Finite Group Databasebeta

← #6560680

Subgroups — #6560680

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#36496C211yesmaximal, characteristic
3#30216C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29350C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#28716C212yesmaximal, complemented
9#27825C214yesmaximal, complemented
10#27815C212yesmaximal, complemented
11#27783C214yesmaximal, complemented
12#27767C212yesmaximal
13#23822C212yesmaximal, complemented
14#23817C211yesmaximal, characteristic, complemented
15#22546C212yesmaximal, complemented
16#22522C212yesmaximal, complemented
17#20573C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×Q82—21no—
20C2×#554C2211yescharacteristic
21C2×#489C2212yes—
22C2×#486C2214yes—
23C2×#446C2212yes—
24#2868C2212yes—
25#2868C2211yescharacteristic
26#2861C2212yes—
27#2858C2211yescharacteristic
28#2567—21no—
29#2567—22no—
30#2567C2211yescharacteristic
31#2566C2212yes—
32#2551—21no—
33#2551—22no—
34#2551C2212yes—
35#2551C2214yes—
36#2544C2212yes—
37#2541C2211yescharacteristic
38#2461—22no—
39#2461—22no—
40#2426—21no—
41#2426—21no—
42#2361—21no—
43#2361—21no—
44Q8×C4⋊C4—21no—
45Q8×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#127C2212yes—
49C4×#124C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2101C2212yes—
52#2100C2214yes—
53#2072C2211yescharacteristic
54#2072C2212yes—
55#2071C2212yes—
56#2071C2212yes—
57#2071C2211yescharacteristic
58#2071C2212yes—
59#2071C2212yes—
60#2042C2212yes—
61#2041C2214yes—
62#2024C2214yes—
63#2024C2211yescharacteristic
64#2023C2214yes—
65#2023C2211yescharacteristic
66#2023C2212yes—
67#2023C2214yes—
68#1993C2212yes—
69#1992C2214yes—
70#1990C2214yes—
71#1990C2212yes—
72#1989C2214yes—
73#1987C2212yes—
74#1983C2214yes—
75#1982C2212yes—
76#1959C2214yes—
77#1957C2212yes—
78#1936C2214yes—
79#1934C2214yes—
80#1933C2214yes—
81#1803C2212yes—
82#1802C2214yes—
83#1802C2214yes—
84#1802C2212yes—
85#1771C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×C4⋊C8C2212yes—
89#1762C2214yes—
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95#559—28no—
96Q82—48no—
97#554—28no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#127C2312yes—
110C2×#125C2312yes—
111C2×#124—21no—
112C2×#124—21no—
113C2×#124—21no—
114C2×#124C2312yes—
115C2×#124C2311yescharacteristic
116C2×#123—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—22no—
119C2×#121—21no—
120C2×#121—21no—
121C2×C4×Q8—22no—
122C2×C4×Q8—21no—
123C2×C4×Q8—22no—
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8D411yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4—22no—
130C2×C4×D4—22no—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×C4×D4C2311yescharacteristic
134C22×C4⋊C4—21no—
135#489—44no—
136#486—48no—
137#446—28no—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#97C2314yes—
151C2×#97C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#396—22no—
155#396—22no—
156#395—24no—
157#395—24no—
158#395—24no—
159#395—22no—
160#394—22no—
161#394—22no—
162#392—22no—
163#391—22no—
164#391—21no—
165#391C2312yes—
166#391C2311yescharacteristic
167#390—22no—
168#390—22no—
169#390—21no—
170#390—21no—
171#390C2311yescharacteristic
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—24no—
176#389—22no—
177#389—22no—
178#389—24no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—21no—
183#389C2314yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—21no—
190#388—21no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384—22no—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2314yes—
203#384C2312yes—
204#382C2311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C43C2312yes—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—24no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4C2412yes—
245C22×D4C2411yescharacteristic
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—42no—
258C4⋊Q8—48no—
259C4⋊Q8—44no—
260C4⋊Q8—416no—
261C4⋊Q8—48no—
262C4⋊Q8—84no—
263C4⋊Q8—82no—
264#127—24no—
265#127—24no—
266#127—42no—
267#125—28no—
268#124—22no—
269#124—28no—
270#124—41no—
271#124—44no—
272#124—82no—
273#124—82no—
274#123—48no—
275C22⋊Q8—44no—
276C22⋊Q8—48no—
277#121—44no—
278#121—44no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—48no—
283C4×Q8—44no—
284C4×Q8—48no—
285C4×Q8#4118yes—
286C4×D4—24no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—44no—
290C4×D4—84no—
291C4×D4—84no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C4⋊C8—24no—
356C4⋊C8—28no—
357#103—24no—
358#103—42no—
359#103—44no—
360#102—24no—
361#102—28no—
362#102—24no—
363#102—28no—
364#102—44no—
365#102—44no—
366#102—44no—
367#97—28no—
368#97—42no—
369C4×C8—24no—
370C4×C8—44no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8—84no—
401C2×Q8—82no—
402C2×Q8—84no—
403C2×Q8—82no—
404C2×Q8#12414yes—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4H(5,𝔽2)12yes—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2511yescharacteristic
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—84no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4#12414yes—
491C4⋊C4#12418yes—
492C4⋊C4AGL(1,ℤ/8)14yes—
493C4⋊C4#13714yes—
494C4⋊C42−1+414yes—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—24no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—22no—
521C42—21no—
522C42—22no—
523C42—48no—
524C42—42no—
525C42—44no—
526C42C2×#4114yes—
527C42C2×#4118yes—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23C23×D411yescharacteristic
543Q8—28no—
544Q8—48no—
545Q8—42no—
546Q8—42no—
547Q8—168no—
548Q8—162no—
549Q8—164no—
550D4—24no—
551D4—42no—
552D4—44no—
553D4—44no—
554D4—88no—
555D4—88no—
556D4—84no—
557D4—162no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—164no—
601C2×C4—164no—
602C2×C4C2×#12414yes—
603C2×C4C2×#12414yes—
604C2×C4C2×#12412yes—
605C2×C4C2×#12412yes—
606C2×C4C2×AGL(1,ℤ/8)14yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×#13714yes—
609C2×C4C2×#13712yes—
610C2×C4#57612yes—
611C2×C4#57712yes—
612C2×C4C22×#4112yes—
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4C2×2−1+412yes—
615C2×C4#58511yescharacteristic
616C8—42no—
617C8—42no—
618C8—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C22×#12411yescharacteristic
634C22#348611yescharacteristic
635C22#348811yescharacteristic
636C22C22×AGL(1,ℤ/8)11yescharacteristic
637C22C22×#13711yescharacteristic
638C22C2×#57611yescharacteristic
639C22C2×#57711yescharacteristic
640C4—24no—
641C4—24no—
642C4—22no—
643C4—22no—
644C4—24no—
645C4—24no—
646C4—22no—
647C4—41no—
648C4—41no—
649C4—44no—
650C4—82no—
651C4—84no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—82no—
656C4#314114yes—
657C4#351812yes—
658C4#352212yes—
659C2—82no—
660C2—162no—
661C2#3594811yescharacteristic
662C2C2×#314111yescharacteristic
663C2#6062211yescharacteristic
664C2#6184311yescharacteristic
665C2#6184411yescharacteristic
666C2#6197911yescharacteristic
667C2#6198011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊Q8C2semidirect
3. #30216C2semidirect
4. #30203C2semidirect
5. #29354C2semidirect
6. #29350C2semidirect
7. #28726C2semidirect
8. #28716C2semidirect
9. #27825C2semidirect
10. #27815C2semidirect
11. #27783C2semidirect
13. #23822C2semidirect
14. #23817C2semidirect
15. #22546C2semidirect
16. #22522C2semidirect
17. #20573C2semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#364962
#302162
#302032
#293542
#293502
#287262
#287162
#278252
#278152
#277832
#277672
#238222
#238172
#225462
#225222
#205732