Finite Group Databasebeta

← #6490586

Subgroups — #6490586

Order 512.

1892 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33149C212yesmaximal, complemented
4#33107C212yesmaximal
5#30150C214yesmaximal, complemented
6#29354C212yesmaximal, complemented
7#29319C212yesmaximal, complemented
8#29287C212yesmaximal, complemented
9#28279C212yesmaximal, complemented
10#27645C212yesmaximal, complemented
11#24607C214yesmaximal, complemented
12#24298C212yesmaximal, complemented
13#22116C214yesmaximal, complemented
14#20810C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529—21no—
17C22×C4×Q8—21no—
18C2×#481C2214yes—
19C2×#474C2214yes—
20C2×#432C2212yes—
21#2861—21no—
22#2860—21no—
23#2860C2212yes—
24#2859C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854—21no—
29#2854C2212yes—
30#2850—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×#127C2211yescharacteristic
41C4×#124—21no—
42C4×#121—21no—
43C4×#121C2212yes—
44C2×#391—21no—
45#2293—21no—
46#2293C2212yes—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2212yes—
52C2×C43C2211yescharacteristic
53#2088C2218yes—
54#2084C2214yes—
55#2071C2214yes—
56#2067C2218yes—
57#2057C2214yes—
58#2023C2214yes—
59#1988C2214yes—
60#1987C2214yes—
61#1982C2214yes—
62#1948C2214yes—
63#1948C2214yes—
64#1947C2214yes—
65#1943C2218yes—
66#1936C2218yes—
67#1926C2218yes—
68#1921C2218yes—
69#1874C2214yes—
70#1852C2214yes—
71#1842C2212yes—
72#1831C2212yes—
73#1824C2214yes—
74#1802C2214yes—
75#1797C2212yes—
76#1797C2212yes—
77#1767C2212yes—
78#1767C2212yes—
79C23×Q8—21no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87#529—48no—
88C2×#126—22no—
89C2×#125—21no—
90C2×#124—21no—
91C2×#124—21no—
92C2×#123—22no—
93C2×C22⋊Q8—21no—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121C2312yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8—48no—
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4C2312yes—
107C2×#117—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C42—22no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#481—48no—
123#474—216no—
124#432—44no—
125C2×#107C2318yes—
126C2×#106C2318yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2314yes—
130C2×#102C2318yes—
131C2×#102C2318yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2314yes—
134#391—22no—
135#391—22no—
136#391—22no—
137#391—22no—
138#391—21no—
139#391—21no—
140#390—21no—
141#390—21no—
142#390—21no—
143#390—21no—
144#390—21no—
145#390—21no—
146#390C2312yes—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389C2314yes—
161#389C2314yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#386—22no—
175#386—22no—
176#384—21no—
177#384—21no—
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#384C2311yescharacteristic
182#382—21no—
183#380—21no—
184#380C2312yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C2×#95—22no—
198C2×#95—22no—
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213C22×Q8—28no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—21no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C2412yes—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2411yescharacteristic
233C22×C8C2418yes—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237C4⋊Q8—24no—
238C4⋊Q8—24no—
239C4⋊Q8#4118yes—
240#127—24no—
241#126—48no—
242#125—44no—
243#124—44no—
244#124—44no—
245#123—48no—
246C22⋊Q8—44no—
247#121—24no—
248#121—24no—
249#121—44no—
250#121—44no—
251C4×Q8—48no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×Q8—44no—
255C4×Q8—48no—
256C4×D4—28no—
257C4×D4—44no—
258C4×D4—48no—
259C4×D4—44no—
260#117—48no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—42no—
314C2×C42—48no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—216no—
344#106—48no—
345C4⋊C8—28no—
346C4⋊C8—28no—
347C4⋊C8—44no—
348#102—216no—
349#102—48no—
350C22⋊C8—28no—
351#97—44no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362C24—22no—
363C24C22×D411yescharacteristic
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—44no—
367C2×Q8—432no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4AGL(1,ℤ/8)14yes—
420C22×C4#13714yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436C22×C4C2511yescharacteristic
437C2×C8—28no—
438C2×C8—28no—
439C2×C8—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4C2×#4118yes—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—21no—
500C42—22no—
501C42—21no—
502C42—21no—
503C42—41no—
504C42—41no—
505C42—48no—
506C42—48no—
507C42—44no—
508C42AGL(1,ℤ/8)14yes—
509C42C2×#4114yes—
510C422−1+414yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23#53712yes—
523C23C2×AGL(1,ℤ/8)12yes—
524C23C2×#13712yes—
525C23C23×D411yescharacteristic
526Q8—416no—
527Q8—416no—
528Q8—84no—
529D4—44no—
530D4—44no—
531D4—84no—
532D4—88no—
533D4—84no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—88no—
585C2×C4—88no—
586C2×C4#52512yes—
587C2×C4#52512yes—
588C2×C4#53612yes—
589C2×C4#53612yes—
590C2×C4#53712yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4#57512yes—
599C2×C4C22×#4112yes—
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+412yes—
602C2×C4C2×2−1+411yescharacteristic
603C8—48no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#325414yes—
618C22C2×#52511yescharacteristic
619C22C2×#53611yescharacteristic
620C22C2×#53711yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C22×#13711yescharacteristic
624C22C2×#57511yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—48no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—84no—
641C4#310914yes—
642C4#323914yes—
643C4#352212yes—
644C4#352312yes—
645C2—24no—
646C2—162no—
647C2#3524711yescharacteristic
648C2C2×#310911yescharacteristic
649C2#6056011yescharacteristic
650C2C2×#323911yescharacteristic
651C2#6086411yescharacteristic
652C2C2×#325411yescharacteristic
653C2#6091611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #33149C2semidirect
5. #30150C2semidirect
6. #29354C2semidirect
7. #29319C2semidirect
8. #29287C2semidirect
9. #28279C2semidirect
10. #27645C2semidirect
11. #24607C2semidirect
12. #24298C2semidirect
13. #22116C2semidirect
14. #20810C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×C4⋊Q82
#331492
#331072
#301502
#293542
#293192
#292872
#282792
#276452
#246072
#242982
#221162
#208102