Finite Group Databasebeta

← #5955468

Subgroups — #5955468

Order 512.

3134 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3#15014C211yesmaximal, characteristic
4#15011C214yesmaximal, complemented
5#14992C214yesmaximal, complemented
6#14955C212yesmaximal
7#14914C211yesmaximal, characteristic
8#14874C212yesmaximal
9#14821C211yesmaximal, characteristic
10#14816C214yesmaximal, complemented
11#14800C214yesmaximal, complemented
12#14720C211yesmaximal, characteristic
13#13796C211yesmaximal, characteristic
14#13782C212yesmaximal
15C4×#385C212yesmaximal
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#398C2212yes—
21C2×#395C2214yes—
22C2×#395C2214yes—
23C2×#395C2212yes—
24#2329—28no—
25C2×#391—28no—
26C2×#391C2212yes—
27C2×#391C2214yes—
28C2×#391C2214yes—
29C2×#389C2212yes—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2214yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2214yes—
46#1446C2214yes—
47#1446C2212yes—
48#1445C2212yes—
49#1444C2212yes—
50#1444C2214yes—
51#1444C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66#1434C2214yes—
67#1434C2214yes—
68#1434C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C4×#95C2214yes—
77C23×Q8—24no—
78C23×Q8—24no—
79C23×Q8—24no—
80C23×Q8—44no—
81C23×Q8—44no—
82C23×Q8C2314yes—
83C23×Q8C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—28no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42—24no—
112C22×C42—44no—
113C22×C42C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2314yes—
117#398—44no—
118#395—44no—
119#395—48no—
120#395—48no—
121#391—28no—
122#391—216no—
123#391—216no—
124#389—28no—
125#387—28no—
126#387—28no—
127#385—28no—
128#385—28no—
129#385—28no—
130#385—28no—
131#385—28no—
132#380—24no—
133C4×C4⋊C4—48no—
134C4×C22⋊C4—28no—
135C4×C22⋊C4Q8116yescomplemented
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95D418yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C25C2411yescharacteristic
171C22×Q8—232no—
172C22×Q8—28no—
173C22×Q8—22no—
174C22×Q8—28no—
175C22×Q8—28no—
176C22×Q8—24no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—416no—
184C22×Q8—416no—
185C22×Q8—432no—
186C22×Q8—432no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×Q818yes—
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C22⋊Q8—232no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
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⋊C4C2×Q818yes—
275C2×C22⋊C4C2×Q818yes—
276C2×C22⋊C4#4118yes—
277C2×C42—28no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—416no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42C2×Q8116yescomplemented
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—42no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95C2×D418yes—
316#95C2×D418yes—
317#95C2×Q8116yescomplemented
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C22×D412yes—
323C24C22×Q814yes—
324C24C2×#4112yes—
325C24C2×#4112yes—
326C24C2×#4114yes—
327C24C2511yescharacteristic
328C2×Q8—232no—
329C2×Q8—28no—
330C2×Q8—416no—
331C2×Q8—416no—
332C2×Q8—416no—
333C2×Q8—416no—
334C2×Q8—4128no—
335C2×Q8—416no—
336C2×Q8—416no—
337C2×Q8—82no—
338C2×Q8—82no—
339C2×Q8—88no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—216no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—216no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—416no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4C4⋊Q8116yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D414yes—
434C22×C4C22×Q818yes—
435C22×C4C22×Q818yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q814yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+414yes—
445C4⋊C4—28no—
446C4⋊C4—216no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—82no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—82no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C22⋊C4—216no—
458C22⋊C4—216no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C42—416no—
462C42—416no—
463C42—84no—
464C42—84no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23#52514yes—
478C23#52514yes—
479C23C2×C4⋊Q814yes—
480C23#53318yes—
481C23#53318yes—
482C23C23×D411yescharacteristic
483C23C23×Q812yes—
484C23C22×#4111yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4112yes—
487C23C2×H(5,𝔽2)12yes—
488C23C2×H(5,𝔽2)12yes—
489C23C2×2−1+412yes—
490C23#58512yes—
491Q8—464no—
492Q8—88no—
493C2×C4—232no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—416no—
528C2×C4—416no—
529C2×C4—416no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
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—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—88no—
571C2×C4—88no—
572C2×C4C2×C4⋊Q818yes—
573C2×C4#53818yes—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—216no—
578C22—24no—
579C22—24no—
580C22—216no—
581C22—24no—
582C22#239918yes—
583C22C2×#52511yescharacteristic
584C22C2×#52511yescharacteristic
585C22C22×C4⋊Q811yescharacteristic
586C22C2×#53312yes—
587C22C2×#53312yes—
588C22C2×#53812yes—
589C22#346112yes—
590C22#348411yescharacteristic
591C22#348411yescharacteristic
592C22#348511yescharacteristic
593C22#348611yescharacteristic
594C22#349012yes—
595C22#350114yes—
596C22#350314yes—
597C22#350712yes—
598C22#351414yes—
599C22#352014yes—
600C4—28no—
601C4—28no—
602C4—44no—
603C4—44no—
604C4—48no—
605C4—42no—
606C4—42no—
607C4—44no—
608C4—44no—
609C4—44no—
610C4—48no—
611C4—82no—
612C4—82no—
613C4—82no—
614C2—28no—
615C2C2×#239911yescharacteristic
616C2#3498811yescharacteristic
617C2#3501614yes—
618C2#3509211yescharacteristic
619C2#3535711yescharacteristic
620C2#3539111yescharacteristic
621C2#3539814yes—
622C2#3545111yescharacteristic
623C2#3557611yescharacteristic
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
4. #15011C2semidirect
5. #14992C2semidirect
10. #14816C2semidirect
11. #14800C2semidirect
135. C4×C22⋊C4Q8semidirect
287. C2×C42C2×Q8semidirect
317. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
C22×#3912
#150142
#150112
#149922
#149552
#149142
#148742
#148212
#148162
#148002
#147202
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