Finite Group Databasebeta

← #4310270

Subgroups — #4310270

Order 512.

3042 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×#2656C211yesmaximal, characteristic
2C2×#2646C211yesmaximal, characteristic, complemented
3#16794C214yesmaximal, complemented
4#16762C214yesmaximal, complemented
5#16373C214yesmaximal, complemented
6#16372C214yesmaximal, complemented
7#14747C214yesmaximal, complemented
8#14747C214yesmaximal, complemented
9#14698C212yesmaximal, complemented
10#14271C212yesmaximal, complemented
11C2×#1434C211yesmaximal, characteristic, complemented
12C22×#121C2212yes—
13C22×#121C2214yes—
14C23×C42C2211yescharacteristic
15#2656—28no—
16#2646—28no—
17C2×#400C2214yes—
18C2×#399C2214yes—
19C2×#398C2214yes—
20C2×#394C2214yes—
21C2×#394C2214yes—
22C2×#392C2214yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#385C2214yes—
30C2×#380C2212yes—
31C22×#95C2211yescharacteristic
32C22×#95C2212yes—
33C22×#95C2212yes—
34C22×#95C2211yescharacteristic
35#1445C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1440C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1440C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2214yes—
51#1438C2214yes—
52#1438C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57#1434C2214yes—
58#1434C2214yes—
59C23×D4C2314yes—
60C23×D4C2312yes—
61C23×D4C2314yes—
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2312yes—
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C2×#121—216no—
68C2×#121—28no—
69C2×#121—28no—
70C2×#121—28no—
71C2×#121—48no—
72C2×#121—48no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C42—24no—
92C22×C42—44no—
93C22×C42C2312yes—
94C22×C42C2312yes—
95#400—216no—
96#399—216no—
97#398—216no—
98#394—48no—
99#394—48no—
100#392—216no—
101#390—44no—
102#390—44no—
103#386—48no—
104#386—48no—
105#386—48no—
106#386—48no—
107#385—216no—
108#380—28no—
109C2×#95—24no—
110C2×#95—44no—
111C2×#95—44no—
112C2×#95D418yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C25C2414yes—
131C25C2411yescharacteristic
132C22×D4—28no—
133C22×D4—28no—
134C22×D4—28no—
135C22×D4—28no—
136C22×D4—28no—
137C22×D4—28no—
138C22×D4—28no—
139C22×D4—28no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—41no—
167C23×C4—41no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C2×D418yes—
179C23×C4C2×D414yes—
180C23×C4#4114yes—
181C23×C4#4114yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191#121—464no—
192#121—432no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
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×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—416no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95C2×D4116yes—
271C24—22no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24C22×D412yes—
293C24C2×#4112yes—
294C24C2×#4112yes—
295C24H(5,𝔽2)14yes—
296C24H(5,𝔽2)14yes—
297C24C2511yescharacteristic
298C2×D4—416no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—416no—
302C2×D4—416no—
303C2×D4—416no—
304C2×D4—416no—
305C2×D4—416no—
306C2×D4—416no—
307C2×D4—416no—
308C2×D4—48no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4—88no—
314C2×D4—88no—
315C2×D4—88no—
316C2×D4—88no—
317C2×D4—88no—
318C2×D4—88no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—216no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
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—44no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4#124116yes—
400C22×C4C22×D418yes—
401C22×C4C22×D414yes—
402C22×C4C2×#4114yes—
403C22×C4C2×#4114yes—
404C22×C4H(5,𝔽2)12yes—
405C22×C4H(5,𝔽2)14yes—
406C22×C42−1+412yes—
407C4⋊C4—416no—
408C4⋊C4—48no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C22⋊C4—416no—
414C22⋊C4—416no—
415C22⋊C4—416no—
416C22⋊C4—416no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C42—48no—
426C42—48no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C2×#12414yes—
481C23#53518yes—
482C23#54018yes—
483C23#54014yes—
484C23#54014yes—
485C23C23×D411yescharacteristic
486C23C22×#4111yescharacteristic
487C23C22×#4111yescharacteristic
488C23C2×H(5,𝔽2)12yes—
489C23C2×H(5,𝔽2)12yes—
490C23C2×H(5,𝔽2)12yes—
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×2−1+411yescharacteristic
493C23#58512yes—
494C23#58511yescharacteristic
495C23#58511yescharacteristic
496D4—816no—
497D4—832no—
498D4—832no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—416no—
516C2×C4—416no—
517C2×C4—416no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
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—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—816no—
561C2×C4—816no—
562C2×C4C2×#12418yes—
563C22—28no—
564C22—24no—
565C22—28no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—28no—
571C22—28no—
572C22—24no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22#264118yes—
589C22C22×#12411yescharacteristic
590C22C2×#53512yes—
591C22C2×#54012yes—
592C22C2×#54011yescharacteristic
593C22C2×#54011yescharacteristic
594C22#345912yes—
595C22#346011yescharacteristic
596C22#346111yescharacteristic
597C22#349011yescharacteristic
598C22#349111yescharacteristic
599C22#349211yescharacteristic
600C22#349311yescharacteristic
601C22#349611yescharacteristic
602C22#349612yes—
603C22#349711yescharacteristic
604C22#353412yes—
605C22#353412yes—
606C22#353612yes—
607C22#353712yes—
608C22#355812yes—
609C22#355912yes—
610C22#356012yes—
611C22#356012yes—
612C4—28no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—48no—
617C4—84no—
618C4—84no—
619C4—84no—
620C4—84no—
621C2—28no—
622C2—164no—
623C2C2×#264111yescharacteristic
624C2#3499311yescharacteristic
625C2#3515811yescharacteristic
626C2#3672711yescharacteristic
627C2#3673111yescharacteristic
628C2#3673912yes—
629C2#3674112yes—
630C2#3697611yescharacteristic
631C2#3697811yescharacteristic
632C2#3745912yes—
633C2#3746512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2656C2
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×#2646C2semidirect
3. #16794C2semidirect
4. #16762C2semidirect
5. #16373C2semidirect
6. #16372C2semidirect
7. #14747C2semidirect
8. #14747C2semidirect
9. #14698C2semidirect
10. #14271C2semidirect
11. C2×#1434C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26562
C2×#26462
#167942
#167622
#163732
#163722
#147472
#146982
#142712
C2×#14342