Finite Group Databasebeta

← #7563412

Subgroups — #7563412

Order 512.

2047 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3384C212yesmaximal, complemented
2#59817C211yesmaximal, characteristic, complemented
3#36180C211yesmaximal, characteristic, complemented
4#30975C214yesmaximal, complemented
5#28817C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28509C211yesmaximal, characteristic, complemented
8#26467C212yesmaximal, complemented
9#26392C214yesmaximal, complemented
10#26044C212yesmaximal, complemented
11#25834C212yesmaximal, complemented
12#25477C212yesmaximal, complemented
13#24327C212yesmaximal, complemented
14#23153C211yesmaximal, characteristic, complemented
15#23001C212yesmaximal
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21#3384—216no—
22C2×#501C2214yes—
23C2×#498C2212yes—
24C2×#478C2214yes—
25C2×#476C2214yes—
26C2×#476C2212yes—
27C2×#476C2212yes—
28C2×#474C2211yescharacteristic
29C2×#459C2211yescharacteristic
30C2×#446C2212yes—
31C2×#442C2214yes—
32C2×C4×D8C2212yes—
33#2868C2211yescharacteristic
34#2857—21no—
35#2683—22no—
36#2567C2212yes—
37#2552C2212yes—
38#2552C2212yes—
39#2551C2212yes—
40#2538—21no—
41#2538C2212yes—
42#2502—22no—
43#2452—22no—
44#2452C2214yes—
45#2440—22no—
46#2440—22no—
47#2409—22no—
48#2366C2212yes—
49D4×C4⋊C4C2212yes—
50D4×C4⋊C4C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127C2211yescharacteristic
54C2×#384—21no—
55C42×D4C2211yescharacteristic
56#2094C2214yes—
57#2088C2214yes—
58#2084C2214yes—
59#2067C2214yes—
60#2059C2214yes—
61#2058C2214yes—
62#2058C2214yes—
63#2025C2214yes—
64#1990C2212yes—
65#1988C2211yescharacteristic
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1957C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1947C2211yescharacteristic
72#1947C2211yescharacteristic
73#1943C2214yes—
74#1939C2212yes—
75#1936C2214yes—
76#1925C2212yes—
77#1918C2212yes—
78#1917C2214yes—
79#1879C2214yes—
80#1879C2212yes—
81#1864C2214yes—
82#1864C2214yes—
83#1864C2214yes—
84#1862C2212yes—
85#1844C2212yes—
86#1802C2211yescharacteristic
87#1802C2211yescharacteristic
88C4×#107C2212yes—
89#1770C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94C23×Q8—22no—
95C22×D8C2312yes—
96#554—28no—
97#554—44no—
98D4×Q8—28no—
99D4×Q8—44no—
100D4×Q8—48no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109C2×#125C2314yes—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2314yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C42—21no—
131#501—216no—
132#498—28no—
133#478—216no—
134#476—28no—
135#476—216no—
136#476—44no—
137#474D418yescomplemented
138#459Q818yescomplemented
139#446—28no—
140#442—216no—
141C4×D8—28no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2311yescharacteristic
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#97C2314yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2312yes—
165#398—24no—
166#398—24no—
167#396—24no—
168#396—24no—
169#395—24no—
170#395—24no—
171#393—24no—
172#391—22no—
173#391—21no—
174#391D412yes—
175#390—22no—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#390C2314yes—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—21no—
184#389D412yes—
185#389D412yes—
186#389C2312yes—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#386—24no—
191#385—24no—
192#385—24no—
193#385—24no—
194#384—22no—
195#384—22no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384D411yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#380—22no—
205#380—22no—
206#380—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2314yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230C22×Q8—21no—
231C22×Q8—24no—
232C22×Q8—22no—
233C22×Q8—48no—
234C22×Q8—48no—
235C22×Q8—84no—
236C22×Q8C2×D412yes—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—24no—
247C2×D8—24no—
248C2×D8—24no—
249C2×D8C2×Q818yescomplemented
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—28no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C4⋊Q8—82no—
262C4⋊Q8—82no—
263C4⋊Q8C2×D414yes—
264#127—22no—
265#127C2×Q814yes—
266#125—28no—
267#125—44no—
268C22⋊Q8—416no—
269C22⋊Q8—416no—
270C22⋊Q8—48no—
271C22⋊Q8—416no—
272C22⋊Q8—48no—
273C22⋊Q8—48no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—44no—
289C4×D4—82no—
290C4×D4C2×D414yes—
291C4×D4C2×Q814yescomplemented
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—44no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2414yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—24no—
349#107—24no—
350#107C2×D418yes—
351#106—28no—
352#106—28no—
353#106—28no—
354#106—44no—
355C4⋊C8—28no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8C2×D414yes—
359C4⋊C8C2×Q814yescomplemented
360#102—24no—
361#102—24no—
362#102—28no—
363#102—24no—
364#102—24no—
365#102—44no—
366#102C2×D418yes—
367#102C2×Q818yescomplemented
368#97—28no—
369C4×C8—24no—
370C4×C8—24no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—21no—
382C24—22no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—84no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8—832no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4C22×D412yes—
411C2×D4C22×Q814yes—
412C2×D4C22×Q812yes—
413C2×D42−1+412yes—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—42no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448D8—216no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8C22×D414yes—
460C2×C8C22×Q814yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—81no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C22×D412yes—
495C4⋊C4C22×D414yes—
496C4⋊C4C22×Q812yes—
497C4⋊C42−1+412yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—84no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—24no—
512C42—21no—
513C42—21no—
514C42—41no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42AGL(1,ℤ/8)14yes—
520C42C22×D412yes—
521C42C22×Q812yes—
522C42H(5,𝔽2)12yes—
523C422−1+412yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C23×D411yescharacteristic
539Q8—48no—
540Q8—48no—
541Q8—816no—
542Q8—164no—
543D4—24no—
544D4—22no—
545D4—22no—
546D4—24no—
547D4—22no—
548D4—48no—
549D4—42no—
550D4—42no—
551D4—84no—
552D4D4×Q814yescomplemented
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—816no—
593C2×C4—88no—
594C2×C4—168no—
595C2×C4D4212yes—
596C2×C4D4212yes—
597C2×C4D4×Q812yes—
598C2×C4D4×Q812yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×AGL(1,ℤ/8)12yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4#57712yes—
603C2×C4#57712yes—
604C2×C4C23×D411yescharacteristic
605C2×C4C23×Q811yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C8—24no—
611C8—24no—
612C8—42no—
613C8—42no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×D4211yescharacteristic
629C22C2×D4×Q811yescharacteristic
630C22#348611yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C2×#57711yescharacteristic
633C22C2×#57711yescharacteristic
634C22#358511yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—44no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—41no—
645C4—82no—
646C4—82no—
647C4—84no—
648C4#327214yes—
649C4#338414yescomplemented
650C4C2×D4×Q811yescharacteristic
651C4#349911yescharacteristic
652C4#350311yescharacteristic
653C4#355211yescharacteristic
654C2—44no—
655C2—84no—
656C2#5912711yescharacteristic
657C2C2×#327211yescharacteristic
658C2#6109511yescharacteristic
659C2C2×#338411yescharacteristic
660C2#6138211yescharacteristic
661C2#6182511yescharacteristic
662C2#6187111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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
1. C2×#3384C2semidirect
2. #59817C2semidirect
3. #36180C2semidirect
4. #30975C2semidirect
5. #28817C2semidirect
6. #28687C2semidirect
7. #28509C2semidirect
8. #26467C2semidirect
9. #26392C2semidirect
10. #26044C2semidirect
11. #25834C2semidirect
12. #25477C2semidirect
13. #24327C2semidirect
14. #23153C2semidirect
16. C4×#474C2semidirect
137. #474D4semidirect
138. #459Q8semidirect
249. C2×D8C2×Q8semidirect
291. C4×D4C2×Q8semidirect
359. C4⋊C8C2×Q8semidirect
367. #102C2×Q8semidirect
552. D4D4×Q8semidirect
649. C4#3384semidirect

Maximal subgroups

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

subgroupindex
C2×#33842
#598172
#361802
#309752
#288172
#286872
#285092
#264672
#263922
#260442
#258342
#254772
#243272
#231532
#230012
C4×#4742
C4×#4592