Finite Group Databasebeta

← #4453591

Subgroups — #4453591

Order 512.

2518 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2398C211yesmaximal, characteristic
2C2×#2327C211yesmaximal, characteristic, complemented
3#16647C214yesmaximal
4#16638C214yesmaximal, complemented
5#16623C214yesmaximal, complemented
6#16614C214yesmaximal, complemented
7#15233C212yesmaximal, complemented
8#15201C212yesmaximal, complemented
9#15178C212yesmaximal, complemented
10#15177C212yesmaximal
11#14350C211yesmaximal, characteristic, complemented
12#13969C212yesmaximal, complemented
13#13969C212yesmaximal, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2398—28no—
21C2×#400C2212yes—
22C2×#397C2212yes—
23C2×#394C2212yes—
24C2×#393C2212yes—
25#2327—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×#376C2211yescharacteristic
37#1446C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8—24no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—44no—
78C2×C22⋊Q8—44no—
79C2×C22⋊Q8—44no—
80C2×C22⋊Q8—44no—
81C2×C4×Q8—28no—
82C2×C4×Q8—24no—
83C2×C4×Q8—42no—
84C2×C4×Q8—42no—
85C2×C4×Q8—44no—
86C2×C4×Q8—44no—
87C2×C4×D4—24no—
88C2×C4×D4—42no—
89C2×C4×D4—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2314yes—
114#400—28no—
115#397—28no—
116#394—28no—
117#393—28no—
118#391—28no—
119#391—44no—
120#391—44no—
121#387—216no—
122#387—216no—
123#382—24no—
124#380—44no—
125#380—44no—
126C4×C22⋊C4—28no—
127C4×C22⋊C4—28no—
128#376D418yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C25C2412yes—
155C22×Q8—24no—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—42no—
166C22×Q8#4118yes—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—42no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C4⋊Q8—232no—
192C22⋊Q8—432no—
193C22⋊Q8—432no—
194C4×Q8—416no—
195C4×Q8—432no—
196C4×D4—416no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
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—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42C2×D414yes—
265C2×C42C2×D414yes—
266C2×C42C2×D414yes—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95C2×D418yes—
291#95C2×D418yes—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C242−1+414yes—
307C24C2511yescharacteristic
308C2×Q8—216no—
309C2×Q8—216no—
310C2×Q8—216no—
311C2×Q8—48no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—48no—
315C2×Q8—44no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×D4—48no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—48no—
324C2×D4—44no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
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—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
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—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4C22×D414yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D414yes—
403C22×C4C22×D412yes—
404C22×C4C2×#4118yes—
405C22×C4C2×#4114yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C42−1+414yes—
410C4⋊C4—232no—
411C4⋊C4—44no—
412C4⋊C4—416no—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—416no—
426C22⋊C4—416no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C42—28no—
430C42—28no—
431C42—28no—
432C42—416no—
433C42—48no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23C23×D411yescharacteristic
470C23C22×#4112yes—
471C23C2×H(5,𝔽2)11yescharacteristic
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×H(5,𝔽2)12yes—
474C23C2×2−1+412yes—
475C23C2×2−1+412yes—
476C23#58512yes—
477C23#58512yes—
478Q8—416no—
479Q8—432no—
480D4—416no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
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—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
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—816no—
555C2×C4#52514yes—
556C2×C4#53514yes—
557C2×C4#53514yes—
558C2×C4#53818yes—
559C2×C4#53818yes—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22C2×#52511yescharacteristic
576C22C2×#53511yescharacteristic
577C22C2×#53511yescharacteristic
578C22C2×#53812yes—
579C22C2×#53812yes—
580C22#348912yes—
581C22#348912yes—
582C22#351612yes—
583C22#351812yes—
584C22#352312yes—
585C22#352412yes—
586C22#353811yescharacteristic
587C22#353811yescharacteristic
588C22#353911yescharacteristic
589C22#353911yescharacteristic
590C22#354112yes—
591C22#354112yes—
592C22#355212yes—
593C22#355412yes—
594C22#356112yes—
595C22#356212yes—
596C4—44no—
597C4—48no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—84no—
603C4—82no—
604C4—84no—
605C4—84no—
606C4—82no—
607C4#263118yes—
608C2—84no—
609C2C2×#263111yescharacteristic
610C2#3566312yes—
611C2#3737511yescharacteristic
612C2#3737511yescharacteristic
613C2#3737912yes—
614C2#4284412yes—
615C2#4285112yes—
616C2#4285812yes—
617C2#4290012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2327C2semidirect
4. #16638C2semidirect
5. #16623C2semidirect
6. #16614C2semidirect
7. #15233C2semidirect
8. #15201C2semidirect
9. #15178C2semidirect
11. #14350C2semidirect
12. #13969C2semidirect
13. #13969C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
C2×#23272
#166472
#166382
#166232
#166142
#152332
#152012
#151782
#151772
#143502
#139692