Finite Group Databasebeta

← #7562740

Subgroups — #7562740

Order 512.

2055 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3373C212yesmaximal
2#59827C211yesmaximal, characteristic, complemented
3D4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#30829C214yesmaximal, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28512C211yesmaximal, characteristic, complemented
7#27776C212yesmaximal, complemented
8#26180C214yesmaximal, complemented
9#26044C212yesmaximal, complemented
10#25652C212yesmaximal, complemented
11#25479C212yesmaximal, complemented
12#24727C212yesmaximal, complemented
13#24330C212yesmaximal, complemented
14#23153C211yesmaximal, characteristic, complemented
15#22670C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#559C2212yes—
19C2×D4×Q8C2212yes—
20C22×C4⋊Q8—21no—
21#3373—216no—
22C2×#489C2214yes—
23C2×#486C2212yes—
24C2×#475C2212yes—
25C2×#475C2212yes—
26C2×#476C2214yes—
27C2×#463C2214yes—
28C2×#461C2211yescharacteristic
29C2×#459C2211yescharacteristic
30C2×#446C2212yes—
31C2×#442C2214yes—
32C2×C4×SD16C2212yes—
33C2×C4×D8C2212yes—
34#2868C2211yescharacteristic
35#2857—21no—
36#2724—22no—
37#2582C2212yes—
38#2551C2212yes—
39#2551C2212yes—
40#2547—21no—
41#2547C2212yes—
42#2546C2212yes—
43#2486—22no—
44#2447—22no—
45#2447C2214yes—
46#2440—22no—
47#2440—22no—
48#2434—22no—
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2212yes—
51D4×C4⋊C4C2212yes—
52D4×C4⋊C4C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C42×D4C2211yescharacteristic
57#2096C2214yes—
58#2088C2214yes—
59#2071C2214yes—
60#2068C2214yes—
61#2058C2214yes—
62#2042C2214yes—
63#2042C2214yes—
64#2024C2214yes—
65#1990C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1981C2211yescharacteristic
69#1956C2212yes—
70#1948C2211yescharacteristic
71#1947C2211yescharacteristic
72#1943C2214yes—
73#1939C2212yes—
74#1935C2214yes—
75#1927C2212yes—
76#1925C2212yes—
77#1917C2214yes—
78#1879C2212yes—
79#1877C2214yes—
80#1877C2212yes—
81#1864C2214yes—
82#1864C2214yes—
83#1864C2214yes—
84#1844C2212yes—
85#1802C2211yescharacteristic
86#1802C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1770C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94C23×Q8—22no—
95C22×SD16C2312yes—
96C22×D8C2312yes—
97#559—28no—
98#559—44no—
99D4×Q8—28no—
100D4×Q8—44no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—48no—
107C2×C4⋊Q8D411yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×#127C2311yescharacteristic
112C2×#124—22no—
113C2×#124C2314yes—
114C2×#123—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—24no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2314yes—
125C2×C4×D4C2314yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C42—21no—
132#489—216no—
133#486—28no—
134#476—216no—
135#475—28no—
136#475—44no—
137#463—216no—
138#461—24no—
139#459D418yes—
140#446—28no—
141#442—216no—
142C4×SD16—28no—
143C4×D8—28no—
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×#106C2312yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#97C2314yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168#398—24no—
169#398—24no—
170#397—24no—
171#397—24no—
172#397—24no—
173#395—24no—
174#395—24no—
175#391—22no—
176#391—22no—
177#391D412yes—
178#391C2312yes—
179#391C2314yes—
180#390—21no—
181#390—21no—
182#390C2312yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389D412yes—
189#389D412yes—
190#389C2314yes—
191#389C2314yes—
192#389C2314yes—
193#385—24no—
194#385—24no—
195#385—24no—
196#385—24no—
197#384—21no—
198#384—21no—
199#384D411yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#381—24no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C43C2311yescharacteristic
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—22no—
232C22×Q8—24no—
233C22×Q8—48no—
234C22×Q8—48no—
235C22×Q8—84no—
236C22×Q8C2×D412yes—
237C22×Q8C2412yes—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—24no—
247C23×C4—24no—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×D8—24no—
252C2×D8—24no—
253C2×D8C2×D418yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269C4⋊Q8—82no—
270#127—22no—
271#127C2×D414yes—
272#124—28no—
273#124—44no—
274#124—48no—
275#123—416no—
276C22⋊Q8—416no—
277C22⋊Q8—48no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—28no—
285C4×D4—28no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—24no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—48no—
293C4×D4—44no—
294C4×D4—82no—
295C4×D4C2×D414yes—
296C4×D4#4114yes—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2414yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—24no—
352#106—28no—
353#106—24no—
354#106—44no—
355C4⋊C8—28no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8C2×D414yes—
360#103—28no—
361#103—24no—
362#103—24no—
363#103—24no—
364#103—44no—
365#102—24no—
366#102—28no—
367#102—28no—
368#102—24no—
369#102—24no—
370#102—24no—
371#102C2×D418yes—
372#97—28no—
373C4×C8—24no—
374C4×C8—24no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95C2×D414yes—
386#95C2×D414yes—
387C24—21no—
388C24—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—84no—
404C2×Q8—832no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4C22×D414yes—
420C2×D4C22×D412yes—
421C2×D4C2×#4112yes—
422C2×D42−1+412yes—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2511yescharacteristic
457SD16—48no—
458D8—216no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8C22×D414yes—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—84no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4C22×D412yes—
503C4⋊C4C2×#4112yes—
504C4⋊C42−1+412yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—24no—
519C42—21no—
520C42—21no—
521C42—41no—
522C42—44no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42AGL(1,ℤ/8)14yes—
527C42C22×D412yes—
528C42C2×#4112yes—
529C42H(5,𝔽2)12yes—
530C422−1+412yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C23×D411yescharacteristic
546Q8—48no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—816no—
552Q8—88no—
553D4—24no—
554D4—22no—
555D4—24no—
556D4—22no—
557D4—22no—
558D4—42no—
559D4—48no—
560D4—42no—
561D4—84no—
562D4#54714yes—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—48no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—816no—
603C2×C4—88no—
604C2×C4—168no—
605C2×C4D4212yes—
606C2×C4D4212yes—
607C2×C4#54812yes—
608C2×C4#54812yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4#57712yes—
613C2×C4#57712yes—
614C2×C4C23×D411yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C8—24no—
621C8—42no—
622C8—42no—
623C8—42no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22C2×D4211yescharacteristic
639C22C2×#54811yescharacteristic
640C22#348611yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C2×#57711yescharacteristic
643C22C2×#57711yescharacteristic
644C22#358611yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—41no—
654C4—84no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#327214yes—
659C4#337314yescomplemented
660C4C2×#54711yescharacteristic
661C4#349911yescharacteristic
662C4#351111yescharacteristic
663C4#355211yescharacteristic
664C2—44no—
665C2—84no—
666C2#5912511yescharacteristic
667C2C2×#327211yescharacteristic
668C2#6109411yescharacteristic
669C2C2×#337311yescharacteristic
670C2#6134611yescharacteristic
671C2#6182711yescharacteristic
672C2#6187111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #59827C2semidirect
3. D4×C4⋊Q8C2semidirect
4. #30829C2semidirect
5. #28716C2semidirect
6. #28512C2semidirect
7. #27776C2semidirect
8. #26180C2semidirect
9. #26044C2semidirect
10. #25652C2semidirect
11. #25479C2semidirect
12. #24727C2semidirect
13. #24330C2semidirect
14. #23153C2semidirect
15. #22670C2semidirect
16. C4×#461C2semidirect
659. C4#3373semidirect

Maximal subgroups

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

subgroupindex
C2×#33732
#598272
D4×C4⋊Q82
#308292
#287162
#285122
#277762
#261802
#260442
#256522
#254792
#247272
#243302
#231532
#226702
C4×#4612
C4×#4592