Finite Group Databasebeta

← #7562701

Subgroups — #7562701

Order 512.

2113 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3362C212yesmaximal, complemented
2#59815C211yesmaximal, characteristic
3Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
4#30813C214yesmaximal, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28509C211yesmaximal, characteristic, complemented
7#27767C212yesmaximal, complemented
8#26163C214yesmaximal, complemented
9#26044C212yesmaximal, complemented
10#25649C212yesmaximal, complemented
11#25470C212yesmaximal, complemented
12#24728C212yesmaximal, complemented
13#24329C212yesmaximal, complemented
14#23149C211yesmaximal, characteristic, complemented
15C4⋊C4×SD16C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×Q82C2212yes—
19C2×#550C2212yes—
20C2×#547—21no—
21#3362—216no—
22C2×#486C2212yes—
23C2×#486C2214yes—
24C2×#476C2212yes—
25C2×#476C2214yes—
26C2×#476C2212yes—
27C2×#460C2214yes—
28C2×#460C2211yescharacteristic
29C2×#460C2211yescharacteristic
30C2×C8×Q8C2212yes—
31C2×C4×SD16C2212yes—
32C2×C4×SD16C2214yes—
33C2×C4×SD16C2212yes—
34#2868—21no—
35#2868C2211yescharacteristic
36#2682—22no—
37#2567—22no—
38#2567C2212yes—
39#2551—22no—
40#2551—22no—
41#2551C2212yes—
42#2551C2212yes—
43#2541C2212yes—
44#2538C2212yes—
45#2537—21no—
46#2445C2214yes—
47#2416—22no—
48#2407—22no—
49Q8×C4⋊C4C2212yes—
50Q8×C4⋊C4C2212yes—
51D4×C4⋊C4C2212yes—
52C4×C4⋊Q8—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C42×Q8C2211yescharacteristic
57#2094C2214yes—
58#2088C2214yes—
59#2071C2214yes—
60#2065C2214yes—
61#2058C2214yes—
62#2041C2214yes—
63#2041C2214yes—
64#2023C2214yes—
65#1990C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2211yescharacteristic
69#1957C2212yes—
70#1948C2211yescharacteristic
71#1948C2211yescharacteristic
72#1941C2214yes—
73#1939C2212yes—
74#1934C2214yes—
75#1926C2212yes—
76#1926C2212yes—
77#1916C2214yes—
78#1880C2214yes—
79#1880C2212yes—
80#1880C2212yes—
81#1865C2214yes—
82#1864C2214yes—
83#1864C2214yes—
84#1844C2212yes—
85#1803C2211yescharacteristic
86#1803C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94C22×#41—22no—
95C22×SD16C2312yes—
96C22×SD16C2312yes—
97Q82—28no—
98Q82—44no—
99#550—28no—
100#550—44no—
101#547—48no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8D411yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×#127C2311yescharacteristic
112C2×#127C2314yes—
113C2×#127C2311yescharacteristic
114C2×#123—24no—
115C2×C22⋊Q8—22no—
116C2×#121—22no—
117C2×#121—24no—
118C2×#121—22no—
119C2×#121—22no—
120C2×C4×Q8—22no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2314yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2314yes—
133C22×C4⋊C4—22no—
134#486—28no—
135#486—216no—
136#476—28no—
137#476—216no—
138#476—44no—
139#460—216no—
140#460—24no—
141#460D418yescomplemented
142C8×Q8—28no—
143C4×SD16—216no—
144C4×SD16—28no—
145C4×SD16—28no—
146C2×#106C2312yes—
147C2×#106C2312yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2314yes—
169C2×C4×C8C2312yes—
170#396—24no—
171#395—24no—
172#395—24no—
173#395—24no—
174#394—24no—
175#394—24no—
176#392—24no—
177#391—22no—
178#391D412yes—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390C2312yes—
184#390C2312yes—
185#390C2314yes—
186#389—22no—
187#389—22no—
188#389—24no—
189#389—24no—
190#389D412yes—
191#389D412yes—
192#389C2314yes—
193#389C2314yes—
194#389C2314yes—
195#386—24no—
196#385—24no—
197#384—22no—
198#384—22no—
199#384—22no—
200#384—21no—
201#384—21no—
202#384D411yescharacteristic
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2312yes—
207#380—22no—
208#380—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C43—21no—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C2×#41—48no—
232C2×#41—48no—
233C2×#41—84no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8C2×D412yes—
237C22×Q8C2411yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—24no—
244C22×D4C2412yes—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—24no—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16C2×D418yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—28no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—48no—
264C4⋊Q8—416no—
265C4⋊Q8—82no—
266C4⋊Q8—84no—
267C4⋊Q8—82no—
268C4⋊Q8—84no—
269C4⋊Q8SD1618yescomplemented
270#127—28no—
271#127—22no—
272#127—22no—
273#127—22no—
274#127—44no—
275#127C2×D414yes—
276#123—416no—
277C22⋊Q8—48no—
278#121—48no—
279#121—416no—
280#121—48no—
281#121—84no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—28no—
289C4×Q8—44no—
290C4×Q8—48no—
291C4×Q8C2×D414yes—
292C4×Q8#4114yescomplemented
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—82no—
302C4×D4—82no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2414yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#106—28no—
358#106—24no—
359#106—24no—
360#106—44no—
361C4⋊C8—28no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8C2×D414yes—
366#103—24no—
367#103—24no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—24no—
372#102—24no—
373#102—24no—
374#102—24no—
375#102—24no—
376#102—44no—
377#102C2×D418yes—
378C4×C8—28no—
379C4×C8—24no—
380C4×C8—24no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95C2×D414yes—
388#95C2×D414yes—
389C24—22no—
390C24—22no—
391#41—832no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—48no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×Q8C2×SD1618yes—
404C2×Q8C22×D412yes—
405C2×Q8C2×#4112yes—
406C2×Q82−1+412yes—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—48no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—81no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4—81no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4C22×D414yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—84no—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D414yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D414yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2511yescharacteristic
462SD16—216no—
463SD16—48no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8C22×D414yes—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—82no—
505C4⋊C4—84no—
506C4⋊C4C2×SD1618yes—
507C4⋊C4C2×SD1618yes—
508C4⋊C4C22×D412yes—
509C4⋊C4C2×#4112yes—
510C4⋊C42−1+412yes—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—24no—
527C42—21no—
528C42—21no—
529C42—41no—
530C42—48no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42C2×SD1614yes—
535C42C22×D412yes—
536C42C2×#4112yes—
537C42H(5,𝔽2)12yes—
538C422−1+412yes—
539C23—24no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C23×D411yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—216no—
557Q8—22no—
558Q8—164no—
559Q8—168no—
560Q8#54714yescomplemented
561D4—24no—
562D4—24no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—48no—
568D4—42no—
569D4—42no—
570D4—84no—
571D4—816no—
572D4—84no—
573D4—84no—
574D4—168no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—816no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4D4212yes—
614C2×C4D4212yes—
615C2×C4#54812yes—
616C2×C4#54812yes—
617C2×C4C22×SD1612yes—
618C2×C4C22×SD1618yes—
619C2×C4C22×SD1612yes—
620C2×C4C22×SD1612yes—
621C2×C4#57612yes—
622C2×C4#57712yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C8—24no—
630C8—42no—
631C8—42no—
632C8—42no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C2×D4211yescharacteristic
648C22C2×#54811yescharacteristic
649C22#348611yescharacteristic
650C22C23×SD1611yescharacteristic
651C22C2×#57611yescharacteristic
652C22C2×#57711yescharacteristic
653C22#358511yescharacteristic
654C4—28no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—24no—
659C4—42no—
660C4—44no—
661C4—42no—
662C4—41no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4D4×SD1614yes—
668C4#336214yescomplemented
669C4C2×#54711yescharacteristic
670C4#349911yescharacteristic
671C4#351111yescharacteristic
672C4#355211yescharacteristic
673C2—84no—
674C2—162no—
675C2#5912511yescharacteristic
676C2C2×D4×SD1611yescharacteristic
677C2#6109311yescharacteristic
678C2C2×#336211yescharacteristic
679C2#6134711yescharacteristic
680C2#6182011yescharacteristic
681C2#6186811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
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×#3362C2semidirect
3. Q8×C4⋊Q8C2semidirect
4. #30813C2semidirect
5. #28716C2semidirect
6. #28509C2semidirect
7. #27767C2semidirect
8. #26163C2semidirect
9. #26044C2semidirect
10. #25649C2semidirect
11. #25470C2semidirect
12. #24728C2semidirect
13. #24329C2semidirect
14. #23149C2semidirect
15. C4⋊C4×SD16C2semidirect
16. C4×#460C2semidirect
17. C4×#460C2semidirect
141. #460D4semidirect
269. C4⋊Q8SD16semidirect
292. C4×Q8#41semidirect
560. Q8#547semidirect
668. C4#3362semidirect

Maximal subgroups

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

subgroupindex
C2×#33622
#598152
Q8×C4⋊Q82
#308132
#287162
#285092
#277672
#261632
#260442
#256492
#254702
#247282
#243292
#231492
C4⋊C4×SD162
C4×#4602