Finite Group Databasebeta

← #4314294

Subgroups — #4314294

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×#2326C211yesmaximal, characteristic
3#16814C212yesmaximal
4#16522C212yesmaximal, complemented
5#16327C214yesmaximal, complemented
6#15430C214yesmaximal, complemented
7#15378C214yesmaximal, complemented
8#15119C212yesmaximal, complemented
9#14956C214yesmaximal, complemented
10#14723C212yesmaximal
11#14501C211yesmaximal, characteristic, complemented
12#14492C212yesmaximal
13#13976C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2811—28no—
19C2×#399C2214yes—
20C2×#398C2214yes—
21C2×#395C2214yes—
22C2×#395C2212yes—
23C2×#395C2214yes—
24C2×#393C2214yes—
25C2×#393C2212yes—
26#2326—28no—
27C2×#391C2212yes—
28C2×#391C2214yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39C2×#376C2211yescharacteristic
40#1446C2212yes—
41#1446C2212yes—
42#1445C2214yes—
43#1444C2212yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2212yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1438C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#124—24no—
75C2×#124—24no—
76C2×#124Q818yescomplemented
77C2×C22⋊Q8—48no—
78C2×C22⋊Q8—44no—
79C2×C4×Q8—24no—
80C2×C4×Q8—42no—
81C2×C4×Q8—42no—
82C2×C4×D4—24no—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110#399—48no—
111#398—48no—
112#395—28no—
113#395—216no—
114#395—48no—
115#393—44no—
116#393—48no—
117#391—28no—
118#391—28no—
119#391—28no—
120#391—216no—
121#391D418yes—
122#390—24no—
123#389—44no—
124#388—216no—
125#385—216no—
126#385—48no—
127#384—24no—
128#384—44no—
129#376—24no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C25C2412yes—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—28no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—44no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8C2×D414yes—
169C22×Q8C2×Q814yes—
170C22×Q8#4114yes—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4C2×Q814yes—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195#124—232no—
196C22⋊Q8—432no—
197C4×Q8—416no—
198C4×D4—416no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D414yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
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⋊C4C2×Q8116yes—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C2×D414yes—
271C2×C42C2×Q814yes—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—42no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95C2×D4116yes—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—22no—
300C24—22no—
301C24—24no—
302C24—24no—
303C24—22no—
304C24—22no—
305C24C22×Q814yes—
306C24C2511yescharacteristic
307C2×Q8—216no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—48no—
315C2×Q8—84no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8—88no—
319C2×Q8—88no—
320C2×Q8—88no—
321C2×Q8—84no—
322C2×Q8—88no—
323C2×Q8C22⋊Q8116yescomplemented
324C2×D4—28no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
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—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—88no—
391C22×C4—84no—
392C22×C4—88no—
393C22×C4C22×D414yes—
394C22×C4C22×D414yes—
395C22×C4C22×D412yes—
396C22×C4C22×D414yes—
397C22×C4C22×Q814yes—
398C22×C4C22×Q812yes—
399C22×C4C22×Q814yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4114yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)18yes—
404C22×C4H(5,𝔽2)12yes—
405C22×C42−1+418yes—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—416no—
410C4⋊C4—48no—
411C4⋊C4—44no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C22⋊C4—232no—
419C22⋊C4—416no—
420C22⋊C4—48no—
421C22⋊C4—416no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C42—28no—
426C42—28no—
427C42—44no—
428C42—48no—
429C42—48no—
430C42—84no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—28no—
435C23—28no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23C23×D411yescharacteristic
455C23C23×Q811yescharacteristic
456C23C22×#4111yescharacteristic
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×H(5,𝔽2)14yes—
460C23C2×2−1+414yes—
461C23#58511yescharacteristic
462C23#58511yescharacteristic
463Q8—416no—
464Q8—816no—
465Q8—816no—
466D4—416no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—816no—
536C2×C4C2×C22⋊Q814yes—
537C2×C4C2×C22⋊Q818yes—
538C22—44no—
539C22—44no—
540C22—48no—
541C22—48no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—84no—
547C22—82no—
548C22—84no—
549C22C22×C22⋊Q811yescharacteristic
550C22#345911yescharacteristic
551C22#345911yescharacteristic
552C22#348411yescharacteristic
553C22#348511yescharacteristic
554C22#348612yes—
555C22#349812yes—
556C22#350111yescharacteristic
557C22#350111yescharacteristic
558C22#350314yes—
559C22#352214yes—
560C22#353212yes—
561C22#353514yes—
562C22#354412yes—
563C22#354912yes—
564C22#355012yes—
565C22#356114yes—
566C4—24no—
567C4—28no—
568C4—42no—
569C4—48no—
570C4—44no—
571C4—42no—
572C4—44no—
573C4—44no—
574C4—48no—
575C4—84no—
576C4—82no—
577C4—84no—
578C2—84no—
579C2#3498511yescharacteristic
580C2#3574812yes—
581C2#3726712yes—
582C2#4212212yes—
583C2#4681612yes—
584C2#4772512yes—
585C2#5160112yes—
586C2#5190211yescharacteristic
587C2#5207211yescharacteristic
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
1. C2×#2811C2semidirect
4. #16522C2semidirect
5. #16327C2semidirect
6. #15430C2semidirect
7. #15378C2semidirect
8. #15119C2semidirect
9. #14956C2semidirect
11. #14501C2semidirect
13. #13976C2semidirect
76. C2×#124Q8semidirect
323. C2×Q8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#23262
#168142
#165222
#163272
#154302
#153782
#151192
#149562
#147232
#145012
#144922
#139762