Finite Group Databasebeta

← #469409

Subgroups — #469409

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11267C212yesmaximal, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11200C212yesmaximal
9#11095C212yesmaximal
10#11042C211yesmaximal, characteristic
11#10806C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#126—21no—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17#2496—24no—
18#2496—24no—
19C2×#400—21no—
20C2×#397—21no—
21C2×#393—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387—21no—
25C2×#384C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2058—24no—
32#1987—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×#126—28no—
56C2×#126—44no—
57C2×#126—44no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—42no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—42no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85#400—28no—
86#397—28no—
87#393—44no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389D414yes—
92#387—28no—
93#384—22no—
94#384—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—24no—
100#380—42no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#340—42no—
124#336—24no—
125#336—22no—
126C25C2×D411yescharacteristic
127C22×Q8—44no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8#4114yes—
168C4⋊Q8—28no—
169C4⋊Q8—48no—
170C4⋊Q8—44no—
171#126—432no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
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—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4SD1614yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#106—48no—
245C4⋊C8—44no—
246C22⋊C8—48no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—44no—
255#95—42no—
256#95—42no—
257#95—44no—
258#95—82no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24C2×SD1612yes—
270C24C22×D411yescharacteristic
271C2×Q8—48no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8—88no—
275C2×Q8—88no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4C2×SD1612yes—
354C22×C4C2×SD1612yes—
355C22×C4AGL(1,ℤ/8)12yes—
356C22×C4AGL(1,ℤ/8)12yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+411yescharacteristic
366C22×C42−1+412yes—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—48no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C22⋊C4—28no—
409C22⋊C4—44no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—88no—
413C42—22no—
414C42—22no—
415C42—24no—
416C42—41no—
417C42—41no—
418C42—42no—
419C42—48no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—42no—
435C23—41no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23C2×#12011yescharacteristic
444C23#53611yescharacteristic
445C23#53711yescharacteristic
446C23#54612yes—
447C23#54712yes—
448C23C22×SD1611yescharacteristic
449C23C2×AGL(1,ℤ/8)11yescharacteristic
450C23C2×AGL(1,ℤ/8)11yescharacteristic
451C23#57511yescharacteristic
452C23#57712yes—
453C23#57812yes—
454Q8—816no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#45014yes—
524C2×C4#52511yescharacteristic
525C2×C4#53211yescharacteristic
526C2×C4#53511yescharacteristic
527C2×C4#53611yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#53611yescharacteristic
530C2×C4#53711yescharacteristic
531C2×C4#53711yescharacteristic
532C2×C4#53812yes—
533C2×C4#54112yes—
534C2×C4#54212yes—
535C2×C4#54212yes—
536C2×C4#54612yes—
537C2×C4#54812yes—
538C2×C4#55212yes—
539C2×C4#55612yes—
540C2×C4#56312yes—
541C2×C4#56312yes—
542C8—44no—
543C8—82no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—44no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—82no—
554C22—82no—
555C22#240411yescharacteristic
556C22#243211yescharacteristic
557C22#261611yescharacteristic
558C22#269812yes—
559C22#271512yes—
560C22C2×#45011yescharacteristic
561C22#300611yescharacteristic
562C22#301011yescharacteristic
563C22#301111yescharacteristic
564C22#322711yescharacteristic
565C22#323511yescharacteristic
566C22#324212yes—
567C22#324611yescharacteristic
568C22#325311yescharacteristic
569C22#325812yes—
570C22#329812yes—
571C22#330812yes—
572C22#331112yes—
573C22#332412yes—
574C22#333512yes—
575C22#334312yes—
576C22#334912yes—
577C22#335412yes—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—82no—
584C4—82no—
585C4—84no—
586C4—82no—
587C4—82no—
588C4—162no—
589C4#260411yescharacteristic
590C4#263711yescharacteristic
591C4#266211yescharacteristic
592C4#270212yes—
593C4#271612yes—
594C4#282111yescharacteristic
595C2—82no—
596C2#1665611yescharacteristic
597C2#2396111yescharacteristic
598C2#2397911yescharacteristic
599C2#2569911yescharacteristic
600C2#2571611yescharacteristic
601C2#2969011yescharacteristic
602C2#2969911yescharacteristic
603C2#3168812yes—
604C2#3171312yes—
605C2#3219812yes—
606C2#3222412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2058C2semidirect
3. C2×#1987C2semidirect
5. #11267C2semidirect
6. #11227C2semidirect
7. #11220C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#20582
C2×#19872
#151252
#112672
#112272
#112202
#112002
#110952
#110422
#108062