Finite Group Databasebeta

← #4309184

Subgroups — #4309184

Order 512.

4006 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic, complemented
3C2×#2621C212yesmaximal, complemented
4#16673C214yesmaximal, complemented
5#16607C212yesmaximal, complemented
6#16558C212yesmaximal, complemented
7#16365C212yesmaximal, complemented
8#16362C212yesmaximal
9#16359C214yesmaximal, complemented
10#15201C212yesmaximal, complemented
11#15177C214yesmaximal, complemented
12#15177C212yesmaximal, complemented
13#14682C211yesmaximal, characteristic, complemented
14#14340C212yesmaximal, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529—28no—
17C2×#529—216no—
18C22×#126C2212yes—
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×#123C2212yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2214yes—
25C22×#121C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C2212yes—
28C22×C4×D4C2211yescharacteristic
29C22×#117C2212yes—
30C23×C42C2211yescharacteristic
31#2857—28no—
32#2621—216no—
33C2×#398C2214yes—
34C2×#398C2214yes—
35C2×#397C2214yes—
36C2×#397C2214yes—
37C2×#391C2212yes—
38C2×#391C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#384C2211yescharacteristic
42C2×#382C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×#380C2212yes—
47#1446C2214yes—
48#1444C2218yes—
49#1444C2214yes—
50#1442C2218yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1438C2214yes—
60#1438C2218yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C23×Q8C2312yes—
68C23×Q8C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—28no—
77#529—2128no—
78C2×#126—28no—
79C2×#126—28no—
80C2×#126—28no—
81C2×#125—28no—
82C2×#125—24no—
83C2×#124—28no—
84C2×#124—24no—
85C2×#123—28no—
86C2×#123—28no—
87C2×#123—28no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—48no—
93C2×C22⋊Q8—48no—
94C2×C22⋊Q8—48no—
95C2×#121—24no—
96C2×#121—28no—
97C2×C4×Q8—24no—
98C2×C4×Q8—28no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4—24no—
104C2×#117—28no—
105C2×#117—28no—
106C2×#117D4116yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2318yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2318yes—
130C22×C42—28no—
131C22×C42D418yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#398—216no—
138#398—216no—
139#397—216no—
140#397—216no—
141#391—28no—
142#391—216no—
143#389—28no—
144#389—28no—
145#384—24no—
146#382—28no—
147#380—28no—
148#380—28no—
149#380—48no—
150#380—44no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C25C2412yes—
164C25C2411yescharacteristic
165C22×Q8—24no—
166C22×Q8—22no—
167C22×Q8—24no—
168C22×Q8—28no—
169C22×Q8—28no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—42no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—42no—
182C22×D4—44no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4C2×D418yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2418yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C4⋊Q8—416no—
216#126—264no—
217#125—232no—
218#124—232no—
219#123—264no—
220C22⋊Q8—232no—
221C22⋊Q8—432no—
222C22⋊Q8—464no—
223#121—232no—
224C4×Q8—232no—
225C4×D4—232no—
226C4×D4—264no—
227#117—264no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4C2×D418yes—
287C2×C22⋊C4C2×D418yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—232no—
294C2×C42C2×D418yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42C2×D418yes—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—216no—
305#95—216no—
306#95—28no—
307#95—44no—
308#95—48no—
309#95—48no—
310#95—44no—
311#95—48no—
312C24—22no—
313C24—22no—
314C24—21no—
315C24—22no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24C22×D414yes—
330C24C22×D412yes—
331C242−1+414yes—
332C24C2511yescharacteristic
333C2×Q8—28no—
334C2×Q8—416no—
335C2×Q8—44no—
336C2×Q8—48no—
337C2×Q8—416no—
338C2×Q8—48no—
339C2×Q8—44no—
340C2×Q8—416no—
341C2×Q8—48no—
342C2×D4—216no—
343C2×D4—28no—
344C2×D4—416no—
345C2×D4—416no—
346C2×D4—44no—
347C2×D4—416no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—44no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4C22×D412yes—
416C22×C4C22×D414yes—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D414yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+414yes—
430C22×C42−1+412yes—
431C4⋊C4—28no—
432C4⋊C4—216no—
433C4⋊C4—28no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—432no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—416no—
444C4⋊C4—48no—
445C22⋊C4—216no—
446C22⋊C4—216no—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—416no—
451C22⋊C4—432no—
452C22⋊C4—88no—
453C22⋊C4—816no—
454C42—28no—
455C42—28no—
456C42—28no—
457C42—216no—
458C42#120116yes—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—28no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23#53614yes—
499C23C23×D411yescharacteristic
500C23C23×D412yes—
501C23C2×H(5,𝔽2)12yes—
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×2−1+412yes—
504C23C2×2−1+412yes—
505C23C2×2−1+411yescharacteristic
506C23#58512yes—
507C23#58512yes—
508Q8—416no—
509Q8—816no—
510D4—416no—
511D4—432no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—416no—
535C2×C4—416no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—416no—
550C2×C4—432no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4C2×#12014yes—
569C2×C4C2×#12014yes—
570C2×C4C2×#12014yes—
571C2×C4#53518yes—
572C2×C4#53514yes—
573C2×C4#53614yes—
574C2×C4#53718yes—
575C2×C4#53714yes—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—48no—
580C22—48no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—48no—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—84no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C22×#12011yescharacteristic
597C22C2×#53512yes—
598C22C2×#53511yescharacteristic
599C22C2×#53611yescharacteristic
600C22C2×#53611yescharacteristic
601C22C2×#53712yes—
602C22C2×#53711yescharacteristic
603C22#344911yescharacteristic
604C22#344911yescharacteristic
605C22#348914yes—
606C22#348914yes—
607C22#355214yes—
608C22#355412yes—
609C22#355512yes—
610C22#356112yes—
611C22#356214yes—
612C22#356312yes—
613C4—24no—
614C4—24no—
615C4—24no—
616C4—42no—
617C4—44no—
618C4—42no—
619C4—44no—
620C4—44no—
621C4—88no—
622C4—84no—
623C4—84no—
624C4#261518yes—
625C2—44no—
626C2—84no—
627C2C2×#261511yescharacteristic
628C2#3530511yescharacteristic
629C2#3530511yescharacteristic
630C2#3737814yes—
631C2#3737914yes—
632C2#3755511yescharacteristic
633C2#3755911yescharacteristic
634C2#3761511yescharacteristic
635C2#3762211yescharacteristic
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
2. C2×#2857C2semidirect
3. C2×#2621C2semidirect
4. #16673C2semidirect
5. #16607C2semidirect
6. #16558C2semidirect
7. #16365C2semidirect
9. #16359C2semidirect
10. #15201C2semidirect
11. #15177C2semidirect
12. #15177C2semidirect
13. #14682C2semidirect
14. #14340C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#28572
C2×#26212
#166732
#166072
#165582
#163652
#163622
#163592
#152012
#151772
#146822
#143402