Finite Group Databasebeta

← #627443

Subgroups — #627443

Order 512.

2714 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3#16060C214yesmaximal, complemented
4#16059C214yesmaximal, complemented
5#15460C214yesmaximal, complemented
6#15438C214yesmaximal, complemented
7#14935C212yesmaximal
8#14728C212yesmaximal
9#14695C212yesmaximal
10#14339C211yesmaximal, characteristic
11#14263C212yesmaximal
12#14128C211yesmaximal, characteristic
13#14128C211yesmaximal, characteristic
14#13870C211yesmaximal, characteristic
15#13870C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#398C2214yes—
24#2317—28no—
25C2×#389—28no—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2214yes—
29C2×#389C2214yes—
30C2×#387C2214yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#384C2212yes—
36C2×#382C2212yes—
37C2×#381C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1444C2212yes—
45#1444C2212yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1442C2212yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—28no—
87C2×#117—44no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4Q818yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—24no—
119C22×C42—44no—
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123#398—48no—
124#389—28no—
125#389—28no—
126#389—216no—
127#389—216no—
128#387—48no—
129#385—28no—
130#385—28no—
131#385—28no—
132#385—44no—
133#384—28no—
134#382—28no—
135#381—24no—
136#381—24no—
137#381—28no—
138#381—44no—
139#380—44no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25C2411yescharacteristic
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2×Q814yes—
199C23×C4C2×Q814yes—
200C23×C4C2×Q814yes—
201C23×C4#4114yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216#117—232no—
217#117—416no—
218C2×C4⋊C4—232no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×Q818yes—
264C2×C4⋊C4C2×Q818yes—
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—24no—
276C2×C22⋊C4#4118yes—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—28no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42—416no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24C22×D412yes—
320C24C22×Q812yes—
321C24C2×#4112yes—
322C24C2×#4114yes—
323C24H(5,𝔽2)14yes—
324C24C2511yescharacteristic
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—216no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—216no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—216no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—416no—
405C22×C4—44no—
406C22×C4—416no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C22⋊Q8116yes—
417C22×C4C22×D414yes—
418C22×C4C22×D414yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×Q814yes—
422C22×C4C22×Q812yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q814yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C42−1+414yes—
431C22×C42−1+414yes—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—416no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C22⋊C4—216no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C42—216no—
454C42—416no—
455C42—48no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23C2×C22⋊Q814yes—
469C23#52514yes—
470C23#53414yes—
471C23#55318yes—
472C23#55418yes—
473C23C23×D411yescharacteristic
474C23C23×Q811yescharacteristic
475C23C22×#4111yescharacteristic
476C23C22×#4112yes—
477C23C2×H(5,𝔽2)12yes—
478C23C2×H(5,𝔽2)12yes—
479C23C2×2−1+412yes—
480C23C2×2−1+412yes—
481C23#58512yes—
482C2×C4—232no—
483C2×C4—232no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
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—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—416no—
513C2×C4—416no—
514C2×C4—416no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
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—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—88no—
561C2×C4—88no—
562C2×C4C2×C22⋊Q814yes—
563C2×C4C2×C22⋊Q814yes—
564C2×C4#54418yes—
565C22—24no—
566C22—216no—
567C22—216no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22#257918yes—
574C22C22×C22⋊Q811yescharacteristic
575C22C2×#52511yescharacteristic
576C22C2×#53411yescharacteristic
577C22C2×#54412yes—
578C22#346112yes—
579C22C2×#55312yes—
580C22C2×#55412yes—
581C22#348411yescharacteristic
582C22#348511yescharacteristic
583C22#348611yescharacteristic
584C22#348611yescharacteristic
585C22#349212yes—
586C22#350212yes—
587C22#351514yes—
588C22#352014yes—
589C22#354214yes—
590C22#354414yes—
591C4—24no—
592C4—24no—
593C4—28no—
594C4—44no—
595C4—44no—
596C4—44no—
597C4—44no—
598C4—44no—
599C4—44no—
600C4—84no—
601C4—84no—
602C4—82no—
603C4—82no—
604C4—82no—
605C2—28no—
606C2C2×#257911yescharacteristic
607C2#3515411yescharacteristic
608C2#3522411yescharacteristic
609C2#3535711yescharacteristic
610C2#3545311yescharacteristic
611C2#3670411yescharacteristic
612C2#3725811yescharacteristic
613C2#3820914yes—
614C2#3896314yes—
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. #16060C2semidirect
4. #16059C2semidirect
5. #15460C2semidirect
6. #15438C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C22×#3892
#160602
#160592
#154602
#154382
#149352
#147282
#146952
#143392
#142632
#141282
#138702