Finite Group Databasebeta

← #618175

Subgroups — #618175

Order 512.

4944 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2591C211yesmaximal, characteristic
3C2×#2436C211yesmaximal, characteristic
4C2×#2618C211yesmaximal, characteristic
5#15783C214yesmaximal, complemented
6#15761C214yesmaximal, complemented
7#15322C218yesmaximal, complemented
8#15091C212yesmaximal, complemented
9#14873C212yesmaximal, complemented
10#14706C212yesmaximal, complemented
11#14256C212yesmaximal, complemented
12#13923C211yesmaximal, characteristic, complemented
13#13922C211yesmaximal, characteristic, complemented
14#13776C211yesmaximal, characteristic, complemented
15C24×Q8C2211yescharacteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8—28no—
18C22×C4⋊Q8—28no—
19C22×C4⋊Q8—28no—
20C22×C4⋊Q8C2214yes—
21C22×C4⋊Q8C2214yes—
22C22×#124C2212yes—
23C22×#123C2212yes—
24C22×#121C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C23×C42C2211yescharacteristic
30#2618—28no—
31#2591—28no—
32#2436—28no—
33C2×#400C2212yes—
34C2×#398C2212yes—
35C2×#396C2214yes—
36C2×#394C2212yes—
37C2×#394C2212yes—
38C2×#393C2214yes—
39C2×#393C2218yes—
40C2×#391C2212yes—
41C2×#391C2214yes—
42C2×#388C2214yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#386C2212yes—
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1446C2218yes—
53#1446C2214yes—
54#1442C2218yes—
55#1441C2214yes—
56#1441C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1437C2218yes—
60#1436C2214yes—
61#1436C2218yes—
62#1436C2214yes—
63#1436C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C23×Q8—24no—
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—24no—
73C23×Q8—24no—
74C23×Q8—48no—
75C23×Q8—48no—
76C23×Q8D418yes—
77C23×Q8C2314yes—
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—2128no—
90C2×C4⋊Q8—216no—
91C2×C4⋊Q8—216no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—216no—
94C2×C4⋊Q8—216no—
95C2×C4⋊Q8—216no—
96C2×#124—28no—
97C2×#124—48no—
98C2×#123—28no—
99C2×#123—216no—
100C2×#121—24no—
101C2×#121—24no—
102C2×#121—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—24no—
113C22×C4⋊C4D418yes—
114C22×C4⋊C4D418yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C42—24no—
135C22×C42—24no—
136C22×C42—24no—
137C22×C42C2314yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140#400—28no—
141#398—28no—
142#396—216no—
143#394—28no—
144#394—28no—
145#393—216no—
146#393—416no—
147#391—28no—
148#391—48no—
149#388—216no—
150#386—28no—
151#386—28no—
152#386—28no—
153#386—28no—
154#386—28no—
155#381—28no—
156#380—24no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159C2×#95C2312yes—
160C2×#95C2318yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C22×Q8—232no—
176C22×Q8—232no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—28no—
184C22×Q8—28no—
185C22×Q8—28no—
186C22×Q8—216no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—48no—
190C22×Q8—432no—
191C22×Q8—432no—
192C22×Q8—432no—
193C22×Q8—432no—
194C22×Q8—432no—
195C22×Q8—432no—
196C22×Q8C2×D418yes—
197C22×Q8C2×D418yes—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—28no—
202C22×D4—28no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—44no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4#4118yes—
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C4⋊Q8—2128no—
249C4⋊Q8—2128no—
250#124—432no—
251#123—432no—
252#121—416no—
253C2×C4⋊C4—232no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—232no—
256C2×C4⋊C4—232no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D418yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—216no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C42—232no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42—44no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—216no—
322#95—216no—
323#95—216no—
324#95—48no—
325#95—48no—
326#95—48no—
327C24—24no—
328C24—24no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—28no—
334C24—28no—
335C24—24no—
336C24C22×D412yes—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C2×#4114yes—
340C242−1+414yes—
341C24C2511yescharacteristic
342C2×Q8—232no—
343C2×Q8—216no—
344C2×Q8—216no—
345C2×Q8—216no—
346C2×Q8—216no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—416no—
352C2×Q8—416no—
353C2×Q8—432no—
354C2×Q8—432no—
355C2×Q8—4128no—
356C2×Q8—4128no—
357C2×Q8—416no—
358C2×Q8—416no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—44no—
366C2×D4—84no—
367C2×D4—84no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4#120116yes—
436C22×C4#124116yescomplemented
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C2×#4118yes—
450C22×C4H(5,𝔽2)18yes—
451C22×C42−1+418yes—
452C4⋊C4—232no—
453C4⋊C4—232no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—44no—
459C4⋊C4—416no—
460C4⋊C4—84no—
461C22⋊C4—416no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—232no—
472C42—48no—
473C42—84no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—216no—
481C23—216no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—416no—
494C23—416no—
495C23C2×#12014yes—
496C23C2×#12414yes—
497C23#53614yes—
498C23#54818yes—
499C23#54818yes—
500C23C23×D411yescharacteristic
501C23C23×D411yescharacteristic
502C23C23×D411yescharacteristic
503C23C22×#4112yes—
504C23C2×H(5,𝔽2)14yes—
505C23C2×2−1+412yes—
506C23C2×2−1+414yes—
507Q8—432no—
508Q8—432no—
509Q8—464no—
510D4—88no—
511D4—816no—
512C2×C4—232no—
513C2×C4—232no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—232no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—416no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—416no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4C2×#12014yes—
570C2×C4C2×#12014yes—
571C2×C4C2×#12418yes—
572C2×C4#53418yes—
573C2×C4#53618yes—
574C22—28no—
575C22—216no—
576C22—216no—
577C22—28no—
578C22—24no—
579C22—24no—
580C22—84no—
581C22—88no—
582C22—88no—
583C22—84no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22#243618yes—
588C22C22×#12011yescharacteristic
589C22C22×#12411yescharacteristic
590C22C2×#53412yes—
591C22C2×#53612yes—
592C22C2×#53611yescharacteristic
593C22C2×#54812yes—
594C22C2×#54812yes—
595C22#348412yes—
596C22#348412yes—
597C22#348612yes—
598C22#348612yes—
599C22#351118yes—
600C22#355218yes—
601C4—28no—
602C4—28no—
603C4—24no—
604C4—24no—
605C4—28no—
606C4—44no—
607C4—44no—
608C4—44no—
609C4—84no—
610C4—84no—
611C4—84no—
612C4—82no—
613C2—28no—
614C2—162no—
615C2C2×#243611yescharacteristic
616C2#3498011yescharacteristic
617C2#3524211yescharacteristic
618C2#3559212yes—
619C2#3723312yes—
620C2#3972014yes—
621C2#3972114yes—
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. C23×C4⋊Q8C2semidirect
5. #15783C2semidirect
6. #15761C2semidirect
7. #15322C2semidirect
8. #15091C2semidirect
9. #14873C2semidirect
10. #14706C2semidirect
11. #14256C2semidirect
12. #13923C2semidirect
13. #13922C2semidirect
14. #13776C2semidirect
436. C22×C4#124semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#25912
C2×#24362
C2×#26182
#157832
#157612
#153222
#150912
#148732
#147062
#142562
#139232
#139222
#137762