Finite Group Databasebeta

← #387721

Subgroups — #387721

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic, complemented
3#13926C211yesmaximal, characteristic, complemented
4#11076C212yesmaximal, complemented
5#11049C212yesmaximal, complemented
6#11008C212yesmaximal
7#10931C212yesmaximal, complemented
8#10925C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10C22×#127C2211yescharacteristic
11C22×#124—21no—
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2548—24no—
20#2548C418yescomplemented
21C2×#394—21no—
22C2×#394—21no—
23C2×#391—21no—
24C2×#390—21no—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1984—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1441—21no—
38#1441—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8D411yescharacteristic
43C23×D4—21no—
44C23×D4—21no—
45C23×D4C2312yes—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#127—22no—
49C2×#127—41no—
50C2×#127—41no—
51C2×#127—41no—
52C2×#127—41no—
53C2×#127C2×C414yes—
54C2×#124—28no—
55C2×#124—44no—
56C2×#124—44no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8—41no—
61C2×C4×Q8C2×C414yes—
62C2×C4×Q8D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—22no—
81C2×C4⋊C8—41no—
82C2×C4⋊C8—41no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#102—42no—
95C2×C4×C8—22no—
96C2×C4×C8—41no—
97C2×C4×C8—41no—
98#394—28no—
99#394—28no—
100#391—44no—
101#390—28no—
102#390—24no—
103#390—24no—
104#390C2×C418yes—
105#390C2×C418yes—
106#384—41no—
107#384—41no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—41no—
110C4×C4⋊C4—41no—
111C4×C4⋊C4C2×C414yes—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118#340—24no—
119#340—24no—
120#340—42no—
121#340—42no—
122#336—24no—
123#336—24no—
124C25—22no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8C22⋊C414yes—
130C22×Q8D814yes—
131C22×Q8SD1614yes—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—82no—
156C22×D4—82no—
157C22×D4C22×C414yes—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8#4114yes—
182#127—44no—
183#127—44no—
184#127—44no—
185#127—44no—
186#124—432no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C22⋊C414yes—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—24no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C22×C412yes—
254C2×C42C22×C412yes—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C4⋊C8—44no—
260#102—48no—
261#102—48no—
262#102—48no—
263#102—48no—
264C4×C8—44no—
265#95—24no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24—24no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24C22×D411yescharacteristic
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8#102116yescomplemented
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—48no—
326C2×D4—88no—
327C2×D4—84no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—88no—
332C2×D4—84no—
333C2×D4—84no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×D812yes—
386C22×C4C2×D812yes—
387C22×C4C2×D812yes—
388C22×C4C2×SD1612yes—
389C22×C4C2×SD1612yes—
390C22×C4C2×SD1612yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C4H(5,𝔽2)12yes—
400C22×C42−1+412yes—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—48no—
449C42—84no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23C22×C22⋊C411yescharacteristic
486C23#53411yescharacteristic
487C23#53411yescharacteristic
488C23#54812yes—
489C23#55012yes—
490C23C22×D811yescharacteristic
491C23C22×SD1611yescharacteristic
492C23#57511yescharacteristic
493C23#57511yescharacteristic
494C23#57612yes—
495C23#57712yes—
496Q8—416no—
497D4—44no—
498D4—44no—
499D4—44no—
500D4—44no—
501D4—816no—
502D4—816no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—82no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4C2×#10214yes—
562C2×C4C2×#10214yes—
563C2×C4C2×#10214yes—
564C2×C4C4×#4112yes—
565C2×C4#51811yescharacteristic
566C2×C4#51811yescharacteristic
567C2×C4#52012yes—
568C2×C4#53411yescharacteristic
569C2×C4#53411yescharacteristic
570C2×C4#53812yes—
571C2×C4#53812yes—
572C2×C4#54212yes—
573C2×C4#54212yes—
574C2×C4#54812yes—
575C2×C4#55012yes—
576C2×C4#55112yes—
577C2×C4#55511yescharacteristic
578C2×C4#55511yescharacteristic
579C2×C4H(3,𝔽4)11yescharacteristic
580C2×C4#56212yes—
581C2×C4#56311yescharacteristic
582C8—44no—
583C8—82no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—162no—
592C22—162no—
593C22—162no—
594C22—162no—
595C22—162no—
596C22—162no—
597C22—162no—
598C22—162no—
599C22#230811yescharacteristic
600C22#234112yes—
601C22#273012yes—
602C22#273111yescharacteristic
603C22#273311yescharacteristic
604C22C22×#10211yescharacteristic
605C22#289611yescharacteristic
606C22#289712yes—
607C22#318611yescharacteristic
608C22#318611yescharacteristic
609C22#318711yescharacteristic
610C22#318711yescharacteristic
611C22#319812yes—
612C22#319911yescharacteristic
613C22#320011yescharacteristic
614C22#335812yes—
615C22#336212yes—
616C22#337012yes—
617C22#337112yes—
618C22#339112yes—
619C22#339212yes—
620C22#340112yes—
621C22#340312yes—
622C4—24no—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—84no—
629C4—84no—
630C4—162no—
631C4#233712yes—
632C4#237211yescharacteristic
633C4#237911yescharacteristic
634C4#273012yes—
635C4#273111yescharacteristic
636C4#273311yescharacteristic
637C2—164no—
638C2#1449211yescharacteristic
639C2#2175811yescharacteristic
640C2#2177011yescharacteristic
641C2#2254012yes—
642C2#2254912yes—
643C2#3259612yes—
644C2#3260012yes—
645C2#3262811yescharacteristic
646C2#3262911yescharacteristic
647C2#3266811yescharacteristic
648C2#3266911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1984C2semidirect
3. #13926C2semidirect
4. #11076C2semidirect
5. #11049C2semidirect
7. #10931C2semidirect
8. #10925C2semidirect
9. #10806C2semidirect
20. #2548C4semidirect
293. C2×Q8#102semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#19842
#139262
#110762
#110492
#110082
#109312
#109252
#108062