Finite Group Databasebeta

← #4414374

Subgroups — #4414374

Order 512.

2318 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2844C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#16653C212yesmaximal, complemented
4#16556C212yesmaximal
5#16327C214yesmaximal, complemented
6#16326C214yesmaximal, complemented
7#14881C212yesmaximal, complemented
8#14832C214yesmaximal, complemented
9#14825C212yesmaximal
10#14770C214yesmaximal, complemented
11#14707C212yesmaximal, complemented
12#14677C211yesmaximal, characteristic, complemented
13#14662C212yesmaximal
14C22×#124C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#2844—28no—
18C2×#400C2212yes—
19C2×#400C2214yes—
20C2×#399C2212yes—
21C2×#396C2214yes—
22C2×#395C2212yes—
23C2×#395C2214yes—
24C2×#395C2212yes—
25C2×#393C2214yes—
26C4×#124—28no—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#382C2212yes—
34C2×#382C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×C4×C22⋊C4C2214yes—
38C2×C43C2211yescharacteristic
39#1446C2214yes—
40#1445C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2212yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1438C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×#124—24no—
74C2×#124—24no—
75C2×#124Q818yescomplemented
76C2×C4×Q8—24no—
77C2×C4×Q8—42no—
78C2×C4×Q8—42no—
79C2×C4×D4—24no—
80C2×C4×D4—42no—
81C2×C4×D4—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#400—28no—
109#400—216no—
110#399—44no—
111#396—48no—
112#395—28no—
113#395—216no—
114#395—44no—
115#393—48no—
116#391—24no—
117#391—44no—
118#390—24no—
119#389—44no—
120#389—44no—
121#388—216no—
122#382—28no—
123#382—28no—
124#382D418yes—
125#380—44no—
126C4×C22⋊C4—216no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C43—24no—
151C25C2412yes—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—44no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—42no—
163C22×Q8C2×Q814yes—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4C2×Q814yes—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—28no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188#124—232no—
189C4×Q8—416no—
190C4×D4—416no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×Q8116yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42C2×D414yes—
265C2×C42C2×Q814yes—
266C2×C42#4114yes—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—28no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
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—42no—
288#95—42no—
289#95C2×D4116yes—
290C24—22no—
291C24—22no—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—22no—
299C24—22no—
300C24C22×Q814yes—
301C24C2511yescharacteristic
302C2×Q8—28no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—48no—
310C2×Q8—84no—
311C2×Q8—88no—
312C2×Q8—84no—
313C2×Q8—88no—
314C2×D4—28no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
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—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—48no—
385C22×C4—416no—
386C22×C4—48no—
387C22×C4—84no—
388C22×C4—88no—
389C22×C4C22×D414yes—
390C22×C4C22×D414yes—
391C22×C4C22×D412yes—
392C22×C4C22×D414yes—
393C22×C4C22×Q814yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q812yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4114yes—
398C22×C4H(5,𝔽2)14yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+414yes—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—88no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C22⋊C4—232no—
416C22⋊C4—416no—
417C22⋊C4—48no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C42—28no—
423C42—48no—
424C42—48no—
425C42—48no—
426C42—44no—
427C42C22⋊Q8116yes—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—28no—
432C23—28no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23C23×D411yescharacteristic
452C23C23×Q811yescharacteristic
453C23C22×#4111yescharacteristic
454C23C2×H(5,𝔽2)11yescharacteristic
455C23C2×H(5,𝔽2)11yescharacteristic
456C23C2×H(5,𝔽2)12yes—
457C23C2×2−1+412yes—
458C23#58511yescharacteristic
459C23#58511yescharacteristic
460C23#58512yes—
461C23#58512yes—
462Q8—416no—
463Q8—816no—
464D4—416no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
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—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—816no—
539C2×C4C2×C22⋊Q818yes—
540C2×C4C2×C22⋊Q814yes—
541C2×C4#56314yes—
542C22—44no—
543C22—48no—
544C22—44no—
545C22—44no—
546C22—48no—
547C22—84no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—84no—
552C22—82no—
553C22C22×C22⋊Q811yescharacteristic
554C22#346011yescharacteristic
555C22#346011yescharacteristic
556C22#346012yes—
557C22#346112yes—
558C22#348411yescharacteristic
559C22#348711yescharacteristic
560C22#348812yes—
561C22#348912yes—
562C22#350111yescharacteristic
563C22#350111yescharacteristic
564C22#350312yes—
565C22#350412yes—
566C22C2×#56311yescharacteristic
567C22#352911yescharacteristic
568C22#353112yes—
569C22#354512yes—
570C22#354511yescharacteristic
571C22#354911yescharacteristic
572C22#355812yes—
573C22#355912yes—
574C22#356112yes—
575C22#356812yes—
576C4—24no—
577C4—24no—
578C4—28no—
579C4—44no—
580C4—44no—
581C4—44no—
582C4—42no—
583C4—44no—
584C4—84no—
585C4—84no—
586C4—84no—
587C4—84no—
588C2—84no—
589C2#3517411yescharacteristic
590C2#3525712yes—
591C2#3669811yescharacteristic
592C2#3759912yes—
593C2#3761111yescharacteristic
594C2#4613412yes—
595C2#4613611yescharacteristic
596C2#4615211yescharacteristic
597C2#4615412yes—
598C2#4622511yescharacteristic
599C2#4624111yescharacteristic
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×#2844C2semidirect
3. #16653C2semidirect
5. #16327C2semidirect
6. #16326C2semidirect
7. #14881C2semidirect
8. #14832C2semidirect
10. #14770C2semidirect
11. #14707C2semidirect
12. #14677C2semidirect
75. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28442
C2×C4×#1242
#166532
#165562
#163272
#163262
#148812
#148322
#148252
#147702
#147072
#146772
#146622