Finite Group Databasebeta

← #6375758

Subgroups — #6375758

Order 512.

1942 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic, complemented
3#34058C211yesmaximal, characteristic
4#22126C214yesmaximal, complemented
5#22122C214yesmaximal, complemented
6#22120C214yesmaximal, complemented
7#22114C214yesmaximal, complemented
8#21314C211yesmaximal, characteristic, complemented
9#21169C211yesmaximal, characteristic, complemented
10#21158C211yesmaximal, characteristic, complemented
11#21157C211yesmaximal, characteristic, complemented
12#21089C211yesmaximal, characteristic
13#21089C211yesmaximal, characteristic
14#21065C211yesmaximal, characteristic
15#21065C211yesmaximal, characteristic
16#21038C211yesmaximal, characteristic, complemented
17#21037C211yesmaximal, characteristic, complemented
18#20646C211yesmaximal, characteristic
19#20431C211yesmaximal, characteristic, complemented
20C22×#125C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#478C4116yescomplemented
23C2×#478C2212yes—
24C2×#478C2212yes—
25C2×#478C2212yes—
26C2×#478C2212yes—
27C2×#427C2211yescharacteristic
28C2×#427C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C2×#423C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#418C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C2×#405C2211yescharacteristic
38C2×#405C2211yescharacteristic
39#2370—22no—
40#2370—22no—
41#2345—22no—
42#2345—22no—
43#2330C2214yescomplemented
44#2319—22no—
45#2319C2214yes—
46#2316C2211yescharacteristic
47#2316C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50#2312—22no—
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53C4×#125C2214yescomplemented
54#2293—22no—
55#2293C2214yes—
56#2285C2211yescharacteristic
57#2280—22no—
58#2274C2211yescharacteristic
59#2274C2211yescharacteristic
60C4×#117C2211yescharacteristic
61C4×#117C2211yescharacteristic
62#1989C2214yescomplemented
63#1989C2214yescomplemented
64#1949C2212yescomplemented
65#1949C2212yescomplemented
66#1948C2212yescomplemented
67#1947C2212yescomplemented
68#1944C2214yescomplemented
69#1943C2214yescomplemented
70#1942C2214yescomplemented
71#1941C2214yescomplemented
72#1936C2214yes—
73#1936C2214yes—
74#1934C2214yes—
75#1934C2214yes—
76#1926C2214yes—
77#1926C2214yes—
78#1925C2214yes—
79#1925C2214yes—
80#1921C2214yescomplemented
81#1920C2214yescomplemented
82#1919C2214yescomplemented
83#1918C2214yescomplemented
84#1804C2214yes—
85#1802C2214yes—
86#1768C2212yescomplemented
87#1768C2212yescomplemented
88#1767C2212yescomplemented
89C4×C4⋊C8C2212yescomplemented
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100C23×D4C2311yescharacteristic
101C24×C4—21no—
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C23×C8C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C2×#125—21no—
107C2×#125—21no—
108C2×#125—24no—
109C2×#125C2×C418yes—
110C2×#125C2312yes—
111C2×#125C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4C2×C418yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—24no—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C42C2311yescharacteristic
132#478C2×C4116yes—
133#478C2×C4116yes—
134#478C2×C4116yes—
135#478C2×C4116yes—
136#427—42no—
137#427—42no—
138C2×#107C2×C418yes—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#106C2×C418yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×#106C2312yes—
147C2×#106C2312yes—
148#423—24no—
149C2×C4⋊C8C2×C418yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2311yescharacteristic
156#418—42no—
157#418—42no—
158C2×#102C2×C418yes—
159C2×#102C2×C418yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166#405—42no—
167#405—42no—
168C2×#97C2312yes—
169C2×#97C2312yes—
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172#390—22no—
173#390—22no—
174#390C2314yes—
175#390C2314yes—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—24no—
181#389—24no—
182#389C2314yes—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#388—24no—
187#388—24no—
188#388—24no—
189#388—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#387—24no—
194#384C2312yes—
195#382C2312yes—
196#381—24no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—24no—
204#379—22no—
205#379—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—24no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—24no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225#376—21no—
226#376C2312yes—
227C43—21no—
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#336C2312yes—
240#336C2312yes—
241#336C2312yes—
242#336C2312yes—
243C25—21no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4C22×C414yes—
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—28no—
257C23×C4—22no—
258C23×C4—84no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8C22×C414yes—
273C22×C8C22×C414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2411yescharacteristic
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280#125—28no—
281#125—28no—
282#125—24no—
283#125—24no—
284#125C22×C418yes—
285#125C22×C418yes—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4C22×C418yes—
291C4×D4C22×C418yes—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—88no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4C22×C414yes—
324C2×C4⋊C4C22×C414yes—
325C2×C4⋊C4C22×C414yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42C22×C414yes—
357C2×C42C2×D414yes—
358C2×C42C2×D414yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42#4114yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—42no—
367#107—42no—
368#107C22×C418yes—
369#107C22×C418yes—
370#106—42no—
371#106—42no—
372#106C22×C418yes—
373#106C22×C418yes—
374C4⋊C8—24no—
375C4⋊C8—24no—
376C4⋊C8C22×C414yes—
377C4⋊C8C22×C414yes—
378C4⋊C8C22×C414yes—
379C4⋊C8C22×C414yes—
380#102—44no—
381#102—44no—
382#102C22×C418yes—
383#102C22×C418yes—
384#102C22×C418yes—
385#102C22×C418yes—
386#97—42no—
387#97—42no—
388C4×C8—42no—
389C4×C8—42no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95C2×D414yes—
401#95C2×D414yes—
402C24—22no—
403C24—24no—
404C24—42no—
405C24—42no—
406C24—82no—
407C24—82no—
408C24C22×D411yescharacteristic
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4C23×C414yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4—816no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4C4×D414yes—
443C22×C4C4×D414yes—
444C22×C4C23×C412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458M16—44no—
459M16—44no—
460C2×C8—28no—
461C2×C8—28no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8C23×C412yes—
477C2×C8C23×C412yes—
478C2×C8C23×C412yes—
479C2×C8C23×C412yes—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4C23×C414yes—
507C4⋊C4C23×C414yes—
508C4⋊C4C23×C414yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—44no—
526C42—44no—
527C42—44no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—82no—
533C42—84no—
534C42—84no—
535C42C4×D414yes—
536C42C4×D414yes—
537C42C23×C412yes—
538C42C23×C412yes—
539C42C2×#4112yes—
540C42C2×#4112yes—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—44no—
545C23—44no—
546C23—48no—
547C23—42no—
548C23—42no—
549C23—84no—
550C23—84no—
551C23—88no—
552C23—88no—
553C23C2×C4×D412yes—
554C23#53612yes—
555C23#53712yes—
556C23C23×D411yescharacteristic
557D4—24no—
558D4—24no—
559D4—48no—
560D4—42no—
561D4—42no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4C2×C4×D412yes—
598C2×C4C2×C4×D412yes—
599C2×C4C2×C4×D412yes—
600C2×C4C2×C4×D412yes—
601C2×C4C2×C4×D412yes—
602C2×C4#51914yes—
603C2×C4#52014yes—
604C2×C4#53612yes—
605C2×C4#53712yes—
606C2×C4#54114yes—
607C2×C4#54214yes—
608C2×C4C24×C411yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C8—22no—
615C8—22no—
616C8—22no—
617C8—22no—
618C8—42no—
619C8—42no—
620C22—24no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—816no—
628C22—84no—
629C22—84no—
630C22#232314yes—
631C22C22×C4×D411yescharacteristic
632C22C2×#53611yescharacteristic
633C22C2×#53711yescharacteristic
634C22#358412yes—
635C22#358512yes—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#232314yes—
650C4C2×#51911yescharacteristic
651C4C2×#52011yescharacteristic
652C4C2×#54111yescharacteristic
653C4C2×#54211yescharacteristic
654C2—24no—
655C2—84no—
656C2C2×#232311yescharacteristic
657C2#6016711yescharacteristic
658C2#6016811yescharacteristic
659C2#6089012yes—
660C2#6093312yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34217C2semidirect
4. #22126C2semidirect
5. #22122C2semidirect
6. #22120C2semidirect
7. #22114C2semidirect
8. #21314C2semidirect
9. #21169C2semidirect
10. #21158C2semidirect
11. #21157C2semidirect
16. #21038C2semidirect
17. #21037C2semidirect
19. #20431C2semidirect
22. C2×#478C4semidirect
43. #2330C22semidirect
53. C4×#125C22semidirect
62. #1989C22semidirect
63. #1989C22semidirect
64. #1949C22semidirect
65. #1949C22semidirect
66. #1948C22semidirect
67. #1947C22semidirect
68. #1944C22semidirect
69. #1943C22semidirect
70. #1942C22semidirect
71. #1941C22semidirect
80. #1921C22semidirect
81. #1920C22semidirect
82. #1919C22semidirect
83. #1918C22semidirect
86. #1768C22semidirect
87. #1768C22semidirect
88. #1767C22semidirect
89. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
#342172
#340582
#221262
#221222
#221202
#221142
#213142
#211692
#211582
#211572
#210892
#210652
#210382
#210372
#206462
#204312