Finite Group Databasebeta

← #360156

Subgroups — #360156

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2C2×C4×#123C211yesmaximal, characteristic
3C2×#1890C212yesmaximal
4C2×#1799C211yesmaximal, characteristic
5#14685C212yesmaximal, complemented
6#11222C212yesmaximal, complemented
7#11070C212yesmaximal, complemented
8#10854C212yesmaximal, complemented
9#10745C212yesmaximal
10C22×#123C2211yescharacteristic
11C2×#518C2212yes—
12C22×C4×D4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#99C2212yes—
15#2349—24no—
16D4×C4⋊C4—24no—
17#2316—21no—
18#2315—21no—
19#2315C2212yes—
20C4×#123—24no—
21C4×#123C418yescomplemented
22C2×#388—21no—
23C2×#387—21no—
24C2×#386C2211yescharacteristic
25C2×#385C2211yescharacteristic
26#2285—21no—
27C2×#381C2214yes—
28C2×#381C2211yescharacteristic
29C2×#380—21no—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4—21no—
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1890—28no—
36#1799—24no—
37C2×#342C2212yes—
38C2×#342C2214yes—
39C2×#342C2214yes—
40C4⋊#95—24no—
41#1434—24no—
42C22⋊#95C2214yes—
43C23×D4C2311yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#123—22no—
49C2×#123—22no—
50C2×#123—22no—
51C2×#123C2×C414yes—
52C2×#123D414yes—
53C2×#123Q814yes—
54#518—24no—
55#518—24no—
56#518—44no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—24no—
60C2×C4×D4—28no—
61C2×C4×D4C2×C414yes—
62C2×C4×D4D414yes—
63C2×#117—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D414yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2311yescharacteristic
77C22×C22⋊C4C2311yescharacteristic
78C22×C22⋊C4C2311yescharacteristic
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42—22no—
84C22×C42—21no—
85C22×C42—24no—
86C22×C42C2×C414yescomplemented
87C22×C42D411yescharacteristic
88C2×#99—24no—
89C2×#99—24no—
90C2×#99—24no—
91#389—28no—
92#388—28no—
93#387—28no—
94#386—22no—
95#386C2×C414yes—
96#385—28no—
97#385—28no—
98#385—22no—
99#385C2×C414yescomplemented
100#384—22no—
101#382—22no—
102#381—22no—
103#381—28no—
104#381—44no—
105#381C2×C414yescomplemented
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—24no—
115#380—24no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—28no—
119C4×C22⋊C4—28no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C414yescomplemented
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—28no—
129C2×#95—22no—
130C2×#95—28no—
131C2×#95D414yes—
132#342—24no—
133#342—28no—
134#342—44no—
135C25C2×D411yescharacteristic
136C25C2411yescharacteristic
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C22×C412yes—
142C22×D4C2×D412yes—
143C22×D4C2×D412yes—
144C22×D4C2×Q812yes—
145C22×D4#4112yes—
146C22×D4#4112yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C22⋊C414yes—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D418yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×Q812yes—
184C23×C4#4114yes—
185#123—28no—
186#123—28no—
187#123—28no—
188#123C4⋊C4116yescomplemented
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—416no—
194C4×D4—416no—
195#117—44no—
196#117—44no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—84no—
213C2×C4⋊C4C22⋊C414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4C22⋊C414yes—
245C2×C22⋊C4C22×C412yes—
246C2×C22⋊C4C22×C412yes—
247C2×C22⋊C4C22×C412yes—
248C2×C22⋊C4C2×D412yes—
249C2×C22⋊C4C2×D412yes—
250C2×C22⋊C4C2×Q812yes—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—28no—
260C2×C42—44no—
261C2×C42—41no—
262C2×C42—42no—
263C2×C42—41no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42C22⋊C414yes—
269#99—48no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—416no—
280#95—416no—
281#95—44no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24C2×C22⋊C412yes—
291C24#12112yes—
292C24C22⋊Q812yes—
293C24C23×C411yescharacteristic
294C24C22×D411yescharacteristic
295C24C22×D412yes—
296C24C22×D411yescharacteristic
297C24C22×D411yescharacteristic
298C24C22×Q811yescharacteristic
299C24C2×#4111yescharacteristic
300C24C2×#4111yescharacteristic
301C24C2×#4112yes—
302C24H(5,𝔽2)12yes—
303C242−1+412yes—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4C2×C4⋊C414yes—
317C2×D4#11714yes—
318C2×D4#12114yes—
319C2×D4C22⋊Q814yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—48no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—84no—
397C22×C4—82no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C4⋊C414yescomplemented
406C22×C4C4×D414yes—
407C22×C4C4×D414yes—
408C22×C4#12011yescharacteristic
409C22×C4#12012yes—
410C22×C4#12011yescharacteristic
411C22×C4#12112yes—
412C22×C4#12114yes—
413C22×C4#12114yes—
414C22×C4C22⋊Q812yes—
415C22×C4#12312yes—
416C22×C4#12312yes—
417C4⋊C4—216no—
418C4⋊C4—28no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C22⋊C4—22no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—216no—
432C22⋊C4—28no—
433C22⋊C4—24no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4C2×C4⋊C414yescomplemented
446C42—24no—
447C42—24no—
448C42—48no—
449C42—48no—
450C42—44no—
451C42—48no—
452C42—44no—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—81no—
482C23—82no—
483C23—82no—
484C23—81no—
485C23#38014yes—
486C23C22×C22⋊C411yescharacteristic
487C23C22×C4⋊C411yescharacteristic
488C23C2×#11711yescharacteristic
489C23C2×C4×D412yes—
490C23#52012yes—
491C23C2×#12011yescharacteristic
492C23C2×#12111yescharacteristic
493C23C2×#12112yes—
494C23C2×#12111yescharacteristic
495C23C2×C22⋊Q811yescharacteristic
496C23C2×C22⋊Q811yescharacteristic
497C23C2×#12311yescharacteristic
498C23#53511yescharacteristic
499C23#53611yescharacteristic
500C23#53712yes—
501C23D4212yes—
502C23#54612yes—
503C23#54712yes—
504C23#54812yes—
505C23D4×Q812yes—
506C23#55212yes—
507C23#55312yes—
508C23#55412yes—
509D4—28no—
510D4—28no—
511D4—28no—
512D4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—84no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—88no—
572C2×C4—88no—
573C2×C4—84no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—168no—
579C2×C4#38014yes—
580C2×C4#38612yes—
581C2×C4#38614yes—
582C2×C4#38612yes—
583C2×C4#45814yes—
584C22—22no—
585C22—24no—
586C22—22no—
587C22—24no—
588C22—22no—
589C22—24no—
590C22—24no—
591C22—22no—
592C22—21no—
593C22—22no—
594C22—21no—
595C22—22no—
596C22—21no—
597C22—22no—
598C22—22no—
599C22—21no—
600C22—24no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#38011yescharacteristic
620C22C2×#38611yescharacteristic
621C22#231111yescharacteristic
622C22D4×C4⋊C412yes—
623C22#235212yes—
624C22#240811yescharacteristic
625C22#240911yescharacteristic
626C22#241612yes—
627C22#241712yes—
628C22#248611yescharacteristic
629C22#248711yescharacteristic
630C22C2×#45812yes—
631C22#302512yes—
632C4—24no—
633C4—22no—
634C4—24no—
635C4—44no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—82no—
641C4—88no—
642C4—164no—
643C4#189014yescomplemented
644C2—21no—
645C2—21no—
646C2—22no—
647C2—21no—
648C2—21no—
649C2—22no—
650C2—22no—
651C2—22no—
652C2—44no—
653C2#1387211yescharacteristic
654C2C2×#189011yescharacteristic
655C2#2095911yescharacteristic
656C2#2426112yes—
657C2#2435112yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #14685C2semidirect
6. #11222C2semidirect
7. #11070C2semidirect
8. #10854C2semidirect
21. C4×#123C4semidirect
86. C22×C42C2×C4semidirect
99. #385C2×C4semidirect
105. #381C2×C4semidirect
121. C4×C22⋊C4C2×C4semidirect
188. #123C4⋊C4semidirect
405. C22×C4C2×C4⋊C4semidirect
445. C22⋊C4C2×C4⋊C4semidirect
643. C4#1890semidirect

Maximal subgroups

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

subgroupindex
#341812
C2×C4×#1232
C2×#18902
C2×#17992
#146852
#112222
#110702
#108542
#107452