Finite Group Databasebeta

← #381867

Subgroups — #381867

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1999C211yesmaximal, characteristic
2C2×#1972C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic, complemented
4#14698C212yesmaximal, complemented
5#11257C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11095C211yesmaximal, characteristic, complemented
8#11022C211yesmaximal, characteristic, complemented
9#10969C212yesmaximal, complemented
10#10927C212yesmaximal, complemented
11#10821C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#121C2212yes—
14C23×C42C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#386—22no—
26C2×#386—22no—
27C2×#380C2212yes—
28#1999C418yescomplemented
29#1972—24no—
30#1749—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1437—22no—
39#1437—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×D4—22no—
43C23×D4C2312yes—
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C22×D8—22no—
47C22×D8—22no—
48C22×D8—42no—
49C22×D8—42no—
50C22×D8C2×C414yescomplemented
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#121—24no—
56C2×#121—42no—
57C2×#121—42no—
58C2×#121—42no—
59C2×#121—42no—
60C2×#121C2×C418yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—42no—
73C22×C42—42no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—22no—
78C2×#106—42no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102C2×C418yescomplemented
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2×C414yes—
94C2×#97—22no—
95C2×#97—42no—
96C2×#97—42no—
97C2×#97D414yescomplemented
98#390—24no—
99#390—24no—
100#390—42no—
101#390—42no—
102#386—48no—
103#386—48no—
104#380—42no—
105#380—42no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111#340—22no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340C2×C414yescomplemented
116#336—22no—
117#336—22no—
118C25—22no—
119C25C2×D411yescharacteristic
120C22×D4—28no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4—44no—
140C22×D4C22×C414yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4C22×C412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C2×D8—28no—
166C2×D8—48no—
167C2×D8—44no—
168C2×D8—44no—
169C2×D8—48no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C22×C412yes—
185C22×C8C22×C412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8#4112yes—
189C22×C8#4112yes—
190#121—48no—
191#121—48no—
192#121—48no—
193#121—48no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C22×C414yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—48no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42#4112yes—
244#106—44no—
245#102—48no—
246#102—48no—
247#102—48no—
248C22⋊C8—28no—
249C22⋊C8—28no—
250#97—44no—
251#97—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—24no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24C4×D412yes—
280C24AGL(1,ℤ/8)14yes—
281C24C22×D411yescharacteristic
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—416no—
289C2×D4—416no—
290C2×D4—416no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—88no—
314C2×D4—88no—
315C2×D4—88no—
316C2×D4—88no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4C4×D412yes—
380C22×C4#12114yes—
381C22×C4AGL(1,ℤ/8)14yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)12yes—
392D8—416no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8C4×D414yes—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C22⋊C4—416no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C42—22no—
432C42—22no—
433C42—24no—
434C42—48no—
435C42—41no—
436C42—41no—
437C42—42no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23#44214yes—
494C23C2×C4×D411yescharacteristic
495C23C2×#12111yescharacteristic
496C23#53211yescharacteristic
497C23#53412yes—
498C23#53512yes—
499C23C2×AGL(1,ℤ/8)12yes—
500C23C2×AGL(1,ℤ/8)12yes—
501C23#57512yes—
502C23#57512yes—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—832no—
508D4—82no—
509D4—82no—
510D4—82no—
511D4—82no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—432no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C4×#4111yescharacteristic
561C2×C4#51812yes—
562C2×C4#51812yes—
563C2×C4#52511yescharacteristic
564C2×C4#52911yescharacteristic
565C2×C4#53411yescharacteristic
566C2×C4#53511yescharacteristic
567C2×C4#53511yescharacteristic
568C2×C4#53511yescharacteristic
569C2×C4#53812yes—
570C2×C4#53812yes—
571C2×C4#53812yes—
572C2×C4#53812yes—
573C2×C4#53812yes—
574C2×C4#54012yes—
575C2×C4#54012yes—
576C2×C4#54012yes—
577C8—24no—
578C8—42no—
579C8—42no—
580C8—42no—
581C22—24no—
582C22—28no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—162no—
598C22—162no—
599C22—162no—
600C22—162no—
601C22—162no—
602C22—162no—
603C22—162no—
604C22—162no—
605C22C4×#12111yescharacteristic
606C22#232112yes—
607C22#261711yescharacteristic
608C22#261911yescharacteristic
609C22#266112yes—
610C22C2×#44211yescharacteristic
611C22#294311yescharacteristic
612C22C4×AGL(1,ℤ/8)12yes—
613C22#305512yes—
614C22#305512yes—
615C22#318012yes—
616C22#318311yescharacteristic
617C22#318411yescharacteristic
618C22#319212yes—
619C22#319412yes—
620C22#319412yes—
621C22#319712yes—
622C22#321312yes—
623C22#321412yes—
624C22#321712yes—
625C22#321912yes—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—84no—
630C4—162no—
631C4—162no—
632C4#232111yescharacteristic
633C4#232811yescharacteristic
634C4#233312yes—
635C4#263111yescharacteristic
636C4#263511yescharacteristic
637C4#264812yes—
638C2—44no—
639C2—164no—
640C2#1431111yescharacteristic
641C2#2152611yescharacteristic
642C2#2153811yescharacteristic
643C2#2204612yes—
644C2#2205112yes—
645C2#2974212yes—
646C2#2981412yes—
647C2#3032212yes—
648C2#3032412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1972C2semidirect
3. C2×#1749C2semidirect
4. #14698C2semidirect
5. #11257C2semidirect
6. #11189C2semidirect
7. #11095C2semidirect
8. #11022C2semidirect
9. #10969C2semidirect
10. #10927C2semidirect
11. #10821C2semidirect
28. #1999C4semidirect
50. C22×D8C2×C4semidirect
87. C2×#102C2×C4semidirect
97. C2×#97D4semidirect
115. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19992
C2×#19722
C2×#17492
#146982
#112572
#111892
#110952
#110222
#109692
#109272
#108212