Finite Group Databasebeta

← #4308099

Subgroups — #4308099

Order 512.

2490 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2C2×C4×#117C211yesmaximal, characteristic
3#14855C214yesmaximal, complemented
4#14819C214yesmaximal
5#14790C214yesmaximal, complemented
6#14744C214yesmaximal
7#14712C211yesmaximal, characteristic, complemented
8#14707C211yesmaximal, characteristic, complemented
9#14699C212yesmaximal, complemented
10#14697C212yesmaximal
11#14348C211yesmaximal, characteristic, complemented
12#14340C211yesmaximal, characteristic
13#14340C211yesmaximal, characteristic
14#14266C212yesmaximal, complemented
15#14264C212yesmaximal
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2647—28no—
20C2×#400C2214yes—
21C2×#398C2214yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#387C2214yes—
28C2×#385C2214yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C4×#117—28no—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×#376C2211yescharacteristic
41C2×#376C2211yescharacteristic
42C2×C43C2211yescharacteristic
43C2×C43C2211yescharacteristic
44#1445C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C2×#117—24no—
76C2×#117—24no—
77C2×#117—24no—
78C2×#117—24no—
79C2×#117—24no—
80C2×#117Q818yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42—24no—
100C22×C42—24no—
101C22×C42—24no—
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#400—216no—
114#398—216no—
115#389—28no—
116#389—28no—
117#389—28no—
118#389—28no—
119#389—48no—
120#387—48no—
121#385—48no—
122#384—24no—
123#384—24no—
124#382—24no—
125#382—24no—
126#380—28no—
127#380—28no—
128C4×C4⋊C4—28no—
129C4×C4⋊C4—28no—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—28no—
132#376—24no—
133#376D418yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C43—24no—
159C43—24no—
160C25C2411yescharacteristic
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4C2×Q814yes—
173C23×C4#4114yes—
174C23×C4#4114yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189#117—232no—
190#117—232no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
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⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×Q818yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4C2×Q818yes—
245C2×C42—232no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C2×D414yes—
272C2×C42C2×D414yes—
273C2×C42C2×D414yes—
274C2×C42C2×Q814yes—
275C2×C42C2×Q814yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95C2×D418yes—
302#95C2×D418yes—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24C22×Q812yes—
312C24C2×#4112yes—
313C24C2×#4112yes—
314C24C2511yescharacteristic
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—216no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
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—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
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—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4#125116yes—
397C22×C4C22×D412yes—
398C22×C4C22×D414yes—
399C22×C4C22×D412yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C22×Q814yes—
403C22×C4C22×Q814yes—
404C22×C4C22×Q812yes—
405C22×C4C22×Q812yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4H(5,𝔽2)14yes—
418C22×C42−1+414yes—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—216no—
422C4⋊C4—416no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—416no—
428C4⋊C4—88no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—216no—
435C22⋊C4—416no—
436C22⋊C4—88no—
437C42—216no—
438C42—28no—
439C42—28no—
440C42—28no—
441C42—28no—
442C42—28no—
443C42—48no—
444C42—48no—
445C42—48no—
446C42—48no—
447C42C22⋊Q8116yes—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23C2×#12514yes—
471C23C23×D411yescharacteristic
472C23C23×Q811yescharacteristic
473C23C22×#4111yescharacteristic
474C23C22×#4111yescharacteristic
475C23C22×#4111yescharacteristic
476C23C22×#4111yescharacteristic
477C23C22×#4111yescharacteristic
478C23C2×H(5,𝔽2)12yes—
479C23C2×2−1+412yes—
480C23#58512yes—
481C23#58512yes—
482C2×C4—22no—
483C2×C4—22no—
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—232no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—432no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
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—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4C2×C22⋊Q818yes—
565C2×C4C2×C22⋊Q814yes—
566C2×C4#52514yes—
567C2×C4#52514yes—
568C2×C4C2×#12514yes—
569C2×C4C2×#12514yes—
570C2×C4#52914yes—
571C2×C4#52914yes—
572C2×C4#53314yes—
573C2×C4#54218yes—
574C22—24no—
575C22—28no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C22×C22⋊Q811yescharacteristic
586C22C2×#52511yescharacteristic
587C22C2×#52511yescharacteristic
588C22C22×#12511yescharacteristic
589C22C2×#52911yescharacteristic
590C22C2×#52911yescharacteristic
591C22C2×#53311yescharacteristic
592C22C2×#54212yes—
593C22#345512yes—
594C22#346012yes—
595C22#346012yes—
596C22#346112yes—
597C22#346112yes—
598C22#348812yes—
599C22#348912yes—
600C22#349412yes—
601C22#349512yes—
602C22#350312yes—
603C22#350412yes—
604C22#350811yescharacteristic
605C22#350911yescharacteristic
606C22#351211yescharacteristic
607C22#351211yescharacteristic
608C4—24no—
609C4—28no—
610C4—28no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—84no—
618C4—84no—
619C4—84no—
620C4—84no—
621C4#285118yes—
622C2—44no—
623C2#3461912yes—
624C2#3525712yes—
625C2#3525911yescharacteristic
626C2#3526012yes—
627C2#3526512yes—
628C2#3528311yescharacteristic
629C2#3528711yescharacteristic
630C2#3530911yescharacteristic
631C2#3532111yescharacteristic
632C2#3533611yescharacteristic
633C2C2×#285111yescharacteristic
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
1. C2×#2647C2semidirect
3. #14855C2semidirect
5. #14790C2semidirect
7. #14712C2semidirect
8. #14707C2semidirect
9. #14699C2semidirect
11. #14348C2semidirect
14. #14266C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×C4×#1172
#148552
#148192
#147902
#147442
#147122
#147072
#146992
#146972
#143482
#143402
#142662
#142642