Finite Group Databasebeta

← #4462726

Subgroups — #4462726

Order 512.

2040 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2396C211yesmaximal, characteristic
3#17192C214yesmaximal, complemented
4#17168C212yesmaximal
5#17144C212yesmaximal, complemented
6#17122C214yesmaximal
7#17106C212yesmaximal
8#17085C212yesmaximal, complemented
9#15235C212yesmaximal, complemented
10#15200C212yesmaximal, complemented
11#15183C212yesmaximal
12#15158C212yesmaximal, complemented
13#15132C212yesmaximal
14#15128C212yesmaximal, complemented
15#14343C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2628—28no—
20#2396—28no—
21C2×#400C2212yes—
22C2×#400C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#397C2214yes—
27C2×#396C2212yes—
28C2×#395C2212yes—
29C2×#395C2211yescharacteristic
30C2×#395C2214yes—
31C2×#395C2214yes—
32C2×#393C2212yes—
33C2×#393C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44#1446C2214yes—
45#1445C2212yes—
46#1445C2212yes—
47#1445C2214yes—
48#1445C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2212yes—
65#1440C2212yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4×Q8—24no—
84C2×C4×Q8—42no—
85C2×C4×Q8—42no—
86C2×#117—42no—
87C2×#117—42no—
88C2×#117D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#400—28no—
116#400—24no—
117#399—28no—
118#399—24no—
119#398—28no—
120#397—48no—
121#396—28no—
122#395—28no—
123#395—24no—
124#395—48no—
125#395—48no—
126#393—28no—
127#393—24no—
128#389—28no—
129#389—44no—
130#389—44no—
131#382—24no—
132#380—28no—
133#380—42no—
134#380—42no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—24no—
137C4×C22⋊C4—28no—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2311yescharacteristic
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25C2411yescharacteristic
169C22×Q8—22no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—44no—
178C22×Q8—44no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4C2×D414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C4⋊Q8—416no—
199C4×Q8—416no—
200#117—416no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4C2×D414yes—
264C2×C22⋊C4C2×D414yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42C2×D414yes—
276C2×C42C2×D414yes—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24C22×D412yes—
316C24C2511yescharacteristic
317C2×Q8—48no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—44no—
323C2×Q8—88no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—88no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4H(5,𝔽2)14yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+414yes—
416C22×C42−1+414yes—
417C22×C42−1+414yes—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—88no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C42—28no—
441C42—28no—
442C42—48no—
443C42—48no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C23×D411yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)12yes—
469C23C2×2−1+412yes—
470C23C2×2−1+412yes—
471C23C2×2−1+412yes—
472C23#58512yes—
473C23#58512yes—
474Q8—416no—
475Q8—816no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—416no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4#53414yes—
559C2×C4#53614yes—
560C2×C4#53614yes—
561C2×C4#56318yes—
562C2×C4#56318yes—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22C2×#53411yescharacteristic
573C22C2×#53611yescharacteristic
574C22C2×#53611yescharacteristic
575C22#348812yes—
576C22#348912yes—
577C22C2×#56312yes—
578C22C2×#56312yes—
579C22#353711yescharacteristic
580C22#354011yescharacteristic
581C22#354212yes—
582C22#354712yes—
583C22#354712yes—
584C22#355011yescharacteristic
585C22#355012yes—
586C22#355111yescharacteristic
587C22#355212yes—
588C22#355412yes—
589C22#355512yes—
590C22#355712yes—
591C22#356312yes—
592C22#356512yes—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—44no—
598C4—42no—
599C4—44no—
600C4—82no—
601C4—84no—
602C4—84no—
603C4—84no—
604C4—82no—
605C4—84no—
606C4#281718yes—
607C2—82no—
608C2C2×#281711yescharacteristic
609C2#3726911yescharacteristic
610C2#3727512yes—
611C2#3727912yes—
612C2#3737611yescharacteristic
613C2#5297412yes—
614C2#5302612yes—
615C2#5311112yes—
616C2#5312412yes—
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
3. #17192C2semidirect
5. #17144C2semidirect
8. #17085C2semidirect
9. #15235C2semidirect
10. #15200C2semidirect
12. #15158C2semidirect
14. #15128C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#23962
#171922
#171682
#171442
#171222
#171062
#170852
#152352
#152002
#151832
#151582
#151322
#151282
#143432