Finite Group Databasebeta

← #454548

Subgroups — #454548

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#2452C211yesmaximal, characteristic
3C2×#2058C212yesmaximal, complemented
4C2×#2058C211yesmaximal, characteristic, complemented
5#15128C211yesmaximal, characteristic, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11202C211yesmaximal, characteristic, complemented
9#11121C212yesmaximal
10#11066C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C22×C4×D4C2211yescharacteristic
16C2×#476—28no—
17C2×#476—24no—
18C22×#106C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23#2452—24no—
24#2452—24no—
25C2×#396—21no—
26C2×#395—21no—
27C2×#393—21no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#385—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#2058—28no—
36#2058—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8—42no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—44no—
65C2×C22⋊Q8—44no—
66C2×C4×D4—24no—
67C2×C4×D4—22no—
68C2×C4×D4—42no—
69C2×C4×D4D414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#476—232no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—42no—
96C2×#106D418yes—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—22no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—42no—
103C2×#102—42no—
104C2×#102Q818yescomplemented
105#396—28no—
106#395—44no—
107#393—28no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—24no—
116#389—42no—
117#389D414yes—
118#385—28no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130#340—24no—
131#340—24no—
132#340—24no—
133#340—42no—
134#336—24no—
135#336—22no—
136#336—42no—
137C25—21no—
138C22×Q8—24no—
139C22×Q8—44no—
140C22×Q8—44no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—82no—
147C22×Q8—82no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4C2×D412yes—
157C22×D4C2×Q814yes—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—82no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C2×D414yes—
188C22×C8C2×Q814yes—
189C4⋊Q8—28no—
190C4⋊Q8—48no—
191C4⋊Q8—44no—
192C4⋊Q8—48no—
193C4⋊Q8—44no—
194C4⋊Q8—44no—
195C22⋊Q8—432no—
196C4×D4—28no—
197C4×D4—44no—
198C4×D4—48no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
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—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4SD1614yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×Q814yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—82no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271#106—216no—
272#106—48no—
273C4⋊C8—28no—
274#102—216no—
275#102—48no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—22no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—42no—
284#95—82no—
285C24—22no—
286C24—24no—
287C24—41no—
288C24—41no—
289C24—42no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—42no—
295C24—42no—
296C24C22×D411yescharacteristic
297C2×Q8—48no—
298C2×Q8—48no—
299C2×Q8—48no—
300C2×Q8—88no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—88no—
305C2×Q8—88no—
306C2×Q8—88no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—42no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4C4⋊Q814yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—48no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—48no—
383C22×C4—84no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4#12012yes—
387C22×C4#12012yes—
388C22×C4C2×SD1612yes—
389C22×C4C2×SD1612yes—
390C22×C4C2×SD1612yes—
391C22×C4AGL(1,ℤ/8)12yes—
392C22×C4AGL(1,ℤ/8)12yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×Q812yes—
397C22×C4H(5,𝔽2)12yes—
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C42−1+411yescharacteristic
400C22×C42−1+412yes—
401C22×C42−1+411yescharacteristic
402C22×C42−1+411yescharacteristic
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—44no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—48no—
440C4⋊C4—88no—
441C4⋊C4#12014yes—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—84no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—42no—
457C23—24no—
458C23—44no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—81no—
471C23—81no—
472C23—82no—
473C23—81no—
474C23—81no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23C2×#12011yescharacteristic
481C23#53611yescharacteristic
482C23#53711yescharacteristic
483C23D4212yes—
484C23D4×Q812yes—
485C23C22×SD1611yescharacteristic
486C23C2×AGL(1,ℤ/8)11yescharacteristic
487C23C2×AGL(1,ℤ/8)11yescharacteristic
488C23#57511yescharacteristic
489C23#57712yes—
490C23#57712yes—
491Q8—816no—
492Q8—816no—
493D4—24no—
494D4—24no—
495D4—42no—
496D4—42no—
497D4—44no—
498D4—44no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—816no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4#45014yes—
563C2×C4C2×#12011yescharacteristic
564C2×C4C2×C4⋊Q811yescharacteristic
565C2×C4#53411yescharacteristic
566C2×C4#53611yescharacteristic
567C2×C4#53611yescharacteristic
568C2×C4#53612yes—
569C2×C4#53611yescharacteristic
570C2×C4#53711yescharacteristic
571C2×C4#53712yes—
572C2×C4#53711yescharacteristic
573C2×C4#54312yes—
574C2×C4#54412yes—
575C2×C4D4212yes—
576C2×C4D4×Q812yes—
577C2×C4D4×Q812yes—
578C2×C4D4×Q812yes—
579C2×C4#56312yes—
580C2×C4#56312yes—
581C8—44no—
582C8—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—82no—
587C22—84no—
588C22—84no—
589C22—82no—
590C22—82no—
591C22—84no—
592C22—82no—
593C22—84no—
594C22#240311yescharacteristic
595C22#243411yescharacteristic
596C22#261611yescharacteristic
597C22#267712yes—
598C22#268312yes—
599C22C2×#45011yescharacteristic
600C22#300611yescharacteristic
601C22#300911yescharacteristic
602C22#301011yescharacteristic
603C22#322711yescharacteristic
604C22#323511yescharacteristic
605C22#324412yes—
606C22#324611yescharacteristic
607C22#325311yescharacteristic
608C22#326312yes—
609C22#326512yes—
610C22#326512yes—
611C22#327512yes—
612C22D4×SD1612yes—
613C22Q8×SD1612yes—
614C22#338412yes—
615C22#338412yes—
616C22#338812yes—
617C4—24no—
618C4—42no—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—81no—
624C4—84no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4#243411yescharacteristic
629C4#261611yescharacteristic
630C4#265911yescharacteristic
631C4#267712yes—
632C4#268312yes—
633C4#281711yescharacteristic
634C2—84no—
635C2#1625411yescharacteristic
636C2#2384911yescharacteristic
637C2#2386611yescharacteristic
638C2#2583411yescharacteristic
639C2#2585111yescharacteristic
640C2#2969011yescharacteristic
641C2#2969711yescharacteristic
642C2#3095412yes—
643C2#3097412yes—
644C2#3124012yes—
645C2#3124712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2058C2semidirect
4. C2×#2058C2semidirect
5. #15128C2semidirect
6. #11227C2semidirect
7. #11220C2semidirect
8. #11202C2semidirect
104. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#24522
C2×#20582
#151282
#112272
#112202
#112022
#111212
#110662
#107882