Finite Group Databasebeta

← #7503484

Subgroups — #7503484

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#482C211yesmaximal, characteristic
2C2×#3081C211yesmaximal, characteristic, complemented
3#35540C211yesmaximal, characteristic
4#35151C211yesmaximal, characteristic, complemented
5#31108C214yesmaximal, complemented
6#31106C214yesmaximal, complemented
7#31084C214yesmaximal, complemented
8#31082C214yesmaximal, complemented
9#23667C211yesmaximal, characteristic, complemented
10#23661C211yesmaximal, characteristic, complemented
11#23638C211yesmaximal, characteristic, complemented
12#23632C211yesmaximal, characteristic, complemented
13#23556C211yesmaximal, characteristic, complemented
14#23480C211yesmaximal, characteristic
15#21089C211yesmaximal, characteristic, complemented
16#21053C211yesmaximal, characteristic, complemented
17#20876C211yesmaximal, characteristic, complemented
18#20845C211yesmaximal, characteristic, complemented
19#20803C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C2×#482—28no—
25C2×#482C2214yes—
26C2×#482C2214yes—
27#3081—28no—
28C2×#478C2212yescomplemented
29C2×#478C2212yescomplemented
30C2×#476C2212yescomplemented
31C2×#474C2212yescomplemented
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C2×#407C2211yescharacteristic
44C22×C22⋊C8C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2855—22no—
47#2626—22no—
48#2483—22no—
49#2441—22no—
50#2441C2214yescomplemented
51#2439—22no—
52#2439C2214yescomplemented
53#2422—22no—
54#2422C2214yes—
55#2420—22no—
56#2420C2214yes—
57#2405—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2317C2211yescharacteristic
62#2316C2211yescharacteristic
63#2309C2211yescharacteristic
64#2280C2211yescharacteristic
65#2280C2211yescharacteristic
66#2280C2211yescharacteristic
67#2087C2214yescomplemented
68#2086C2214yescomplemented
69#2085C2214yescomplemented
70#2084C2214yescomplemented
71#2077C2212yescomplemented
72#2077C2212yescomplemented
73#2076C2212yescomplemented
74#2075C2212yescomplemented
75#2073C2214yescomplemented
76#2071C2214yescomplemented
77#2059C2214yescomplemented
78#2058C2214yescomplemented
79#2034C2214yes—
80#2033C2214yes—
81#1936C2214yescomplemented
82#1934C2214yescomplemented
83#1924C2214yescomplemented
84#1923C2214yescomplemented
85#1864C2214yescomplemented
86#1862C2214yescomplemented
87#1854C2214yescomplemented
88#1854C2214yescomplemented
89#1847C2212yescomplemented
90#1847C2212yescomplemented
91#1846C2212yescomplemented
92#1845C2212yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C23×D4—21no—
102C23×D4C2311yescharacteristic
103C24×C4C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—42no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2312yes—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#123—24no—
115C2×C22⋊Q8—22no—
116C2×#121—22no—
117C2×#121—24no—
118C2×#121—42no—
119C2×#121C2314yes—
120#518—24no—
121#518—42no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127C2×#117—24no—
128C2×#117—42no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2314yes—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C42—21no—
143#482—216no—
144#482—216no—
145#478—28no—
146#478—28no—
147#476—28no—
148#474—28no—
149#428—24no—
150#428—42no—
151#428—42no—
152#427—24no—
153C2×#107—24no—
154C2×#107C2311yescharacteristic
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2311yescharacteristic
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#106—24no—
162C2×#106C2311yescharacteristic
163C2×#106C2312yes—
164C2×#106C2314yes—
165C2×#106C2311yescharacteristic
166C2×#106C2312yes—
167C2×#106C2312yes—
168#424—24no—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171#418—24no—
172#418—24no—
173C2×#102—24no—
174C2×#102D418yes—
175C2×#102C2312yes—
176C2×#102C2314yes—
177C2×#102C2312yes—
178C2×#102C2314yes—
179C2×#102C2312yes—
180C2×#102C2312yes—
181#407—24no—
182C2×C22⋊C8—24no—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2312yes—
185C2×C22⋊C8C2314yes—
186#405—24no—
187C2×#97C2312yes—
188C2×C4×C8C2312yes—
189#399—24no—
190#397—24no—
191#397—24no—
192#396—24no—
193#396—24no—
194#394—24no—
195#394—24no—
196#393—24no—
197#391—24no—
198#390—22no—
199#390C2314yes—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—24no—
204#389C2314yes—
205#389C2314yes—
206#389C2314yes—
207#387—24no—
208#387—24no—
209#386—24no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#385—24no—
215#384—21no—
216#384—21no—
217#384C2312yes—
218#384C2312yes—
219#382—21no—
220#382—21no—
221#382C2312yes—
222#382C2312yes—
223#381—21no—
224#381—21no—
225#381—24no—
226#381D412yes—
227#380—22no—
228#380D412yes—
229#380D412yes—
230#380C2312yes—
231#380C2312yes—
232#380C2312yes—
233#380C2312yes—
234#379—21no—
235#379—21no—
236#379—22no—
237#379D412yes—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4C2314yes—
241#340C2312yes—
242#340C2314yes—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#336C2314yes—
251#336C2314yes—
252C25—21no—
253C25C2×D411yescharacteristic
254C22×Q8—22no—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—44no—
263C22×D4—84no—
264C22×D4C2×D414yes—
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D412yes—
279C23×C4C2412yes—
280C23×C4C2411yescharacteristic
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16#4114yes—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8C2×D414yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C22×C8C2412yes—
298C4⋊Q8—24no—
299C4⋊Q8—42no—
300C4⋊Q8—44no—
301#125—24no—
302#125—42no—
303#125—44no—
304#123—88no—
305C22⋊Q8—48no—
306#121—28no—
307#121—44no—
308#121—48no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—42no—
312C4×D4—42no—
313C4×D4—44no—
314C4×D4—44no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—28no—
328#117—44no—
329#117—48no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—48no—
374C2×C22⋊C4—44no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D414yes—
379C2×C22⋊C4C2×D414yes—
380C2×C42—24no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—44no—
386C2×C42C2412yes—
387C2×C42C2412yes—
388#107—24no—
389#107—24no—
390#107—22no—
391#107—22no—
392#107—24no—
393#107—42no—
394#107—42no—
395#106—22no—
396#106—22no—
397#106—28no—
398#106—24no—
399#106—42no—
400#106—42no—
401C4⋊C8—24no—
402C4⋊C8—24no—
403#102—28no—
404#102—24no—
405#102—24no—
406#102—28no—
407#102C2×D418yes—
408#102C2×D418yes—
409C22⋊C8—24no—
410C22⋊C8—24no—
411C22⋊C8—28no—
412#97—24no—
413C4×C8—24no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95C2×D414yes—
424#95C2×D414yes—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—24no—
429C24—22no—
430C24—42no—
431C24—42no—
432C24—42no—
433C24—42no—
434C24—82no—
435C24—82no—
436C24C22×D412yes—
437C24C22×D411yescharacteristic
438C2×Q8—44no—
439C2×Q8—84no—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—48no—
449C2×D4—48no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—84no—
455C2×D4—816no—
456C2×D4C22×D414yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D414yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C42−1+412yes—
501C22×C42−1+412yes—
502C22×C42−1+412yes—
503C22×C4C2511yescharacteristic
504M16—28no—
505M16—44no—
506C2×C8—28no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—22no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—44no—
520C2×C8C22×D412yes—
521C2×C8C22×D412yes—
522C2×C8C2×#4114yes—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—84no—
550C4⋊C4—84no—
551C4⋊C4—84no—
552C4⋊C4—84no—
553C4⋊C4—84no—
554C4⋊C4C22×D414yes—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—48no—
563C22⋊C4—44no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—84no—
569C22⋊C4—84no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—48no—
578C42—42no—
579C42—42no—
580C42—42no—
581C42—42no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—24no—
591C23—22no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—42no—
597C23—44no—
598C23—42no—
599C23—48no—
600C23—84no—
601C23—84no—
602C23—84no—
603C23—84no—
604C23—164no—
605C23—164no—
606C23#53612yes—
607C23D4212yes—
608C23#54712yes—
609C23C23×D411yescharacteristic
610Q8—164no—
611D4—24no—
612D4—24no—
613D4—48no—
614D4—42no—
615D4—42no—
616D4—816no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—164no—
657C2×C4—164no—
658C2×C4#53614yes—
659C2×C4#53612yes—
660C2×C4#53914yes—
661C2×C4D4212yes—
662C2×C4#54712yes—
663C2×C4#54714yes—
664C2×C4#56314yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C8—24no—
672C8—22no—
673C8—22no—
674C8—42no—
675C8—42no—
676C22—24no—
677C22—22no—
678C22—22no—
679C22—24no—
680C22—42no—
681C22—42no—
682C22—44no—
683C22—42no—
684C22—42no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—88no—
691C22—84no—
692C22—84no—
693C22—168no—
694C22#268014yes—
695C22C2×#53611yescharacteristic
696C22C2×D4211yescharacteristic
697C22C2×#54711yescharacteristic
698C22#358412yes—
699C22#358512yes—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—42no—
705C4—82no—
706C4—81no—
707C4—81no—
708C4—81no—
709C4—82no—
710C4—82no—
711C4—81no—
712C4—82no—
713C4#268014yes—
714C4C2×#53611yescharacteristic
715C4C2×#53911yescharacteristic
716C4C2×#54711yescharacteristic
717C4C2×#56311yescharacteristic
718C2—24no—
719C2—82no—
720C2—84no—
721C2C2×#268011yescharacteristic
722C2#6087512yes—
723C2#6105111yescharacteristic
724C2#6105911yescharacteristic
725C2#6127612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3081C2semidirect
4. #35151C2semidirect
5. #31108C2semidirect
6. #31106C2semidirect
7. #31084C2semidirect
8. #31082C2semidirect
9. #23667C2semidirect
10. #23661C2semidirect
11. #23638C2semidirect
12. #23632C2semidirect
13. #23556C2semidirect
15. #21089C2semidirect
16. #21053C2semidirect
17. #20876C2semidirect
18. #20845C2semidirect
19. #20803C2semidirect
28. C2×#478C22semidirect
29. C2×#478C22semidirect
30. C2×#476C22semidirect
31. C2×#474C22semidirect
50. #2441C22semidirect
52. #2439C22semidirect
67. #2087C22semidirect
68. #2086C22semidirect
69. #2085C22semidirect
70. #2084C22semidirect
71. #2077C22semidirect
72. #2077C22semidirect
73. #2076C22semidirect
74. #2075C22semidirect
75. #2073C22semidirect
76. #2071C22semidirect
77. #2059C22semidirect
78. #2058C22semidirect
81. #1936C22semidirect
82. #1934C22semidirect
83. #1924C22semidirect
84. #1923C22semidirect
85. #1864C22semidirect
86. #1862C22semidirect
87. #1854C22semidirect
88. #1854C22semidirect
89. #1847C22semidirect
90. #1847C22semidirect
91. #1846C22semidirect
92. #1845C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4822
C2×#30812
#355402
#351512
#311082
#311062
#310842
#310822
#236672
#236612
#236382
#236322
#235562
#234802
#210892
#210532
#208762
#208452
#208032