Finite Group Databasebeta

← #6816554

Subgroups — #6816554

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3387C211yesmaximal, characteristic, complemented
2C2×#3356C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic
4#39617C211yesmaximal, characteristic
5#33310C211yesmaximal, characteristic, complemented
6#31091C214yesmaximal, complemented
7#31089C214yesmaximal, complemented
8#29869C211yesmaximal, characteristic, complemented
9#29866C211yesmaximal, characteristic, complemented
10#28713C211yesmaximal, characteristic
11#28684C211yesmaximal, characteristic, complemented
12#26078C214yesmaximal, complemented
13#25837C211yesmaximal, characteristic, complemented
14#25836C211yesmaximal, characteristic, complemented
15#25702C211yesmaximal, characteristic
16#25701C211yesmaximal, characteristic
17#25047C214yesmaximal, complemented
18#21631C211yesmaximal, characteristic
19#21276C211yesmaximal, characteristic, complemented
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3387—28no—
25#3356—28no—
26C2×#501C2212yes—
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#485C2212yes—
32C2×#483C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#477C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#475C2211yescharacteristic
38C2×#475C2212yes—
39C2×#470C2212yes—
40C2×#446C2211yescharacteristic
41C2×#441C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#428C2212yes—
45#2868C2211yescharacteristic
46#2868C2211yescharacteristic
47#2721—24no—
48#2677—24no—
49#2660—21no—
50#2658—21no—
51#2567C2211yescharacteristic
52#2565—21no—
53#2563C2211yescharacteristic
54#2547C2211yescharacteristic
55#2546—21no—
56#2546C2211yescharacteristic
57#2545—21no—
58#2542—21no—
59#2441C2214yes—
60#2440C2214yes—
61#2423C2214yes—
62#2423C2214yes—
63#2327—21no—
64#2326—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67#2295C2211yescharacteristic
68#2088C2214yes—
69#2084C2214yes—
70#2083C2214yes—
71#2077C2214yes—
72#2072C2214yes—
73#2072C2214yes—
74#2059C2214yes—
75#2058C2214yes—
76#2035C2214yes—
77#2035C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1987C2211yescharacteristic
82#1984C2211yescharacteristic
83#1969C2212yes—
84#1954C2211yescharacteristic
85#1943C2214yes—
86#1942C2212yes—
87#1933C2212yes—
88#1933C2212yes—
89#1923C2214yes—
90#1865C2214yes—
91#1865C2214yes—
92#1854C2214yes—
93#1853C2214yes—
94#1844C2214yes—
95#1844C2214yes—
96#1831C2211yescharacteristic
97#1809C2214yes—
98C4×#107C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#103C2211yescharacteristic
101#1753C2214yes—
102C42×C8C2211yescharacteristic
103C22×Q16C2311yescharacteristic
104C22×M16C2312yes—
105Q82—24no—
106Q82—42no—
107#557—24no—
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—22no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×#126—22no—
126C2×#125—21no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#124—21no—
130C2×#124—21no—
131C2×#124D411yescharacteristic
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—24no—
134C2×C22⋊Q8C2312yes—
135C2×C22⋊Q8C2312yes—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×D4—21no—
145C2×C4×D4C2311yescharacteristic
146C2×#117C2312yes—
147C22×C4⋊C4C2312yes—
148#501—28no—
149#498—24no—
150#498—24no—
151#498—42no—
152#498—42no—
153#485—28no—
154#483—28no—
155#479—28no—
156#477—24no—
157#477—24no—
158#475—24no—
159#475—24no—
160#475—28no—
161#470—28no—
162#446—24no—
163#441—28no—
164C4×Q16D418yes—
165#436—24no—
166#428—28no—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×#106C2314yes—
178C2×#106C2314yes—
179C2×C4⋊C8C2311yescharacteristic
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2311yescharacteristic
189C2×#103C2311yescharacteristic
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2314yes—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194C2×C4×C8C2312yes—
195C2×C4×C8C2311yescharacteristic
196C2×C4×C8C2311yescharacteristic
197#399—24no—
198#399—24no—
199#399—24no—
200#398—22no—
201#398—24no—
202#398—24no—
203#398—24no—
204#397—22no—
205#396—24no—
206#395—22no—
207#395—24no—
208#393—22no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391D411yescharacteristic
213#391C2312yes—
214#391C2312yes—
215#390—21no—
216#390D411yescharacteristic
217#389—24no—
218#389C2314yes—
219#389C2314yes—
220#389C2312yes—
221#389C2312yes—
222#388—24no—
223#388—24no—
224#388D414yes—
225#387—22no—
226#387—22no—
227#387—24no—
228#387—24no—
229#387—24no—
230#385—24no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384—22no—
235#384C2314yes—
236#384C2314yes—
237#382—21no—
238#382D411yescharacteristic
239#380C2314yes—
240#380C2314yes—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4C2314yes—
246#376—21no—
247#376—21no—
248C43C2311yescharacteristic
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2411yescharacteristic
265C22×Q8C2411yescharacteristic
266C22×D4C2×D411yescharacteristic
267C23×C4—22no—
268C23×C4C2412yes—
269C2×Q16—24no—
270C2×Q16C2×D414yes—
271C2×M16—24no—
272C2×M16—28no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2411yescharacteristic
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C4⋊Q8—24no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—42no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—48no—
291C4⋊Q8—44no—
292C4⋊Q8—84no—
293C4⋊Q8—82no—
294C4⋊Q8—82no—
295#126—48no—
296#125—24no—
297#125—42no—
298#125—44no—
299#125—48no—
300#124—24no—
301#124—44no—
302#124—82no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—48no—
306C22⋊Q8—42no—
307C22⋊Q8—42no—
308C22⋊Q8—88no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—42no—
316C4×Q8—42no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8C2×D414yes—
321C4×Q8C2×D414yes—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—44no—
325#117—24no—
326#117—42no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D414yes—
369C2×C22⋊C4C2×D414yes—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—24no—
385#107—24no—
386#107C2×D414yes—
387#106—24no—
388#106—24no—
389#106—24no—
390#106—24no—
391#106—24no—
392#106—24no—
393#106—44no—
394#106—44no—
395C4⋊C8—24no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103C2×D414yes—
405#103C2×D414yes—
406C22⋊C8—24no—
407C22⋊C8—28no—
408#97—24no—
409C4×C8—24no—
410C4×C8—24no—
411C4×C8—22no—
412C4×C8C2×D414yes—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—21no—
420#95—21no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24C22×D412yes—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—84no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×Q8C22×D412yes—
450C2×Q8C22×D412yes—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—84no—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C2511yescharacteristic
478Q16—28no—
479M16—48no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8C22×D412yes—
488C2×C8C22×D412yes—
489C2×C82−1+414yes—
490C2×C82−1+414yes—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
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—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C22×D412yes—
531C4⋊C4C22×D412yes—
532C22⋊C4—24no—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—44no—
549C42—44no—
550C42—44no—
551C42—42no—
552C42—42no—
553C42#13714yes—
554C42C22×D412yes—
555C42H(5,𝔽2)12yes—
556C23—22no—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23C23×D411yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—22no—
566Q8—44no—
567Q8—42no—
568Q8—42no—
569Q8—84no—
570Q8—84no—
571Q8—162no—
572Q8—162no—
573Q8—164no—
574D4—44no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—164no—
615C2×C4D4212yes—
616C2×C4D4212yes—
617C2×C4C2×#13714yes—
618C2×C4C2×#13712yes—
619C2×C4#57714yes—
620C2×C4#57712yes—
621C2×C4C23×D411yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+412yes—
624C2×C4C2×2−1+412yes—
625C8—22no—
626C8—24no—
627C8—42no—
628C8—42no—
629C8#53614yes—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22C2×D4211yescharacteristic
636C22#349912yes—
637C22C22×#13711yescharacteristic
638C22C2×#57712yes—
639C22C2×#57711yescharacteristic
640C4—24no—
641C4—24no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#327814yes—
657C4C2×#53611yescharacteristic
658C4#349911yescharacteristic
659C4#355212yes—
660C2—82no—
661C2#3936011yescharacteristic
662C2C2×#327811yescharacteristic
663C2#6108711yescharacteristic
664C2#6247812yes—
665C2#6248612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3387C2semidirect
5. #33310C2semidirect
6. #31091C2semidirect
7. #31089C2semidirect
8. #29869C2semidirect
9. #29866C2semidirect
11. #28684C2semidirect
12. #26078C2semidirect
13. #25837C2semidirect
14. #25836C2semidirect
17. #25047C2semidirect
19. #21276C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33872
C2×#33562
#397442
#396172
#333102
#310912
#310892
#298692
#298662
#287132
#286842
#260782
#258372
#258362
#257022
#257012
#250472
#216312
#212762