Finite Group Databasebeta

← #6444615

Subgroups — #6444615

Order 512.

2082 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C212yesmaximal
2#34938C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic
4#26474C214yesmaximal, complemented
5#26473C214yesmaximal, complemented
6#26466C212yesmaximal, complemented
7#26465C212yesmaximal, complemented
8#26462C212yesmaximal, complemented
9#26461C212yesmaximal, complemented
10#23619C212yesmaximal
11#23455C212yesmaximal
12#21259C211yesmaximal, characteristic
13#21202C211yesmaximal, characteristic
14#21170C211yesmaximal, characteristic
15#21146C211yesmaximal, characteristic
16#20874C212yesmaximal
17C4×#428C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3078—216no—
22C2×#479C2214yes—
23C2×#479C2214yes—
24C2×#477C2214yes—
25C2×#475C2214yes—
26C2×#428C2212yes—
27C2×#428C2211yescharacteristic
28C2×#428C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#423C2212yes—
31C2×#419C2212yes—
32C2×#419C2212yes—
33C2×#419C2212yes—
34C2×#419C2212yes—
35C2×C4×M16C2211yescharacteristic
36#2552C2211yescharacteristic
37#2552C2211yescharacteristic
38#2551C2211yescharacteristic
39#2551C2211yescharacteristic
40#2550C2212yes—
41#2550C2212yes—
42#2549—22no—
43#2440—21no—
44#2440—21no—
45#2399—22no—
46#2365—21no—
47#2365—21no—
48#2365—22no—
49#2365C2212yes—
50#2365C2212yes—
51#2365C2214yes—
52C4×C4⋊Q8—21no—
53C4×C4⋊Q8—21no—
54#2296C2212yes—
55C2×#391C2212yes—
56C2×#389—21no—
57C2×#387—22no—
58#2285C2211yescharacteristic
59C4×#117C2211yescharacteristic
60C2×C4×C4⋊C4—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63#2068C2214yes—
64#2068C2214yes—
65#2066C2214yes—
66#2066C2214yes—
67#2026C2214yes—
68#2026C2214yes—
69#2024C2214yes—
70#2024C2214yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1981C2212yes—
75#1960C2214yes—
76#1956C2212yes—
77#1956C2212yes—
78#1949C2214yes—
79#1948C2212yes—
80#1947C2212yes—
81#1940C2214yes—
82#1939C2212yes—
83#1939C2212yes—
84#1865C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1863C2214yes—
88#1776C2212yes—
89#1776C2212yes—
90C4×#107C2212yes—
91C4×#106C2212yes—
92#1736C2212yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1728C2212yes—
96C23×Q8C2312yes—
97C22×M16C2311yescharacteristic
98C22×M16C2312yes—
99C22×M16C2311yescharacteristic
100#533—28no—
101#533—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—24no—
107C2×C4⋊Q8Q818yescomplemented
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×#125C2312yes—
113C2×#125C2312yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8—28no—
121C2×C4×Q8—44no—
122C2×C4×Q8C2314yes—
123C2×C4×Q8C2314yes—
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42—22no—
135C22×C42—21no—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#479—216no—
139#479—216no—
140#477—216no—
141#475—216no—
142#428—24no—
143#428—28no—
144#428D418yes—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151#423—28no—
152#423—24no—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157#419—28no—
158#419—28no—
159#419—28no—
160#419—44no—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C4×M16—24no—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#398—22no—
173#398—22no—
174#398—22no—
175#398—22no—
176#395—22no—
177#395—22no—
178#395—22no—
179#395—22no—
180#391—22no—
181#391—22no—
182#391—22no—
183#391—22no—
184#391C2314yes—
185#391C2314yes—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—22no—
191#389—22no—
192#389—24no—
193#389—22no—
194#389—22no—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#387—24no—
200#387—24no—
201#387—24no—
202#387—24no—
203#385—22no—
204#385—22no—
205#384—21no—
206#384—21no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382C2312yes—
210#380—21no—
211#380—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D412yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2312yes—
233C4×C22⋊C4—21no—
234C4×C22⋊C4—21no—
235#376—21no—
236C2×#95—22no—
237C2×#95—21no—
238C2×#95—22no—
239C2×#95D412yes—
240C43—21no—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2314yes—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—24no—
253C22×Q8—24no—
254C22×Q8—24no—
255C22×Q8—24no—
256C22×Q8—24no—
257C22×Q8C2×Q818yes—
258C22×Q8C2414yes—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—24no—
270C2×M16—22no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16C2×D414yes—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C22×C8C2412yes—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—28no—
282C4⋊Q8—28no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—42no—
286C4⋊Q8—42no—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C4⋊Q8C2×Q818yes—
290C4⋊Q8C2×Q818yes—
291#125—24no—
292#125—24no—
293#125—42no—
294#125—42no—
295C22⋊Q8—48no—
296C22⋊Q8—48no—
297C4×Q8—28no—
298C4×Q8—28no—
299C4×Q8—48no—
300C4×Q8—48no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—48no—
304C4×Q8—48no—
305#117—22no—
306#117—22no—
307#117—24no—
308#117—24no—
309#117C2×D414yes—
310#117#4114yes—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—42no—
340C2×C4⋊C4—48no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—48no—
373C2×C42—44no—
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×Q814yes—
379C2×C42#4114yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—28no—
385#107—24no—
386#107C2×D418yes—
387#106—28no—
388#106—24no—
389#106C2×D418yes—
390C4⋊C8—28no—
391C4⋊C8—28no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394#103—28no—
395#103—28no—
396#103—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—44no—
401#103—44no—
402#97—24no—
403C4×C8—24no—
404#95—24no—
405#95—24no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95C2×D412yes—
421#95C2×D412yes—
422#95C2×D412yes—
423#95C2×D412yes—
424C24C22×D411yescharacteristic
425C2×Q8—28no—
426C2×Q8—24no—
427C2×Q8—216no—
428C2×Q8—216no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—48no—
437C2×Q8—48no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—82no—
442C2×Q8—82no—
443C2×Q8C22×Q818yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4C22⋊Q814yes—
473C22×C4C22⋊Q814yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×Q812yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+412yes—
489C22×C4C2511yescharacteristic
490M16—28no—
491M16—48no—
492M16—44no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8C22×D414yes—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—81no—
537C4⋊C4—81no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4#13714yes—
541C4⋊C4C22×D412yes—
542C4⋊C4C22×D412yes—
543C4⋊C4C22×D414yes—
544C4⋊C4C2×#4112yes—
545C4⋊C4C2×#4112yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C42—24no—
550C42—24no—
551C42—24no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—48no—
558C42—48no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42—44no—
563C42—42no—
564C42—81no—
565C42—81no—
566C42C22⋊Q814yes—
567C42C22⋊Q814yes—
568C42C22×Q812yes—
569C42C22×Q812yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23C2×C22⋊Q812yes—
577C23D4212yes—
578C23#54812yes—
579C23C23×D411yescharacteristic
580Q8—28no—
581Q8—28no—
582Q8—44no—
583Q8—44no—
584Q8—416no—
585Q8—88no—
586Q8—88no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—416no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4C2×C22⋊Q812yes—
639C2×C4C2×C22⋊Q812yes—
640C2×C4C2×C22⋊Q812yes—
641C2×C4C2×C22⋊Q812yes—
642C2×C4C2×C22⋊Q812yes—
643C2×C4#53214yes—
644C2×C4#54214yes—
645C2×C4D4212yes—
646C2×C4#54812yes—
647C2×C4D4×Q814yes—
648C2×C4#55414yes—
649C2×C4C2×#13714yes—
650C2×C4C2×#13712yes—
651C2×C4#57512yes—
652C2×C4C23×D411yescharacteristic
653C2×C4C23×Q811yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C8—24no—
659C8—44no—
660C8—42no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22#245214yes—
668C22C22×C22⋊Q811yescharacteristic
669C22C2×D4211yescharacteristic
670C22C2×#54811yescharacteristic
671C22C22×#13711yescharacteristic
672C22C2×#57511yescharacteristic
673C22#358611yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—24no—
679C4—48no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#245214yes—
692C4#307814yescomplemented
693C4C2×#53211yescharacteristic
694C4C2×#54211yescharacteristic
695C4C2×D4×Q811yescharacteristic
696C4C2×#55411yescharacteristic
697C2—24no—
698C2C2×#245211yescharacteristic
699C2C2×#307811yescharacteristic
700C2#6049111yescharacteristic
701C2#6106711yescharacteristic
702C2D4×#13711yescharacteristic
703C2#6132211yescharacteristic
704C2#6132311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #26474C2semidirect
5. #26473C2semidirect
6. #26466C2semidirect
7. #26465C2semidirect
8. #26462C2semidirect
9. #26461C2semidirect
107. C2×C4⋊Q8Q8semidirect
692. C4#3078semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#349382
C2×#23652
#264742
#264732
#264662
#264652
#264622
#264612
#236192
#234552
#212592
#212022
#211702
#211462
#208742
C4×#4282