Finite Group Databasebeta

← #365591

Subgroups — #365591

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal, complemented
5C2×#1854C212yesmaximal, complemented
6C2×#1847C211yesmaximal, characteristic, complemented
7#13905C211yesmaximal, characteristic
8#11096C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2211yescharacteristic
22#2354—24no—
23#2354—24no—
24C2×#390—21no—
25C2×#388—22no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1955C418yescomplemented
35#1949—24no—
36#1854—28no—
37#1854—28no—
38#1847—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1445—21no—
46#1438—21no—
47#1437—22no—
48C23×D4D411yescharacteristic
49C24×C4C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4Q814yes—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—42no—
64C2×#117—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—22no—
87C2×#107—42no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—22no—
92C2×#106—42no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2×C414yes—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8C2×C418yes—
101C2×C22⋊C8C2×C418yes—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97C2×C414yes—
105#390—28no—
106#388—216no—
107#384—22no—
108#384D414yes—
109#382—22no—
110#382—22no—
111#380—24no—
112#380—24no—
113#380—24no—
114#380—24no—
115#380—42no—
116#380—42no—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4D414yes—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—21no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—22no—
133#340—22no—
134#340D418yescomplemented
135C25C2×D412yes—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4C4⋊C414yes—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C22×C414yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×Q814yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C22×C412yes—
178C22×C8C22×C412yes—
179C22×C8C22×C414yes—
180C22×C8C2×D414yes—
181C22×C8#4114yes—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—28no—
186#117—48no—
187#117—48no—
188#117—48no—
189#117—48no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
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—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4C4⋊C414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
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—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4C4⋊C418yes—
241C2×C42—28no—
242C2×C42—24no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—81no—
248C2×C42—81no—
249C2×C42C22×C412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255#107—48no—
256#107—44no—
257#106—48no—
258#106—44no—
259C4⋊C8—28no—
260C22⋊C8—216no—
261C22⋊C8—216no—
262#97—28no—
263#95—28no—
264#95—24no—
265#95—44no—
266#95—44no—
267#95—42no—
268#95—42no—
269#95—42no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24C2×C4⋊C414yes—
285C24C22×D411yescharacteristic
286C2×D4—28no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—44no—
290C2×D4—44no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4C2×C4⋊C414yes—
366C22×C4C2×C4⋊C412yes—
367C22×C4C2×C4⋊C412yes—
368C22×C4#12012yes—
369C22×C4#12012yes—
370C22×C4#12312yes—
371C22×C4#12312yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×Q811yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8C4×D418yes—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—28no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—84no—
421C4⋊C4#12714yes—
422C22⋊C4—216no—
423C22⋊C4—416no—
424C22⋊C4—88no—
425C22⋊C4—88no—
426C22⋊C4—88no—
427C42—48no—
428C42—416no—
429C42—84no—
430C42C4×D414yes—
431C42C4×D414yes—
432C42C22⋊Q814yes—
433C42C22⋊Q814yes—
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—
469C23C22×C4⋊C411yescharacteristic
470C23C2×#12011yescharacteristic
471C23C2×#12311yescharacteristic
472C23D4212yes—
473C23#54712yes—
474C23#57512yes—
475C23#57512yes—
476C23#57611yescharacteristic
477C23#57711yescharacteristic
478C23#57711yescharacteristic
479C23#57811yescharacteristic
480D4—416no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#38514yes—
541C2×C4#38514yes—
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×C4×D412yes—
544C2×C4C2×C4×D411yescharacteristic
545C2×C4#52012yes—
546C2×C4C2×C22⋊Q811yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×#12711yescharacteristic
549C2×C4#53211yescharacteristic
550C2×C4#53212yes—
551C2×C4#53511yescharacteristic
552C2×C4#53611yescharacteristic
553C2×C4#54212yes—
554C2×C4D4212yes—
555C2×C4#54712yes—
556C2×C4#54812yes—
557C2×C4D4×Q812yes—
558C2×C4#55312yes—
559C2×C4#55412yes—
560C8—28no—
561C8—42no—
562C8—42no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22C2×#38511yescharacteristic
579C22D4×C4⋊C412yes—
580C22#242212yes—
581C22#243211yescharacteristic
582C22#243311yescharacteristic
583C22#291112yes—
584C22#291211yescharacteristic
585C22#291311yescharacteristic
586C22#300511yescharacteristic
587C22#300511yescharacteristic
588C22#301211yescharacteristic
589C22#301311yescharacteristic
590C22#309312yes—
591C22#309611yescharacteristic
592C22#309711yescharacteristic
593C22#326712yes—
594C22#326812yes—
595C22#326912yes—
596C22#327112yes—
597C22#334712yes—
598C22#334812yes—
599C22#334912yes—
600C22#335012yes—
601C4—24no—
602C4—24no—
603C4—44no—
604C4—44no—
605C4—82no—
606C4—84no—
607C4—84no—
608C4—161no—
609C4—161no—
610C4C4×#12711yescharacteristic
611C4#232611yescharacteristic
612C4#236112yes—
613C4#245212yes—
614C4#254211yescharacteristic
615C4#256611yescharacteristic
616C2—84no—
617C2#1397711yescharacteristic
618C2#2084511yescharacteristic
619C2#2084911yescharacteristic
620C2#2266012yes—
621C2#2266212yes—
622C2#2502312yes—
623C2#2502512yes—
624C2#2571011yescharacteristic
625C2#2571111yescharacteristic
626C2#2576411yescharacteristic
627C2#2576511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1949C2semidirect
4. C2×#1854C2semidirect
5. C2×#1854C2semidirect
6. C2×#1847C2semidirect
8. #11096C2semidirect
9. #11095C2semidirect
10. #10849C2semidirect
11. #10703C2semidirect
34. #1955C4semidirect
134. #340D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#19552
C2×#19492
C2×#18542
C2×#18472
#139052
#110962
#110952
#108492
#107032