Finite Group Databasebeta

← #6683885

Subgroups — #6683885

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3386C211yesmaximal, characteristic
2C2×#3364C211yesmaximal, characteristic
3#38954C211yesmaximal, characteristic
4#36593C211yesmaximal, characteristic, complemented
5#33033C211yesmaximal, characteristic, complemented
6#30976C214yesmaximal, complemented
7#30811C214yesmaximal
8#30279C211yesmaximal, characteristic, complemented
9#29864C211yesmaximal, characteristic
10#28675C211yesmaximal, characteristic, complemented
11#28646C211yesmaximal, characteristic, complemented
12#26381C214yesmaximal, complemented
13#26177C214yesmaximal
14#25931C211yesmaximal, characteristic, complemented
15#25840C211yesmaximal, characteristic, complemented
16#25838C211yesmaximal, characteristic
17#25650C211yesmaximal, characteristic
18#22240C211yesmaximal, characteristic, complemented
19#21120C211yesmaximal, characteristic
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3386—28no—
25#3364—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#488C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2212yes—
34C2×#475C2211yescharacteristic
35C2×#478C2212yes—
36C2×#474C2211yescharacteristic
37C2×#474C2211yescharacteristic
38C2×#464C2212yes—
39C2×#462C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×C4×Q16C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2868—21no—
46#2868C2211yescharacteristic
47#2862C2211yescharacteristic
48#2698—24no—
49#2621—21no—
50#2568C2211yescharacteristic
51#2552—24no—
52#2552C2214yes—
53#2546—21no—
54#2546—21no—
55#2546C2211yescharacteristic
56#2546C2211yescharacteristic
57#2546C2211yescharacteristic
58#2499—21no—
59#2451C2214yes—
60#2447C2214yes—
61#2434—21no—
62Q8×C4⋊C4C2214yes—
63#2310—21no—
64C4×C4⋊Q8—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2293C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2095C2214yes—
69#2095C2214yes—
70#2086C2214yes—
71#2083C2214yes—
72#2071C2214yes—
73#2068C2214yes—
74#2059C2214yes—
75#2057C2214yes—
76#2042C2214yes—
77#2026C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1981C2211yescharacteristic
84#1942C2214yes—
85#1935C2214yes—
86#1927C2214yes—
87#1925C2212yes—
88#1920C2212yes—
89#1917C2214yes—
90#1916C2214yes—
91#1880C2212yes—
92#1878C2214yes—
93#1878C2214yes—
94#1863C2214yes—
95#1863C2212yes—
96#1863C2214yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C23×Q8—21no—
104C22×Q16C2312yes—
105C22×SD16C2311yescharacteristic
106#557—24no—
107#557—24no—
108#557—42no—
109#557—42no—
110#556—24no—
111#556—42no—
112D4×Q8—24no—
113D4×Q8—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8D411yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2311yescharacteristic
126C2×C4⋊Q8C2312yes—
127C2×#126—22no—
128C2×#125—22no—
129C2×#125—22no—
130C2×#125C2312yes—
131C2×#125C2312yes—
132C2×#124C2312yes—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8—21no—
135C2×C22⋊Q8—22no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×D4C2312yes—
146C2×C4×D4C2311yescharacteristic
147C2×#117—21no—
148C2×#117—21no—
149#501—28no—
150#500—24no—
151#500—42no—
152#488—28no—
153#487—24no—
154#487—42no—
155#479—28no—
156#478—28no—
157#477—28no—
158#475—24no—
159#474—24no—
160#474—24no—
161#464—28no—
162#462—24no—
163#446—24no—
164C8×Q8—24no—
165#440—28no—
166C4×Q16—28no—
167C4×SD16D418yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2314yes—
174C2×#106C2312yes—
175C2×#106C2311yescharacteristic
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2311yescharacteristic
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2312yes—
186C2×#103C2314yes—
187C2×#103C2311yescharacteristic
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2311yescharacteristic
192C2×#97C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2312yes—
195C2×C4×C8C2311yescharacteristic
196#400—22no—
197#398—22no—
198#398—22no—
199#398—22no—
200#398—24no—
201#398—24no—
202#398—24no—
203#397—24no—
204#397—24no—
205#395—24no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—24no—
213#391D411yescharacteristic
214#391D411yescharacteristic
215#391C2314yes—
216#391C2314yes—
217#390C2312yes—
218#389—24no—
219#389—24no—
220#389—24no—
221#389D414yes—
222#389C2314yes—
223#389C2314yes—
224#389C2312yes—
225#389C2314yes—
226#388—22no—
227#387—24no—
228#385—24no—
229#384—24no—
230#384—24no—
231#384—22no—
232#384D411yescharacteristic
233#384C2314yes—
234#382—21no—
235#382—22no—
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#340C2314yes—
249#340C2312yes—
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×Q8—22no—
262C22×Q8—44no—
263C22×Q8—44no—
264C22×Q8—82no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2412yes—
268C22×Q8C2411yescharacteristic
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—22no—
272C2×Q16—24no—
273C2×Q16—28no—
274C2×SD16—24no—
275C2×SD16C2×D414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—42no—
291C4⋊Q8—42no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—82no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298C4⋊Q8—82no—
299#126—48no—
300#125—24no—
301#125—24no—
302#125—42no—
303#125—42no—
304#125—48no—
305#125—48no—
306#124—24no—
307#124—42no—
308C22⋊Q8—48no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8C2×D414yes—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—42no—
330C4×D4C2×D414yes—
331#117—44no—
332#117—82no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388#107—24no—
389#107—24no—
390#107—24no—
391#107—24no—
392#107—24no—
393#107—44no—
394#106—24no—
395#106C2×D414yes—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—28no—
406#103—44no—
407#103C2×D414yes—
408#102—24no—
409#102—24no—
410#102—24no—
411#102C2×D414yes—
412#97—24no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8C2×D414yes—
416#95—22no—
417#95—21no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—21no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95C2×D414yes—
428#95C2×D414yes—
429C24—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—82no—
454C2×Q8—816no—
455C2×Q8C22×D412yes—
456C2×Q82−1+414yes—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4C22×D412yes—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—82no—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C2511yescharacteristic
485Q16—48no—
486SD16—28no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8C22×D412yes—
497C2×C8C22×D412yes—
498C4⋊C4—28no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—28no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D412yes—
538C4⋊C42−1+414yes—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—82no—
545C42—28no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—24no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—44no—
562C42C22×D412yes—
563C42H(5,𝔽2)12yes—
564C23—22no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23C23×D411yescharacteristic
570Q8—22no—
571Q8—24no—
572Q8—44no—
573Q8—44no—
574Q8—42no—
575Q8—42no—
576Q8—44no—
577Q8—88no—
578Q8—84no—
579Q8—84no—
580Q8—84no—
581Q8#53614yes—
582D4—22no—
583D4—22no—
584D4—42no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—88no—
627C2×C4D4212yes—
628C2×C4D4212yes—
629C2×C4#57512yes—
630C2×C4#57612yes—
631C2×C4#57814yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+412yes—
635C2×C4C2×2−1+412yes—
636C8—22no—
637C8—22no—
638C8—42no—
639C8—42no—
640C8—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—82no—
645C22—82no—
646C22C2×D4211yescharacteristic
647C22#348612yes—
648C22C2×#57511yescharacteristic
649C22C2×#57611yescharacteristic
650C22C2×#57812yes—
651C4—24no—
652C4—24no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4#326814yes—
668C4C2×#53611yescharacteristic
669C4#349911yescharacteristic
670C4#355212yes—
671C2—82no—
672C2#3767511yescharacteristic
673C2C2×#326811yescharacteristic
674C2#6109111yescharacteristic
675C2#6184212yes—
676C2#6187012yes—
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. #33033C2semidirect
6. #30976C2semidirect
8. #30279C2semidirect
10. #28675C2semidirect
11. #28646C2semidirect
12. #26381C2semidirect
14. #25931C2semidirect
15. #25840C2semidirect
18. #22240C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33862
C2×#33642
#389542
#365932
#330332
#309762
#308112
#302792
#298642
#286752
#286462
#263812
#261772
#259312
#258402
#258382
#256502
#222402
#211202