Finite Group Databasebeta

← #7604174

Subgroups — #7604174

Order 512.

2070 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3373C212yesmaximal, complemented
2#59827C211yesmaximal, characteristic, complemented
3#59817C211yesmaximal, characteristic, complemented
4#31160C214yesmaximal, complemented
5#28283C211yesmaximal, characteristic, complemented
6#28251C211yesmaximal, characteristic
7#28195C212yesmaximal, complemented
8#27776C212yesmaximal, complemented
9#26842C214yesmaximal, complemented
10#26044C212yesmaximal, complemented
11#25367C212yesmaximal, complemented
12#24330C212yesmaximal, complemented
13#24186C212yesmaximal, complemented
14#23001C212yesmaximal, complemented
15#21712C211yesmaximal, characteristic, complemented
16C4×#492C211yesmaximal, characteristic, complemented
17C4×#486C211yesmaximal, characteristic, complemented
18C2×#559C2212yes—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21#3373—216no—
22C2×#492C2211yescharacteristic
23C2×#489C2214yes—
24C2×#486C2211yescharacteristic
25C2×#475C2212yes—
26C2×#476C2214yes—
27C2×#463C2214yes—
28C2×#459C2212yes—
29C2×#459C2212yes—
30C2×#446C2212yes—
31C2×#442C2214yes—
32C2×C4×SD16C2212yes—
33C2×C4×SD16C2212yes—
34#2724—22no—
35#2683—22no—
36#2582C2212yes—
37#2551C2212yes—
38#2547—21no—
39#2547C2212yes—
40#2538—21no—
41#2538C2212yes—
42#2486—22no—
43#2452—22no—
44#2452C2214yes—
45#2447—22no—
46#2447C2214yes—
47#2409—22no—
48#2366C2212yes—
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C4×#127C2211yescharacteristic
56#2088C2214yes—
57#2066C2214yes—
58#2058C2214yes—
59#2055C2214yes—
60#2054C2214yes—
61#2042C2214yes—
62#2024C2214yes—
63#2012C2214yes—
64#1990C2211yescharacteristic
65#1984C2212yes—
66#1981C2212yes—
67#1968C2212yes—
68#1968C2212yes—
69#1957C2211yescharacteristic
70#1957C2211yescharacteristic
71#1956C2211yescharacteristic
72#1948C2212yes—
73#1939C2212yes—
74#1939C2212yes—
75#1936C2214yes—
76#1935C2214yes—
77#1934C2212yes—
78#1934C2212yes—
79#1879C2214yes—
80#1877C2214yes—
81#1864C2214yes—
82#1864C2214yes—
83#1844C2211yescharacteristic
84#1842C2211yescharacteristic
85#1834C2211yescharacteristic
86#1803C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1770C2212yes—
90#1770C2212yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C42×C8C2211yescharacteristic
94C23×Q8—22no—
95C22×SD16C2312yes—
96C22×SD16C2312yes—
97C22×D8C2314yes—
98#559—28no—
99#559—44no—
100D4×Q8—28no—
101D4×Q8—44no—
102D4×Q8—48no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#124—22no—
111C2×#124C2314yes—
112C2×#123—24no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2314yes—
129C2×C4×D4C2314yes—
130C22×C4⋊C4—22no—
131#492—24no—
132#489—216no—
133#486D418yescomplemented
134#476—216no—
135#475—28no—
136#463—216no—
137#459—28no—
138#459—28no—
139#446—28no—
140#442—216no—
141C4×SD16—28no—
142C4×SD16—44no—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#97C2314yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#398—24no—
170#397—24no—
171#397—24no—
172#397—24no—
173#396—24no—
174#396—24no—
175#395—24no—
176#393—24no—
177#391—22no—
178#391—21no—
179#391D412yes—
180#391C2312yes—
181#391C2314yes—
182#390—22no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390C2312yes—
188#390C2312yes—
189#390C2314yes—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389D412yes—
196#389D412yes—
197#389C2312yes—
198#389C2314yes—
199#389C2314yes—
200#386—24no—
201#385—24no—
202#385—24no—
203#385—24no—
204#384D411yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#381—24no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—48no—
232C22×Q8—48no—
233C22×Q8—84no—
234C22×Q8C2×D412yes—
235C22×Q8C2412yes—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—24no—
246C2×SD16—24no—
247C2×SD16—24no—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C2×SD16—42no—
252C2×D8—28no—
253C2×D8—28no—
254C2×D8—44no—
255C22×C8C2414yes—
256C22×C8C2411yescharacteristic
257C22×C8C2411yescharacteristic
258C22×C8C2411yescharacteristic
259C22×C8C2411yescharacteristic
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267C4⋊Q8C2×D414yes—
268#127—22no—
269#127—22no—
270#127C2×D414yes—
271#127#4114yes—
272#124—28no—
273#124—44no—
274#124—48no—
275#123—416no—
276C22⋊Q8—416no—
277C22⋊Q8—48no—
278C22⋊Q8—416no—
279C22⋊Q8—48no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—44no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—82no—
298C4×D4—82no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2414yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#106—24no—
350#106—28no—
351#106—42no—
352C4⋊C8—28no—
353C4⋊C8—24no—
354C4⋊C8—24no—
355C4⋊C8—42no—
356#103—24no—
357#103—28no—
358#103—44no—
359#103—42no—
360#102—28no—
361#102—28no—
362#102—24no—
363#102—24no—
364#102—42no—
365#102—42no—
366#102C2×D418yes—
367#102C2×D418yes—
368#97—28no—
369C4×C8—22no—
370C4×C8—22no—
371C4×C8—22no—
372C4×C8—22no—
373C4×C8—41no—
374C4×C8C2×D414yes—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95C2×D414yes—
384#95C2×D414yes—
385C24—22no—
386C24—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—84no—
402C2×Q8—832no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—81no—
425C2×D4C22×D414yes—
426C2×D4C2×#4114yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454SD16—48no—
455SD16—48no—
456D8—416no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—24no—
469C2×C8C22×D412yes—
470C2×C8C22×D412yes—
471C2×C82−1+412yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—84no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4C22×D414yes—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—84no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—24no—
520C42—21no—
521C42—21no—
522C42—41no—
523C42—44no—
524C42—44no—
525C42AGL(1,ℤ/8)14yes—
526C42C22×D412yes—
527C42C2×#4112yes—
528C42H(5,𝔽2)12yes—
529C422−1+412yes—
530C23—24no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23C23×D411yescharacteristic
545Q8—48no—
546Q8—48no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—816no—
552D4—24no—
553D4—24no—
554D4—24no—
555D4—24no—
556D4—48no—
557D4—42no—
558D4—42no—
559D4—48no—
560D4—42no—
561D4—42no—
562D4—84no—
563D4—84no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—816no—
598C2×C4—88no—
599C2×C4—88no—
600C2×C4—168no—
601C2×C4D4212yes—
602C2×C4D4212yes—
603C2×C4#54812yes—
604C2×C4#54812yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4#57712yes—
609C2×C4#57712yes—
610C2×C4C23×D411yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C2×C4C2×2−1+411yescharacteristic
615C2×C4#58511yescharacteristic
616C8—22no—
617C8—22no—
618C8—22no—
619C8—44no—
620C8—41no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22C2×D4211yescharacteristic
636C22C2×#54811yescharacteristic
637C22#349711yescharacteristic
638C22C22×AGL(1,ℤ/8)11yescharacteristic
639C22C2×#57711yescharacteristic
640C22C2×#57711yescharacteristic
641C22#358411yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—41no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4#327214yes—
655C4#337314yescomplemented
656C4D4×#4111yescharacteristic
657C4#349911yescharacteristic
658C4#352011yescharacteristic
659C4#354711yescharacteristic
660C2—84no—
661C2—84no—
662C2#5965411yescharacteristic
663C2C2×#327211yescharacteristic
664C2#6108511yescharacteristic
665C2C2×#337311yescharacteristic
666C2#6132611yescharacteristic
667C2#6236511yescharacteristic
668C2#6237211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28251C2
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
1. C2×#3373C2semidirect
2. #59827C2semidirect
3. #59817C2semidirect
4. #31160C2semidirect
5. #28283C2semidirect
7. #28195C2semidirect
8. #27776C2semidirect
9. #26842C2semidirect
10. #26044C2semidirect
11. #25367C2semidirect
12. #24330C2semidirect
13. #24186C2semidirect
14. #23001C2semidirect
15. #21712C2semidirect
16. C4×#492C2semidirect
17. C4×#486C2semidirect
133. #486D4semidirect
655. 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
#598172
#311602
#282832
#282512
#281952
#277762
#268422
#260442
#253672
#243302
#241862
#230012
#217122
C4×#4922
C4×#4862