492C23#58511yescharacteristic
493Q8—232no—
494Q8—448no—
495Q8—448no—
496D4—232no—
497D4—88no—
498D4—88no—
499C2×C4—26no—
500C2×C4—26no—
501C2×C4—26no—
502C2×C4—212no—
503C2×C4—26no—
504C2×C4—212no—
505C2×C4—26no—
506C2×C4—26no—
507C2×C4—26no—
508C2×C4—26no—
509C2×C4—26no—
510C2×C4—26no—
511C2×C4—26no—
512C2×C4—26no—
513C2×C4—26no—
514C2×C4—26no—
515C2×C4—26no—
516C2×C4—26no—
517C2×C4—26no—
518C2×C4—26no—
519C2×C4—26no—
520C2×C4—26no—
521C2×C4—448no—
522C2×C4—448no—
523C2×C4—448no—
524C2×C4—41no—
525C2×C4—43no—
526C2×C4—43no—
527C2×C4—43no—
528C2×C4—46no—
529C2×C4—46no—
530C2×C4—46no—
531C2×C4—46no—
532C2×C4—43no—
533C2×C4—46no—
534C2×C4—43no—
535C2×C4—43no—
536C2×C4—46no—
537C2×C4—43no—
538C2×C4—43no—
539C2×C4—43no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—43no—
543C2×C4—43no—
544C2×C4—43no—
545C2×C4—43no—
546C2×C4—43no—
547C2×C4—43no—
548C2×C4—43no—
549C2×C4—43no—
550C2×C4—46no—
551C2×C4—43no—
552C2×C4—43no—
553C2×C4—46no—
554C2×C4—46no—
555C2×C4—46no—
556C2×C4—88no—
557C2×C4C2×#12014yes—
558C2×C4D42112yes—
559C22—212no—
560C22—212no—
561C22—212no—
562C22—212no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—46no—
569C22—46no—
570C22—46no—
571C22—46no—
572C22—83no—
573C22—81no—
574C22—83no—
575C22—83no—
576C22—83no—
577C22—81no—
578C22—81no—
579C22#261418yescomplemented
580C22C22×#12011yescharacteristic
581C22C2×#53613yes—
582C22C2×#53613yes—
583C22C2×D4213yes—
584C22#348413yes—
585C22#348613yes—
586C22#348911yescharacteristic
587C22#348913yes—
588C22#349913yes—
589C22#353813yes—
590C22#354113yes—
591C22#355216yes—
592C4—24no—
593C4—212no—
594C4—46no—
595C4—46no—
596C4—46no—
597C4—46no—
598C4—46no—
599C4—42no—
600C4—81no—
601C4—83no—
602C4—83no—
603C4—81no—
604C2—28no—
605C2—44no—
606C2—161no—
607C2C2×#261411yescharacteristic
608C2#3608511yescharacteristic
609C2D4×#12011yescharacteristic
610C2#3765113yes—
611C2#3767513yes—
612C2#4049513yes—
613C2#4059113yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×D4×Q8C2semidirect
3. C2×#2614C2semidirect
4. C22×#395C2semidirect
5. #16511C2semidirect
6. #16507C2semidirect
7. #16262C2semidirect
8. #16251C2semidirect
9. #15176C2semidirect
10. #14874C2semidirect
11. #14873C2semidirect
12. #14759C2semidirect
13. #14720C2semidirect
14. #14302C2semidirect
15. #13894C2semidirect
133. #395D4semidirect
579. C22#2614semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#24432
C2×#26142
C22×#3952
#165112
#165072
#162622
#162512
#151762
#148742
#148732
#147592
#147202
#143022
#138942