Finite Group Databasebeta

← #361256

Subgroups — #361256

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2C2×#1879C212yesmaximal
3#13916C211yesmaximal, characteristic, complemented
4D4×#95C211yesmaximal, characteristic, complemented
5#11379C212yesmaximal, complemented
6#11189C212yesmaximal, complemented
7#10820C211yesmaximal, characteristic, complemented
8#10797C211yesmaximal, characteristic, complemented
9#10788C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal, complemented
11#10788C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×C4×D8—28no—
17C2×C4×D8C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#390C2212yes—
25C2×#381—21no—
26C22×#95—21no—
27#1879—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1440—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—21no—
41C22⋊#95—21no—
42C22⋊#95—22no—
43C4×#95C2211yescharacteristic
44C23×D4C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—22no—
47C24×C4—21no—
48C24×C4—21no—
49C22×D8—22no—
50C22×D8—24no—
51C22×D8—22no—
52C22×D8C2×C414yes—
53C22×D8D418yes—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—42no—
66C2×C4×D4—42no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C2×C4×D4C2×C414yes—
70C2×C4×D4C2×C414yescomplemented
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C4×D8—232no—
83C2×#107—24no—
84C2×#107C2×C414yescomplemented
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102C2×C414yes—
94C2×#102C2×C414yescomplemented
95C2×C4×C8—24no—
96C2×C4×C8C2×C414yescomplemented
97#390—24no—
98#390—42no—
99#381—44no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—44no—
109C2×#95D411yescharacteristic
110C2×#95D412yes—
111C2×#95D411yescharacteristic
112#340—24no—
113#340—24no—
114#340—42no—
115#340—42no—
116#336—24no—
117#336—22no—
118#336—42no—
119C25—21no—
120C25—21no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C22×C412yes—
135C22×D4C22×C412yes—
136C22×D4C2×D414yes—
137C22×D4C2×D414yes—
138C22×D4#4112yes—
139C22×D4#4114yes—
140C22×D4#4112yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—28no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×D8—28no—
168C2×D8—28no—
169C2×D8—216no—
170C2×D8—216no—
171C2×D8C22⋊C4116yes—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8C22×C412yes—
187C22×C8C22×C412yes—
188C22×C8C2×D414yes—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—44no—
192C4×D4—44no—
193C4×D4—48no—
194C4×D4—48no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4—48no—
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—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
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×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42C22×C412yes—
241C2×C42#4112yes—
242#107—28no—
243#102—28no—
244#102—28no—
245#102—48no—
246#102—48no—
247C4×C8—28no—
248#95—22no—
249#95—42no—
250#95—44no—
251#95—42no—
252#95—44no—
253#95—42no—
254#95—81no—
255#95—81no—
256#95—82no—
257#95—82no—
258#95D814yescomplemented
259C24—22no—
260C24—24no—
261C24—24no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—24no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24—42no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24C22×D411yescharacteristic
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4C2×C22⋊C414yes—
313C2×D4C2×C22⋊C414yes—
314C2×D4#11714yes—
315C2×D4#11714yes—
316C2×D4#12318yes—
317C2×D4#12318yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—416no—
343C22×C4—416no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
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—48no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C4×D412yes—
379C22×C4C4×D412yes—
380C22×C4C2×D814yes—
381C22×C4C2×D812yes—
382C22×C4#13512yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C42−1+411yescharacteristic
393D8—232no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8C2×C22⋊C414yes—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
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—48no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C42—22no—
431C42—22no—
432C42—24no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C23—24no—
438C23—24no—
439C23—28no—
440C23—28no—
441C23—24no—
442C23—24no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—42no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—82no—
474C23—82no—
475C23—81no—
476C23—81no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—81no—
481C23—81no—
482C23—82no—
483C23C2×C4×D411yescharacteristic
484C23#52512yes—
485C23D4212yes—
486C23#54611yescharacteristic
487C23#54811yescharacteristic
488C23C22×D811yescharacteristic
489C23C2×#13511yescharacteristic
490C23#57512yes—
491C23#57612yes—
492C23#57711yescharacteristic
493C23#57711yescharacteristic
494D4—24no—
495D4—24no—
496D4—24no—
497D4—24no—
498D4—28no—
499D4—28no—
500D4—24no—
501D4—42no—
502D4—42no—
503D4—44no—
504D4—44no—
505D4#38118yescomplemented
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—816no—
539C2×C4—816no—
540C2×C4—832no—
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×C4C4×D814yes—
557C2×C4C22×C22⋊C411yescharacteristic
558C2×C4C2×#11711yescharacteristic
559C2×C4C2×#11711yescharacteristic
560C2×C4C4×#4111yescharacteristic
561C2×C4#51811yescharacteristic
562C2×C4#52011yescharacteristic
563C2×C4C2×#12312yes—
564C2×C4C2×#12312yes—
565C2×C4#52912yes—
566C2×C4#53812yes—
567C2×C4#53912yes—
568C2×C4#54612yes—
569C2×C4#54612yes—
570C2×C4#54712yes—
571C2×C4#55112yes—
572C2×C4#55211yescharacteristic
573C2×C4#55312yes—
574C2×C4#55311yescharacteristic
575C8—24no—
576C8—42no—
577C8—82no—
578C22—28no—
579C22—48no—
580C22—48no—
581C22—48no—
582C22—48no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22#229011yescharacteristic
601C22D4×C22⋊C411yescharacteristic
602C22#235211yescharacteristic
603C22#241812yes—
604C22#244312yes—
605C22C2×C4×D811yescharacteristic
606C22#294311yescharacteristic
607C22#294811yescharacteristic
608C22#295211yescharacteristic
609C22#310012yes—
610C22#310212yes—
611C22#311812yes—
612C22#311912yes—
613C22#326712yes—
614C22#326712yes—
615C22D4×D812yes—
616C22#328012yes—
617C22#329411yescharacteristic
618C22#330311yescharacteristic
619C22#332212yes—
620C22#335811yescharacteristic
621C22#336011yescharacteristic
622C22#337012yes—
623C4—24no—
624C4—44no—
625C4—42no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—162no—
631C4C2×#38111yescharacteristic
632C4#231811yescharacteristic
633C4#233911yescharacteristic
634C4#235311yescharacteristic
635C4#241912yes—
636C4#244412yes—
637C2—48no—
638C2—84no—
639C2#1389411yescharacteristic
640C2#2111311yescharacteristic
641C2#2113411yescharacteristic
642C2#2243811yescharacteristic
643C2C22⋊C4×D811yescharacteristic
644C2#2276411yescharacteristic
645C2#2276611yescharacteristic
646C2#2489312yes—
647C2#2489512yes—
648C2#2615812yes—
649C2#2616012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13916C2semidirect
4. D4×#95C2semidirect
5. #11379C2semidirect
6. #11189C2semidirect
7. #10820C2semidirect
8. #10797C2semidirect
9. #10788C2semidirect
10. #10788C2semidirect
11. #10788C2semidirect
17. C2×C4×D8C4semidirect
70. C2×C4×D4C2×C4semidirect
84. C2×#107C2×C4semidirect
94. C2×#102C2×C4semidirect
96. C2×C4×C8C2×C4semidirect
258. #95D8semidirect
505. D4#381semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#18792
#139162
D4×#952
#113792
#111892
#108202
#107972
#107882