Finite Group Databasebeta

← #393373

Subgroups — #393373

Order 512.

1707 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic, complemented
4#21329C211yesmaximal, characteristic, complemented
5#20646C211yesmaximal, characteristic
6#10903C214yesmaximal, complemented
7#10856C211yesmaximal, characteristic, complemented
8#10836C211yesmaximal, characteristic, complemented
9#10785C214yesmaximal
10C2×#533C2211yescharacteristic
11C2×#518C2212yes—
12C22×C4×D4C2211yescharacteristic
13#3077—22no—
14#3077—22no—
15#3077—22no—
16#3077—22no—
17C2×#478C414yescomplemented
18C2×#476C412yescomplemented
19C2×#474C412yescomplemented
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#418C2212yes—
24C2×#418C2212yes—
25#2285C2211yescharacteristic
26#2275—22no—
27#2275—22no—
28#2274C2212yes—
29#2273—22no—
30C42×D4—22no—
31C42×D4—22no—
32C4×#117C2211yescharacteristic
33C4×#117C2211yescharacteristic
34C2×C4×C22⋊C4—22no—
35#1996—22no—
36#1996—22no—
37#1995—22no—
38#1994—22no—
39#1804—22no—
40#1802—22no—
41C2×#344C2214yes—
42#1740C2214yes—
43#1736C2212yes—
44#1733C2214yes—
45C2×H(3,ℤ/4)C2214yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C22×M16C2312yes—
49C22×M16C2312yes—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58C2×C4⋊Q8C2×C411yescharacteristic, complemented
59C2×C4⋊Q8C2×C411yescharacteristic, complemented
60C2×#125C2×C412yescomplemented
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63#518—24no—
64#518—24no—
65#518—24no—
66#518Q818yescomplemented
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×D4—24no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—24no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C412yes—
79C2×C4×D4D412yes—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×C4×D4D412yes—
83C2×#117C2312yes—
84C2×#117C2312yes—
85C2×#117C2312yes—
86C2×#117C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—24no—
90C22×C42—24no—
91C22×C42C2311yescharacteristic
92#478—28no—
93#478—28no—
94#476—24no—
95#476—24no—
96#474—24no—
97#474—24no—
98#428—24no—
99#428—24no—
100C2×#107C2×C414yescomplemented
101C2×#106C2×C414yescomplemented
102#423—21no—
103#423—21no—
104#423—21no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423—21no—
109#423—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8C2×C412yescomplemented
113C2×C4⋊C8C2×C412yescomplemented
114#418—24no—
115#418—24no—
116#418—28no—
117C2×#102—24no—
118C2×#102C2×C414yes—
119C2×#102C2×C414yes—
120#384—21no—
121#382—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4D412yes—
140C4×C22⋊C4D412yes—
141#376—21no—
142#376D412yes—
143C2×#95—24no—
144C43—21no—
145C43D412yes—
146#344—28no—
147#342—28no—
148#340—24no—
149#339—28no—
150H(3,ℤ/4)—28no—
151C25—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4C22×C412yes—
160C22×D4C2×D412yes—
161C22×D4C2×D412yes—
162C22×D4C2×Q814yes—
163C22×D4#4114yes—
164C23×C4—28no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—84no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—24no—
179C2×M16—24no—
180C2×M16—22no—
181C2×M16—24no—
182C2×M16—24no—
183C2×M16—22no—
184C22×C8—22no—
185C22×C8C22×C414yes—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194#125—22no—
195#125—22no—
196#125—22no—
197#125—22no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×D4—216no—
215C4×D4—216no—
216C4×D4—216no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—216no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4C22⋊C414yes—
231C4×D4C22⋊C414yes—
232C4×D4C22⋊C414yes—
233C4×D4C22⋊C414yes—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—24no—
243#117—24no—
244#117C2×Q814yes—
245#117C2×Q814yes—
246#117#4114yes—
247#117#4114yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C2×D414yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—28no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—88no—
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D414yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42#4111yescharacteristic
297C2×C42#4111yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42#4111yescharacteristic
300#107—28no—
301#106—28no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—41no—
307C4⋊C8—41no—
308C4⋊C8—41no—
309C4⋊C8—41no—
310#102—28no—
311#102—28no—
312#102—48no—
313#95—28no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—41no—
319#95—41no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—82no—
326C24—82no—
327C24C22×D411yescharacteristic
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4C2×C22⋊C414yes—
348C2×D4C4×Q818yes—
349C2×D4C22⋊Q818yes—
350C2×D4#12414yes—
351C2×D4C4⋊Q814yes—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—232no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—48no—
366C22×C4—42no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—81no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—84no—
375C22×C4—82no—
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4#12011yescharacteristic
381C22×C4#12011yescharacteristic
382C22×C4#12011yescharacteristic
383C22×C4#12011yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4#12111yescharacteristic
387C22×C4#12111yescharacteristic
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×Q812yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4111yescharacteristic
398M16—44no—
399M16—42no—
400M16—42no—
401M16—44no—
402M16—44no—
403C2×C8—24no—
404C2×C8—28no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4C2×C22⋊C414yes—
432C22⋊C4—232no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4C22⋊Q818yes—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—44no—
448C42—44no—
449C42—42no—
450C42—42no—
451C42—44no—
452C42—82no—
453C42—82no—
454C42—88no—
455C42C2×C22⋊C411yescharacteristic
456C42C2×C22⋊C411yescharacteristic
457C42C2×C22⋊C411yescharacteristic
458C42C2×C22⋊C411yescharacteristic
459C42C4×D411yescharacteristic
460C42C4×D411yescharacteristic
461C42C4×D411yescharacteristic
462C42C4×D411yescharacteristic
463C42#12111yescharacteristic
464C42#12111yescharacteristic
465C42#12111yescharacteristic
466C42#12111yescharacteristic
467C42C22⋊Q818yes—
468C42#12311yescharacteristic
469C42#12311yescharacteristic
470C42#12311yescharacteristic
471C42#12311yescharacteristic
472C23—28no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—41no—
478C23—41no—
479C23—44no—
480C23—44no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—88no—
486C23—88no—
487C23C2×C4×D411yescharacteristic
488C23C2×#12011yescharacteristic
489C23C2×#12111yescharacteristic
490C23C2×C22⋊Q812yes—
491C23#52512yes—
492Q8—82no—
493Q8—82no—
494Q8—82no—
495Q8—82no—
496D4—24no—
497D4—28no—
498D4—24no—
499D4—24no—
500D4—28no—
501D4#39118yes—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—832no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4#38611yescharacteristic
537C2×C4#38611yescharacteristic
538C2×C4#38611yescharacteristic
539C2×C4#38611yescharacteristic
540C2×C4#38611yescharacteristic
541C2×C4#38611yescharacteristic
542C2×C4#38611yescharacteristic
543C2×C4#38611yescharacteristic
544C2×C4C22×C22⋊C411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×Q812yes—
547C2×C4C4×#4112yes—
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×C22⋊Q812yes—
550C2×C4C2×C22⋊Q812yes—
551C2×C4C2×#12311yescharacteristic
552C2×C4#52512yes—
553C2×C4#52512yes—
554C2×C4C2×#12411yescharacteristic
555C2×C4#52912yes—
556C2×C4#52912yes—
557C2×C4C2×C4⋊Q811yescharacteristic
558C2×C4#53211yescharacteristic
559C2×C4#53211yescharacteristic
560C2×C4#53312yes—
561C2×C4#53312yes—
562C8—44no—
563C8—82no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—28no—
571C22—42no—
572C22—42no—
573C22—416no—
574C22—44no—
575C22—44no—
576C22—84no—
577C22—84no—
578C22—88no—
579C22C2×#38611yescharacteristic
580C22C4×C22⋊Q812yes—
581C22#284812yes—
582C22#285011yescharacteristic
583C22#285711yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—48no—
592C4—48no—
593C4—44no—
594C4—82no—
595C4—82no—
596C4—162no—
597C4#190412yes—
598C4C2×#39111yescharacteristic
599C4#229711yescharacteristic
600C4C4×C22⋊Q812yes—
601C4#285011yescharacteristic
602C4#285512yes—
603C4#285711yescharacteristic
604C2—22no—
605C2—21no—
606C2—21no—
607C2—22no—
608C2—48no—
609C2#1470211yescharacteristic
610C2C2×#190411yescharacteristic
611C2#2099811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21332C2semidirect
4. #21329C2semidirect
6. #10903C2semidirect
7. #10856C2semidirect
8. #10836C2semidirect
17. C2×#478C4semidirect
18. C2×#476C4semidirect
19. C2×#474C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
60. C2×#125C2×C4semidirect
66. #518Q8semidirect
100. C2×#107C2×C4semidirect
101. C2×#106C2×C4semidirect
112. C2×C4⋊C8C2×C4semidirect
113. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
C4×#5182
#213322
#213292
#206462
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