Finite Group Databasebeta

← #4308793

Subgroups — #4308793

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3#16418C214yesmaximal, complemented
4#16407C214yesmaximal, complemented
5#16325C214yesmaximal, complemented
6#16311C214yesmaximal, complemented
7#15073C212yesmaximal
8#15019C211yesmaximal, characteristic
9#14934C211yesmaximal, characteristic
10#14919C212yesmaximal
11#14821C211yesmaximal, characteristic
12#14720C211yesmaximal, characteristic
13#14695C212yesmaximal
14#14342C211yesmaximal, characteristic
15#14263C212yesmaximal
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2627—28no—
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#389—28no—
30C2×#389C2214yes—
31C2×#389C2214yes—
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1446C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1445C2212yes—
47#1445C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2212yes—
60#1442C2214yes—
61#1442C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1434C2214yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—44no—
87C22×C4⋊C4—24no—
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⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42—24no—
115C22×C42—44no—
116C22×C42C2314yes—
117#400—28no—
118#398—28no—
119#398—28no—
120#398—44no—
121#395—28no—
122#395—44no—
123#389—216no—
124#389—216no—
125#387—48no—
126#385—28no—
127#385—28no—
128#385—44no—
129#381—28no—
130#381—28no—
131#381—44no—
132#381—44no—
133#380—24no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C22×Q8—22no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4#4114yes—
203C23×C4#4114yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C22⋊Q8—416no—
219C2×C4⋊C4—232no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D418yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
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×C42—28no—
273C2×C42—28no—
274C2×C42—44no—
275C2×C42—416no—
276#95—24no—
277#95—24no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—42no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—42no—
307#95—44no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C22×D412yes—
313C24C2×#4112yes—
314C24C2×#4112yes—
315C24H(5,𝔽2)14yes—
316C242−1+414yes—
317C24C2511yescharacteristic
318C2×Q8—44no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×Q8—88no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—216no—
325C22×C4—28no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—216no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—216no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
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—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—416no—
407C22×C4—44no—
408C22×C4—416no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4114yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C42−1+414yes—
428C22×C42−1+414yes—
429C4⋊C4—232no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—416no—
438C4⋊C4—84no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4—88no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C42—416no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23#52514yes—
461C23#53414yes—
462C23#53614yes—
463C23#53918yes—
464C23#54218yes—
465C23C23×D411yescharacteristic
466C23C22×#4111yescharacteristic
467C23C22×#4111yescharacteristic
468C23C2×H(5,𝔽2)12yes—
469C23C2×H(5,𝔽2)12yes—
470C23C2×2−1+412yes—
471C23C2×2−1+412yes—
472C23C2×2−1+412yes—
473C23#58512yes—
474Q8—88no—
475C2×C4—232no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—416no—
499C2×C4—416no—
500C2×C4—416no—
501C2×C4—416no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
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—88no—
556C2×C4—88no—
557C2×C4#53918yes—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—216no—
563C22—24no—
564C22—24no—
565C22—216no—
566C22#263818yes—
567C22C2×#52511yescharacteristic
568C22C2×#53411yescharacteristic
569C22C2×#53611yescharacteristic
570C22C2×#53912yes—
571C22C2×#53912yes—
572C22C2×#54212yes—
573C22#346112yes—
574C22#348411yescharacteristic
575C22#348511yescharacteristic
576C22#348611yescharacteristic
577C22#348611yescharacteristic
578C22#349212yes—
579C22#350712yes—
580C22#353814yes—
581C22#354214yes—
582C22#354414yes—
583C22#356314yes—
584C4—28no—
585C4—44no—
586C4—44no—
587C4—44no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4—84no—
598C2—28no—
599C2C2×#263811yescharacteristic
600C2#3498911yescharacteristic
601C2#3509211yescharacteristic
602C2#3670811yescharacteristic
603C2#3671511yescharacteristic
604C2#3682414yes—
605C2#3695911yescharacteristic
606C2#3725811yescharacteristic
607C2#3750314yes—
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. #16418C2semidirect
4. #16407C2semidirect
5. #16325C2semidirect
6. #16311C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3892
#164182
#164072
#163252
#163112
#150732
#150192
#149342
#149192
#148212
#147202
#146952
#143422
#142632