Finite Group Databasebeta

← #627027

Subgroups — #627027

Order 512.

2320 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3#15766C214yesmaximal, complemented
4#15765C214yesmaximal
5#15396C214yesmaximal
6#15388C214yesmaximal, complemented
7#15137C212yesmaximal, complemented
8#15135C212yesmaximal
9#14709C211yesmaximal, characteristic, complemented
10#14692C211yesmaximal, characteristic, complemented
11#14340C211yesmaximal, characteristic
12#14277C211yesmaximal, characteristic, complemented
13#14029C212yesmaximal, complemented
14#13876C212yesmaximal
15#13761C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#117C2211yescharacteristic
18#2629—28no—
19C2×#399C2214yes—
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#396C2214yes—
23#2330—28no—
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2212yes—
33C2×#384C2211yescharacteristic
34C2×#384C2214yes—
35C2×#384C2212yes—
36C2×#384C2214yes—
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44C2×C43C2211yescharacteristic
45#1445C2214yes—
46#1445C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2212yes—
60#1437C2212yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4×#95C2211yescharacteristic
72C4×#95C2211yescharacteristic
73C24×C4C2311yescharacteristic
74C2×#125—24no—
75C2×#125—24no—
76C2×#125D418yes—
77C2×#117—24no—
78C2×#117—24no—
79C2×#117—24no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110#399—48no—
111#398—216no—
112#398—48no—
113#396—216no—
114#389—216no—
115#389—28no—
116#389—28no—
117#389—44no—
118#388—48no—
119#387—216no—
120#387—48no—
121#384—24no—
122#384—24no—
123#384—28no—
124#384—44no—
125#384—48no—
126#384—48no—
127#382—24no—
128#380—28no—
129#380—44no—
130C4×C4⋊C4—28no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4Q818yes—
133#376—24no—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C43—24no—
159C25C2411yescharacteristic
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4C2×Q814yes—
167C23×C4C2414yes—
168C23×C4C2411yescharacteristic
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2411yescharacteristic
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181#125—232no—
182#117—416no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C2×D418yes—
227C2×C4⋊C4C2×D4116yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4C2×Q818yes—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—28no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42C2×D414yes—
266C2×C42C2×Q814yes—
267C2×C42C2×Q814yes—
268C2×C42#4114yes—
269C2×C42#4114yes—
270C2×C42#4114yes—
271#95—24no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—22no—
278#95—22no—
279#95—28no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95C2×Q818yes—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24C22×Q812yes—
306C24C2511yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4C22×D418yes—
385C22×C4C22×D414yes—
386C22×C4C22×D412yes—
387C22×C4C22×Q812yes—
388C22×C4C22×Q814yes—
389C22×C4C22×Q814yes—
390C22×C4C22×Q812yes—
391C22×C4C2×#4114yes—
392C22×C4C2×#4114yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)14yes—
399C22×C4H(5,𝔽2)14yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+414yes—
402C22×C42−1+414yes—
403C22×C42−1+412yes—
404C4⋊C4—232no—
405C4⋊C4—216no—
406C4⋊C4—48no—
407C4⋊C4—416no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C4⋊C4—88no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C4⋊C4—88no—
419C22⋊C4—216no—
420C22⋊C4—48no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C42—216no—
426C42—216no—
427C42—28no—
428C42—28no—
429C42—28no—
430C42—44no—
431C42—44no—
432C42—88no—
433C42#123116yes—
434C23—22no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23C23×D411yescharacteristic
458C23C23×Q811yescharacteristic
459C23C22×#4111yescharacteristic
460C23C22×#4111yescharacteristic
461C23C22×#4111yescharacteristic
462C23C2×H(5,𝔽2)12yes—
463C23C2×H(5,𝔽2)11yescharacteristic
464C23C2×H(5,𝔽2)12yes—
465C23C2×2−1+412yes—
466C23C2×2−1+411yescharacteristic
467C23C2×2−1+412yes—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—416no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4C2×#12318yes—
549C2×C4C2×#12314yes—
550C2×C4#53314yes—
551C2×C4#54014yes—
552C2×C4#54114yes—
553C2×C4#54214yes—
554C2×C4#54418yes—
555C2×C4D4×Q814yes—
556C2×C4#55014yes—
557C2×C4#55214yes—
558C2×C4#55414yes—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—82no—
565C22—82no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22C22×#12311yescharacteristic
571C22C2×#53311yescharacteristic
572C22C2×#54011yescharacteristic
573C22C2×#54111yescharacteristic
574C22C2×#54211yescharacteristic
575C22C2×#54412yes—
576C22C2×D4×Q811yescharacteristic
577C22C2×#55011yescharacteristic
578C22C2×#55211yescharacteristic
579C22C2×#55411yescharacteristic
580C22#349212yes—
581C22#349212yes—
582C22#349812yes—
583C22#349912yes—
584C22#350312yes—
585C22#350811yescharacteristic
586C22#350911yescharacteristic
587C22#351112yes—
588C22#351112yes—
589C22#354312yes—
590C22#354412yes—
591C22#355212yes—
592C22#355212yes—
593C4—28no—
594C4—28no—
595C4—24no—
596C4—24no—
597C4—42no—
598C4—44no—
599C4—48no—
600C4—82no—
601C4—84no—
602C4—81no—
603C4—81no—
604C4—84no—
605C4—84no—
606C4#242318yes—
607C2—82no—
608C2C2×#242311yescharacteristic
609C2#3528411yescharacteristic
610C2#3529411yescharacteristic
611C2#3553012yes—
612C2#3732712yes—
613C2#3787412yes—
614C2#3788211yescharacteristic
615C2#3788311yescharacteristic
616C2#3936912yes—
617C2#3938511yescharacteristic
618C2#3938611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2330C2semidirect
3. #15766C2semidirect
6. #15388C2semidirect
7. #15137C2semidirect
9. #14709C2semidirect
10. #14692C2semidirect
12. #14277C2semidirect
13. #14029C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#23302
#157662
#157652
#153962
#153882
#151372
#151352
#147092
#146922
#143402
#142772
#140292
#138762
#137612