Finite Group Databasebeta

← #362897

Subgroups — #362897

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#14682C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#11381C212yesmaximal
5#11353C212yesmaximal
6#10912C214yesmaximal
7#10846C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C4×C22⋊Q8—21no—
12C4×C22⋊Q8—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C2×#391C411yescharacteristic, complemented
20C2×#389C411yescharacteristic, complemented
21C2×#387C412yescomplemented
22C2×#385C2212yes—
23C2×#382—21no—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C412yescomplemented
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2212yes—
32C2×#342C2218yes—
33C2×#342C2218yes—
34C4⋊#95—21no—
35C4⋊#95—21no—
36#1434—21no—
37#1434—21no—
38#1434—21no—
39C22⋊#95C2212yes—
40C22⋊#95C2212yes—
41C22⋊#95C2212yes—
42C4×#95—21no—
43C23×Q8C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—28no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—44no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2312yes—
69C22×C42—22no—
70C22×C42—44no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C412yes—
73C22×C42C2×C412yes—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85#391—21no—
86#391—21no—
87#391—21no—
88#391—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#382—44no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380—24no—
107#380—28no—
108#380Q818yescomplemented
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—44no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—44no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137#342—216no—
138#342—48no—
139C25C2411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—24no—
143C22×Q8—42no—
144C22×Q8—42no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×Q814yes—
179C23×C4#4114yes—
180C22⋊Q8—28no—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C22⋊Q8—28no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8—44no—
187C4×Q8—44no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4C2×Q818yes—
228C2×C22⋊C4#4118yes—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—28no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24C23×C411yescharacteristic
290C24C22×D411yescharacteristic
291C24C22×D411yescharacteristic
292C24C22×Q812yes—
293C24C2×#4114yes—
294C24C2×#4112yes—
295C24H(5,𝔽2)12yes—
296C242−1+412yes—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—42no—
300C2×Q8—416no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—416no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4#11714yes—
402C22×C4#11714yes—
403C22×C4C4×Q814yes—
404C22×C4C4×Q814yes—
405C22×C4C22⋊Q814yes—
406C22×C4C22⋊Q814yes—
407C22×C4#12412yes—
408C22×C4#12412yes—
409C22×C4C4⋊Q812yes—
410C22×C4C4⋊Q812yes—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—42no—
416C4⋊C4—416no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—88no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—216no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—44no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—88no—
454C42#9914yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23C22×C22⋊C411yescharacteristic
478C23C2×#11712yes—
479C23C2×C4×Q812yes—
480C23#52012yes—
481C23C2×C22⋊Q812yes—
482C23#52512yes—
483C23#52512yes—
484C23C2×#12411yescharacteristic
485C23C2×C4⋊Q811yescharacteristic
486C23#53511yescharacteristic
487C23#53611yescharacteristic
488C23#53712yes—
489C23#55414yes—
490C23#55614yes—
491C23#55714yes—
492C23#55914yes—
493Q8—48no—
494Q8—48no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
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—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4#39112yes—
576C2×C4#39112yes—
577C2×C4#39112yes—
578C2×C4#39112yes—
579C2×C4C2×#9912yes—
580C2×C4C2×#9912yes—
581C2×C4C2×#9912yes—
582C2×C4#41012yes—
583C2×C4#41012yes—
584C2×C4#41012yes—
585C22—22no—
586C22—21no—
587C22—22no—
588C22—21no—
589C22—22no—
590C22—21no—
591C22—22no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22#228011yescharacteristic
612C22C2×#39111yescharacteristic
613C22#231111yescharacteristic
614C22#235514yes—
615C22#258012yes—
616C22#258112yes—
617C22#258412yes—
618C22#259212yes—
619C22C22×#9911yescharacteristic
620C22C2×#41011yescharacteristic
621C22C2×#41011yescharacteristic
622C22C2×#41011yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—44no—
635C4—44no—
636C4—88no—
637C4—82no—
638C4—82no—
639C4—164no—
640C4#199614yes—
641C2—22no—
642C2—22no—
643C2—22no—
644C2—22no—
645C2—24no—
646C2#1392511yescharacteristic
647C2#2072211yescharacteristic
648C2#2072311yescharacteristic
649C2C2×#199611yescharacteristic
650C2#2133211yescharacteristic
651C2#2187411yescharacteristic
652C2#2187511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×C4×Q8C4semidirect
19. C2×#391C4semidirect
20. C2×#389C4semidirect
21. C2×#387C4semidirect
28. C2×C4×C4⋊C4C4semidirect
29. C2×C4×C4⋊C4C4semidirect
108. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#146822
C4×#3802
#113812
#113532
#109122
#108462