Finite Group Databasebeta

← #7502457

Subgroups — #7502457

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic
3#35640C211yesmaximal, characteristic, complemented
4#35174C211yesmaximal, characteristic
5#30975C214yesmaximal, complemented
6#30971C214yesmaximal, complemented
7#30959C214yesmaximal, complemented
8#30955C214yesmaximal, complemented
9#23735C211yesmaximal, characteristic
10#23666C211yesmaximal, characteristic, complemented
11#23661C211yesmaximal, characteristic, complemented
12#23556C211yesmaximal, characteristic, complemented
13#23553C211yesmaximal, characteristic, complemented
14#23456C211yesmaximal, characteristic
15#21091C211yesmaximal, characteristic
16#21063C211yesmaximal, characteristic
17#21037C211yesmaximal, characteristic, complemented
18#21024C211yesmaximal, characteristic, complemented
19#20873C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#3164—28no—
25C2×#501C2212yescomplemented
26C2×#501C2212yescomplemented
27C2×#500C2212yescomplemented
28C2×#499C2212yescomplemented
29C2×#478—28no—
30C2×#478C2214yes—
31C2×#478C2214yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C2×#405C2211yescharacteristic
44#2848—22no—
45#2626—22no—
46#2583—22no—
47#2552—22no—
48#2552C2214yescomplemented
49#2550—22no—
50#2550C2214yescomplemented
51#2502—22no—
52#2454—22no—
53#2454C2214yes—
54#2452—22no—
55#2452C2214yes—
56#2317C2211yescharacteristic
57#2317C2211yescharacteristic
58#2316C2211yescharacteristic
59#2309C2211yescharacteristic
60#2309C2211yescharacteristic
61#2296C2211yescharacteristic
62#2296C2211yescharacteristic
63#2296C2211yescharacteristic
64#2274C2211yescharacteristic
65#2094C2212yes—
66#2093C2212yes—
67#2092C2212yes—
68#2092C2212yes—
69#2088C2212yescomplemented
70#2088C2212yescomplemented
71#2087C2212yescomplemented
72#2086C2212yescomplemented
73#2085C2214yescomplemented
74#2084C2214yescomplemented
75#2059C2214yescomplemented
76#2059C2214yescomplemented
77#2059C2214yescomplemented
78#2058C2214yescomplemented
79#2025C2214yes—
80#2023C2214yes—
81#1936C2214yes—
82#1934C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1921C2214yescomplemented
86#1918C2214yescomplemented
87#1917C2214yescomplemented
88#1916C2214yescomplemented
89#1864C2214yes—
90#1862C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1735C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C23×D4C2311yescharacteristic
102C24×C4—21no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#544—24no—
108#544—24no—
109#544—42no—
110#544—42no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#125—21no—
116C2×#125—21no—
117C2×#125—24no—
118C2×#125—22no—
119C2×#125—42no—
120C2×#125C2314yes—
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×C4×D4—24no—
127C2×C4×D4—22no—
128C2×C4×D4—42no—
129C2×C4×D4C2314yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C22⋊C4—21no—
142C22×C42C2311yescharacteristic
143#501—28no—
144#501—28no—
145#500—28no—
146#499—28no—
147#478—216no—
148#478—216no—
149#428—24no—
150#428—24no—
151#428—42no—
152#427—24no—
153C2×#107D418yes—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2311yescharacteristic
157C2×#107C2311yescharacteristic
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#106—24no—
162C2×#106C2314yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165C2×#106C2311yescharacteristic
166C2×#106C2311yescharacteristic
167C2×#106C2312yes—
168C2×C4⋊C8—24no—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171#418—42no—
172#418Q818yescomplemented
173C2×#102—24no—
174C2×#102—24no—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×#102C2314yes—
178C2×#102C2312yes—
179C2×#102C2312yes—
180C2×#102C2314yes—
181#405—24no—
182C2×#97C2312yes—
183C2×C4×C8C2312yes—
184#399—24no—
185#398—24no—
186#398—24no—
187#398—24no—
188#396—24no—
189#396—24no—
190#395—24no—
191#393—24no—
192#391—24no—
193#390—21no—
194#390—22no—
195#390—21no—
196#390C2312yes—
197#390C2312yes—
198#390C2314yes—
199#389—22no—
200#389—22no—
201#389—24no—
202#389—24no—
203#389D412yes—
204#389D412yes—
205#389C2314yes—
206#389C2312yes—
207#389C2312yes—
208#389C2314yes—
209#389C2312yes—
210#389C2312yes—
211#385—21no—
212#385—21no—
213#385—21no—
214#385—21no—
215#385—24no—
216#385—22no—
217#385—22no—
218#385—24no—
219#385D412yes—
220#385D412yes—
221#384—21no—
222#384—24no—
223#384C2312yes—
224#382—21no—
225#382C2312yes—
226#380—22no—
227#380—24no—
228#379—22no—
229#379—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4C2314yes—
234C4×C4⋊C4C2314yes—
235C4×C22⋊C4—22no—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2312yes—
242#340C2312yes—
243#340C2312yes—
244#340C2312yes—
245#336C2314yes—
246#336C2314yes—
247#336C2314yes—
248#336C2312yes—
249#336C2312yes—
250C25—21no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4C2×Q814yes—
260C22×D4C2412yes—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—24no—
265C23×C4—44no—
266C23×C4—84no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16C2×Q814yes—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8C2×D414yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2411yescharacteristic
285C22×C8C2411yescharacteristic
286C22×C8C2412yes—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—42no—
290C4⋊Q8—42no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293#125—28no—
294#125—24no—
295#125—24no—
296#125—44no—
297#125—42no—
298#125—42no—
299#125—48no—
300#125—44no—
301#125—44no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304C22⋊Q8—88no—
305C4×D4—28no—
306C4×D4—44no—
307C4×D4—48no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117C2×Q814yes—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
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×C4⋊C4C2412yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—24no—
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383#107—22no—
384#107—22no—
385#107—24no—
386#107—24no—
387#107—42no—
388#107C2×D418yes—
389#107C2×D418yes—
390#106—22no—
391#106—22no—
392#106—28no—
393#106—24no—
394#106—24no—
395#106—42no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398#102—24no—
399#102—24no—
400#102—28no—
401#102—42no—
402#102—42no—
403#102C2×Q8116yes—
404#97—24no—
405C4×C8—24no—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95C2×D414yes—
415#95C2×D414yes—
416C24—22no—
417C24—24no—
418C24—42no—
419C24—42no—
420C24—82no—
421C24—82no—
422C24C22×D411yescharacteristic
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—84no—
426C2×Q8—84no—
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4C22×Q814yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—42no—
458C22×C4—84no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—816no—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×Q812yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476M16—28no—
477M16—44no—
478C2×C8—28no—
479C2×C8—22no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—44no—
492C2×C8C22×D412yes—
493C2×C8C22×D412yes—
494C2×C8C22×Q814yes—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4C22×D414yes—
534C4⋊C4C22×D414yes—
535C4⋊C4C22×Q814yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—48no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—42no—
557C42—42no—
558C42—84no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—44no—
563C23—44no—
564C23—48no—
565C23—42no—
566C23—42no—
567C23—88no—
568C23—84no—
569C23—84no—
570C23—88no—
571C23#53612yes—
572C23D4212yes—
573C23D4×Q812yes—
574C23C23×D411yescharacteristic
575Q8—164no—
576Q8—164no—
577D4—24no—
578D4—24no—
579D4—48no—
580D4—42no—
581D4—42no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—816no—
620C2×C4—84no—
621C2×C4#53612yes—
622C2×C4#53614yes—
623C2×C4#54414yes—
624C2×C4D4212yes—
625C2×C4D4×Q812yes—
626C2×C4D4×Q814yes—
627C2×C4#56314yes—
628C2×C4C23×D411yescharacteristic
629C2×C4C23×Q811yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C8—22no—
635C8—22no—
636C8—24no—
637C8—42no—
638C8—42no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—44no—
644C22—48no—
645C22—816no—
646C22—82no—
647C22—82no—
648C22—84no—
649C22#267714yes—
650C22C2×#53611yescharacteristic
651C22C2×D4211yescharacteristic
652C22C2×D4×Q811yescharacteristic
653C22#358412yes—
654C22#358512yes—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—81no—
664C4—81no—
665C4—82no—
666C4—81no—
667C4—81no—
668C4—82no—
669C4#267714yes—
670C4C2×#53611yescharacteristic
671C4C2×#54411yescharacteristic
672C4C2×D4×Q811yescharacteristic
673C4C2×#56311yescharacteristic
674C2—24no—
675C2—84no—
676C2C2×#267711yescharacteristic
677C2#6089012yes—
678C2#6105411yescharacteristic
679C2#6105811yescharacteristic
680C2#6138512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3164C2semidirect
3. #35640C2semidirect
5. #30975C2semidirect
6. #30971C2semidirect
7. #30959C2semidirect
8. #30955C2semidirect
10. #23666C2semidirect
11. #23661C2semidirect
12. #23556C2semidirect
13. #23553C2semidirect
17. #21037C2semidirect
18. #21024C2semidirect
25. C2×#501C22semidirect
26. C2×#501C22semidirect
27. C2×#500C22semidirect
28. C2×#499C22semidirect
48. #2552C22semidirect
50. #2550C22semidirect
69. #2088C22semidirect
70. #2088C22semidirect
71. #2087C22semidirect
72. #2086C22semidirect
73. #2085C22semidirect
74. #2084C22semidirect
75. #2059C22semidirect
76. #2059C22semidirect
77. #2059C22semidirect
78. #2058C22semidirect
85. #1921C22semidirect
86. #1918C22semidirect
87. #1917C22semidirect
88. #1916C22semidirect
172. #418Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C22×#4782
#356402
#351742
#309752
#309712
#309592
#309552
#237352
#236662
#236612
#235562
#235532
#234562
#210912
#210632
#210372
#210242
#208732