Finite Group Databasebeta

← #6380094

Subgroups — #6380094

Order 512.

2690 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2935C211yesmaximal, characteristic
2C2×#2913C212yesmaximal, complemented
3#34207C211yesmaximal, characteristic
4#34206C211yesmaximal, characteristic
5#22665C212yesmaximal, complemented
6#22663C212yesmaximal, complemented
7#22661C214yesmaximal, complemented
8#22660C214yesmaximal, complemented
9#22657C212yesmaximal, complemented
10D4×#107C212yesmaximal, complemented
11#21169C211yesmaximal, characteristic, complemented
12C2×#1947C211yesmaximal, characteristic, complemented
13#20849C212yesmaximal, complemented
14#20847C212yesmaximal, complemented
15#20800C211yesmaximal, characteristic, complemented
16#20664C212yesmaximal, complemented
17C4×#428C211yesmaximal, characteristic, complemented
18C22×#131C2212yes—
19C2×#532C2211yescharacteristic
20C2×#520C2212yes—
21C2×#520C2212yes—
22#2935C4116yescomplemented
23C2×#436C2212yes—
24C2×#435C2214yes—
25C2×C8×D4C2212yes—
26#2913—216no—
27C2×#428C2212yes—
28C2×#428C2214yes—
29C2×#428C2214yes—
30C2×#428C2211yescharacteristic
31C2×#428C2212yes—
32C2×#428C2211yescharacteristic
33C2×#427C2212yes—
34C2×#427C2214yes—
35C22×#107C2212yes—
36C22×#107C2214yes—
37C22×C4⋊C8C2211yescharacteristic
38C2×#408C2212yes—
39C2×C4×M16C2211yescharacteristic
40#2367—22no—
41#2366—21no—
42#2365—21no—
43#2361—22no—
44#2361—22no—
45#2354C2212yes—
46#2354C2212yes—
47#2354C2212yes—
48#2352C2214yes—
49D4×C4⋊C4C2212yes—
50D4×C4⋊C4C2212yes—
51D4×C4⋊C4C2212yes—
52#2326—22no—
53#2325—22no—
54#2311C2212yes—
55#2311C2212yes—
56C4×C4⋊Q8—21no—
57C4×#127—21no—
58C4×#121—22no—
59#2285C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C4×#117C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63#1949C2212yes—
64#1949C2212yes—
65#1948C2211yescharacteristic
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1947C2212yes—
69#1947C2211yescharacteristic
70#1947C2214yes—
71#1947C2212yes—
72#1854C2214yes—
73#1854C2214yes—
74#1854C2214yes—
75#1854C2214yes—
76#1853C2214yes—
77#1852C2214yes—
78#1852C2214yes—
79#1852C2214yes—
80#1847C2214yes—
81#1845C2212yes—
82#1844C2212yes—
83#1809C2214yes—
84#1809C2214yes—
85#1808C2214yes—
86#1807C2214yes—
87#1776C2212yes—
88#1776C2212yes—
89C4×#107C2212yes—
90C4×#106C2212yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2212yes—
94C2×#340C2212yes—
95C22×#41C2312yes—
96C2×#131—28no—
97C2×#131—28no—
98C2×#131C2×C4116yescomplemented
99C22×M16C2314yes—
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2314yes—
103C23×C8C2312yes—
104#532C2×C418yes—
105#532Q818yes—
106C2×C4⋊Q8D411yescharacteristic
107C2×#127D411yescharacteristic
108C2×#124D412yes—
109C2×#121D412yes—
110C2×#121D412yes—
111#520—28no—
112#520—28no—
113#520—44no—
114#520—44no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×D4—22no—
118C2×C4×D4—24no—
119C2×C4×D4—24no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×#117C2311yescharacteristic
127C2×#117C2312yes—
128C2×#117C2314yes—
129C2×#117C2314yes—
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140#436C2×C4116yescomplemented
141#435C2×C4132yescomplemented
142C8×D4C2×C4116yescomplemented
143#428—28no—
144#428—24no—
145#428—216no—
146#428—216no—
147#428—44no—
148#428D418yescomplemented
149#427—28no—
150#427—216no—
151C2×#107—28no—
152C2×#107—216no—
153C2×#107C2312yes—
154C2×#107C2314yes—
155C2×#107C2312yes—
156C2×#107C2314yes—
157C2×#107C2314yes—
158C2×#107C2312yes—
159C2×#107C2314yes—
160C2×#107C2312yes—
161C2×#106C2312yes—
162C2×#106C2314yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165C2×#106C2314yes—
166C2×#106C2312yes—
167C2×C4⋊C8C2×C418yescomplemented
168C2×C4⋊C8C2311yescharacteristic
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2311yescharacteristic
171#408C2×C4116yescomplemented
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2314yes—
174C4×M16C2×C418yescomplemented
175C2×#97C2312yes—
176C2×C4×C8C2312yes—
177#391—22no—
178#391—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—22no—
183#390—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#388—24no—
188#388—24no—
189#388—24no—
190#388—24no—
191#387—24no—
192#386—24no—
193#386—24no—
194#386—24no—
195#385—24no—
196#385—24no—
197#385—24no—
198#384—21no—
199#384—21no—
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#384C2311yescharacteristic
204#382C2312yes—
205#380—21no—
206#380—21no—
207#380C2314yes—
208#380C2314yes—
209#380C2314yes—
210#380C2314yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—24no—
227#376—21no—
228C2×#95—21no—
229C43—21no—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#340C2312yes—
239#340C2312yes—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41C22×C418yes—
245C2×#41C2×Q818yes—
246C22×Q8C2×D412yes—
247C22×D4C2×D412yes—
248C22×D4C2×D414yes—
249C22×D4C2×D412yes—
250C22×D4C2×D412yes—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2414yes—
259C23×C4C2411yescharacteristic
260#131—264no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—28no—
264C2×M16—28no—
265C2×M16C22×C4116yes—
266C2×M16C22×C414yes—
267C2×M16C22×C414yes—
268C2×M16C2×D414yes—
269C2×M16#4114yes—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—28no—
273C22×C8—28no—
274C22×C8C22×C418yes—
275C22×C8C22×C4116yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8C4⋊C418yes—
281#127C4⋊C418yes—
282#124C4⋊C4116yes—
283#121C4⋊C4116yes—
284#121C4⋊C4116yes—
285C4×Q8—48no—
286C4×Q8—48no—
287C4×D4—48no—
288C4×D4—416no—
289C4×D4—48no—
290C4×D4—416no—
291C4×D4—84no—
292C4×D4—84no—
293C4×D4—84no—
294C4×D4C22×C418yes—
295C4×D4C22×C418yes—
296C4×D4C2×Q818yes—
297C4×D4C2×Q818yes—
298#117—22no—
299#117—24no—
300#117—22no—
301#117—24no—
302#117—22no—
303#117—28no—
304#117—28no—
305#117—44no—
306#117—44no—
307#117C2×D414yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4C2×D414yes—
354C2×C22⋊C4C2×D414yes—
355C2×C42—21no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—82no—
367C2×C42C22×C414yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×Q814yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—28no—
374#107—24no—
375#107—24no—
376#107—28no—
377#107—28no—
378#107—24no—
379#107—44no—
380#107C2×D418yes—
381#106—28no—
382#106—24no—
383#106—24no—
384#106—24no—
385#106—44no—
386#106C2×D418yes—
387C4⋊C8C22×C414yes—
388C4⋊C8C22×C418yes—
389C4⋊C8C22×C414yes—
390C22⋊C8C22×C4116yes—
391C22⋊C8C22×C4116yes—
392#97C22×C418yes—
393C4×C8C22×C418yes—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—24no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406C24C22×D414yes—
407C24C22×D411yescharacteristic
408#41—216no—
409#41—216no—
410#41—216no—
411#41—216no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8C2×C4⋊C418yes—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4C2×C4⋊C418yes—
424C2×D4C2×C4⋊C4116yes—
425C2×D4C2×C4⋊C418yes—
426C2×D4C2×C4⋊C418yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4C2×C4⋊C414yes—
460C22×C4C2×C4⋊C414yes—
461C22×C4C23×C418yes—
462C22×C4C23×C412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D414yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×Q818yes—
471C22×C4C22×Q812yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476M16—28no—
477M16—232no—
478M16C4×D4116yescomplemented
479C2×C8—216no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—232no—
487C2×C8C23×C414yes—
488C2×C8C23×C414yes—
489C2×C8C23×C414yes—
490C2×C8C23×C414yes—
491C2×C8C22×D414yes—
492C2×C8C2×#4114yes—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4C2×C4⋊C418yes—
522C4⋊C4C2×C4⋊C418yes—
523C4⋊C4C22×D412yes—
524C4⋊C4C22×D412yes—
525C4⋊C4C22×D414yes—
526C22⋊C4—24no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4C2×C4⋊C4116yes—
536C22⋊C4C2×C4⋊C4116yes—
537C42—24no—
538C42—48no—
539C42—48no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42—42no—
544C42—44no—
545C42—42no—
546C42—42no—
547C42—82no—
548C42C2×C4⋊C414yes—
549C42C2×C4⋊C414yes—
550C42C23×C412yes—
551C42C23×C412yes—
552C42C22×Q812yes—
553C42C22×Q812yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23C22×C4⋊C418yes—
565C23C22×C4⋊C412yes—
566C23D4212yes—
567C23#54712yes—
568C23C23×D411yescharacteristic
569Q8—216no—
570D4—232no—
571D4—216no—
572D4—216no—
573D4—48no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—168no—
618C2×C4—168no—
619C2×C4—164no—
620C2×C4C22×C4⋊C412yes—
621C2×C4C22×C4⋊C412yes—
622C2×C4C22×C4⋊C412yes—
623C2×C4C22×C4⋊C418yes—
624C2×C4C22×C4⋊C412yes—
625C2×C4C22×C4⋊C412yes—
626C2×C4C2×C4×D414yes—
627C2×C4#52014yes—
628C2×C4D4212yes—
629C2×C4#54712yes—
630C2×C4D4×Q814yes—
631C2×C4#55414yes—
632C2×C4#57612yes—
633C2×C4#57812yes—
634C2×C4C24×C411yescharacteristic
635C2×C4C23×D411yescharacteristic
636C2×C4C23×Q811yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C8—24no—
641C8—24no—
642C8—24no—
643C8C2×C4×D418yes—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—28no—
651C22—28no—
652C22—28no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22D4×C4⋊C414yes—
658C22C23×C4⋊C411yescharacteristic
659C22C2×D4211yescharacteristic
660C22C2×#54711yescharacteristic
661C22C2×#57611yescharacteristic
662C22C2×#57811yescharacteristic
663C22#358511yescharacteristic
664C22#358611yescharacteristic
665C4—28no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—82no—
674C4—81no—
675C4—84no—
676C4—82no—
677C4—82no—
678C4D4×C4⋊C414yes—
679C4#291314yescomplemented
680C4C22×C4×D411yescharacteristic
681C4C2×#52011yescharacteristic
682C4C2×D4×Q811yescharacteristic
683C4C2×#55411yescharacteristic
684C2—24no—
685C2—48no—
686C2C2×D4×C4⋊C411yescharacteristic
687C2C2×#291311yescharacteristic
688C2#6010911yescharacteristic
689C2#6109311yescharacteristic
690C2#6109711yescharacteristic
691C2#6127811yescharacteristic
692C2#6128211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2913C2semidirect
5. #22665C2semidirect
6. #22663C2semidirect
7. #22661C2semidirect
8. #22660C2semidirect
9. #22657C2semidirect
10. D4×#107C2semidirect
11. #21169C2semidirect
12. C2×#1947C2semidirect
13. #20849C2semidirect
14. #20847C2semidirect
15. #20800C2semidirect
16. #20664C2semidirect
17. C4×#428C2semidirect
22. #2935C4semidirect
98. C2×#131C2×C4semidirect
140. #436C2×C4semidirect
141. #435C2×C4semidirect
142. C8×D4C2×C4semidirect
148. #428D4semidirect
167. C2×C4⋊C8C2×C4semidirect
171. #408C2×C4semidirect
174. C4×M16C2×C4semidirect
478. M16C4×D4semidirect
679. C4#2913semidirect

Maximal subgroups

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

subgroupindex
C2×#29352
C2×#29132
#342072
#342062
#226652
#226632
#226612
#226602
#226572
D4×#1072
#211692
C2×#19472
#208492
#208472
#208002
#206642
C4×#4282