Finite Group Databasebeta

← #6683845

Subgroups — #6683845

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3381C211yesmaximal, characteristic
2C2×#3364C211yesmaximal, characteristic
3#39490C211yesmaximal, characteristic
4#36593C211yesmaximal, characteristic, complemented
5#33027C211yesmaximal, characteristic, complemented
6#30960C214yesmaximal, complemented
7#30819C214yesmaximal
8#30280C211yesmaximal, characteristic, complemented
9#29864C211yesmaximal, characteristic
10#28677C211yesmaximal, characteristic, complemented
11#28639C211yesmaximal, characteristic, complemented
12#26371C214yesmaximal, complemented
13#26161C214yesmaximal
14#25931C211yesmaximal, characteristic, complemented
15#25836C211yesmaximal, characteristic, complemented
16#25835C211yesmaximal, characteristic
17#25650C211yesmaximal, characteristic
18#22238C211yesmaximal, characteristic, complemented
19#21115C211yesmaximal, characteristic
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3381—28no—
25#3364—28no—
26C2×#499C2212yes—
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#488C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#478C2212yes—
36C2×#474C2211yescharacteristic
37C2×#474C2211yescharacteristic
38C2×#464C2212yes—
39C2×#462C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×C4×Q16C2212yes—
43C2×C4×SD16C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2868—21no—
46#2868C2211yescharacteristic
47#2862C2211yescharacteristic
48#2689—24no—
49#2621—21no—
50#2568C2211yescharacteristic
51#2567C2211yescharacteristic
52#2552—24no—
53#2552C2214yes—
54#2546—21no—
55#2546—21no—
56#2546C2211yescharacteristic
57#2546C2211yescharacteristic
58#2503—21no—
59#2499—21no—
60#2453C2214yes—
61#2447C2214yes—
62Q8×C4⋊C4C2214yes—
63#2317—21no—
64C4×C4⋊Q8—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2293C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2096C2214yes—
69#2096C2214yes—
70#2088C2214yes—
71#2083C2214yes—
72#2071C2214yes—
73#2066C2214yes—
74#2059C2214yes—
75#2059C2214yes—
76#2042C2214yes—
77#2026C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1981C2211yescharacteristic
84#1941C2214yes—
85#1933C2214yes—
86#1928C2214yes—
87#1926C2212yes—
88#1919C2212yes—
89#1916C2214yes—
90#1916C2214yes—
91#1880C2212yes—
92#1878C2214yes—
93#1878C2214yes—
94#1865C2212yes—
95#1865C2214yes—
96#1864C2214yes—
97#1863C2214yes—
98#1831C2211yescharacteristic
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2312yes—
104C22×SD16C2311yescharacteristic
105#557—24no—
106#557—24no—
107#557—42no—
108#557—42no—
109#556—24no—
110#556—42no—
111#554—24no—
112#554—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8D411yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×#126—22no—
125C2×#125—22no—
126C2×#125—22no—
127C2×#125C2312yes—
128C2×#125C2312yes—
129C2×#125C2312yes—
130C2×#124C2312yes—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—22no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×D4C2312yes—
144C2×C4×D4C2311yescharacteristic
145C2×#117—21no—
146C2×#117—21no—
147C22×C4⋊C4—21no—
148#499—28no—
149#498—24no—
150#498—42no—
151#488—28no—
152#487—24no—
153#487—42no—
154#479—28no—
155#479—28no—
156#478—28no—
157#477—24no—
158#474—24no—
159#474—24no—
160#464—28no—
161#462—24no—
162C8×Q8—24no—
163C8×Q8—24no—
164C4×Q16—28no—
165C4×SD16—28no—
166C4×SD16D418yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2314yes—
174C2×#106C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2311yescharacteristic
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2312yes—
185C2×#103C2314yes—
186C2×#103C2311yescharacteristic
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2312yes—
193C2×C4×C8C2312yes—
194C2×C4×C8C2311yescharacteristic
195#400—22no—
196#400—24no—
197#398—24no—
198#398—22no—
199#398—22no—
200#398—24no—
201#398—24no—
202#397—24no—
203#397—24no—
204#395—22no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—22no—
210#391D411yescharacteristic
211#391D411yescharacteristic
212#391C2314yes—
213#391C2314yes—
214#390C2312yes—
215#389—24no—
216#389—24no—
217#389—24no—
218#389—24no—
219#389D414yes—
220#389C2314yes—
221#389C2314yes—
222#389C2312yes—
223#389C2314yes—
224#388—22no—
225#387—24no—
226#385—24no—
227#384—21no—
228#384—24no—
229#384—24no—
230#384—22no—
231#384D411yescharacteristic
232#384C2314yes—
233#382—21no—
234#382—21no—
235#382—21no—
236#381—24no—
237C4×C4⋊C4—24no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2314yes—
244C4×C22⋊C4—21no—
245C43—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2412yes—
264C22×Q8C2411yescharacteristic
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—22no—
268C2×Q16—24no—
269C2×Q16—28no—
270C2×SD16—24no—
271C2×SD16C2×D414yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2411yescharacteristic
276C22×C8C2411yescharacteristic
277C4⋊Q8—24no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—44no—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C4⋊Q8—82no—
290C4⋊Q8—82no—
291C4⋊Q8—82no—
292#126—48no—
293#125—24no—
294#125—24no—
295#125—24no—
296#125—42no—
297#125—42no—
298#125—42no—
299#125—48no—
300#125—48no—
301#124—24no—
302#124—42no—
303C22⋊Q8—48no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—42no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8C2×D414yes—
322C4×D4—24no—
323C4×D4—22no—
324C4×D4—42no—
325C4×D4C2×D414yes—
326#117—44no—
327#117—44no—
328C2×C4⋊C4—21no—
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—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D412yes—
380C2×C42C2×D411yescharacteristic
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385#107—24no—
386#107—24no—
387#107—24no—
388#106—24no—
389#106—24no—
390#106—24no—
391#106—44no—
392#106C2×D414yes—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—28no—
403#103—44no—
404#103C2×D414yes—
405#102—24no—
406#102—24no—
407#102—24no—
408#102C2×D414yes—
409C4×C8—24no—
410C4×C8—24no—
411C4×C8—22no—
412C4×C8C2×D414yes—
413#95—22no—
414#95—21no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95—21no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95C2×D414yes—
425#95C2×D414yes—
426C24—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—24no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8C22×D412yes—
448C2×Q82−1+414yes—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4C22×D412yes—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—84no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476Q16—48no—
477SD16—28no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8C22×D412yes—
488C2×C8C22×D412yes—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C22×D412yes—
529C4⋊C4C22×D412yes—
530C4⋊C42−1+414yes—
531C22⋊C4—24no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—82no—
537C42—28no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—44no—
551C42—42no—
552C42—42no—
553C42#13514yes—
554C42C22×D412yes—
555C42H(5,𝔽2)12yes—
556C23—22no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—41no—
561C23C23×D411yescharacteristic
562Q8—22no—
563Q8—24no—
564Q8—44no—
565Q8—44no—
566Q8—42no—
567Q8—42no—
568Q8—44no—
569Q8—84no—
570Q8—84no—
571Q8—162no—
572Q8#53614yes—
573D4—22no—
574D4—22no—
575D4—42no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—164no—
617C2×C4D4212yes—
618C2×C4D4212yes—
619C2×C4C2×#13514yes—
620C2×C4C2×#13512yes—
621C2×C4#57712yes—
622C2×C4C23×D411yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+412yes—
625C2×C4C2×2−1+412yes—
626C8—22no—
627C8—22no—
628C8—42no—
629C8—42no—
630C8—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22C2×D4211yescharacteristic
637C22#348612yes—
638C22C22×#13511yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358612yes—
641C4—24no—
642C4—24no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#328414yes—
658C4C2×#53611yescharacteristic
659C4#349911yescharacteristic
660C4#355212yes—
661C2—82no—
662C2#3767511yescharacteristic
663C2C2×#328411yescharacteristic
664C2#6108311yescharacteristic
665C2#6182212yes—
666C2#6185812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
4. #36593C2semidirect
5. #33027C2semidirect
6. #30960C2semidirect
8. #30280C2semidirect
10. #28677C2semidirect
11. #28639C2semidirect
12. #26371C2semidirect
14. #25931C2semidirect
15. #25836C2semidirect
18. #22238C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33812
C2×#33642
#394902
#365932
#330272
#309602
#308192
#302802
#298642
#286772
#286392
#263712
#261612
#259312
#258362
#258352
#256502
#222382
#211152