Finite Group Databasebeta

← #470020

Subgroups — #470020

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic
2C2×#2071C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic, complemented
4#15196C211yesmaximal, characteristic, complemented
5#11327C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#11202C211yesmaximal, characteristic, complemented
8#11121C212yesmaximal
9#11076C211yesmaximal, characteristic
10#11066C212yesmaximal
11#10806C211yesmaximal, characteristic, complemented
12C22×#126—21no—
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17#2512—24no—
18#2512—24no—
19C2×#397—21no—
20C2×#396—21no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#388—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2071—24no—
32#1948—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1444—21no—
42#1444—21no—
43#1441—21no—
44#1441—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#126—28no—
51C2×#126—44no—
52C2×#126—44no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#106—24no—
77C2×#106—42no—
78C2×#106—42no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87#397—28no—
88#396—28no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390—22no—
93#390—22no—
94#390D414yes—
95#389—22no—
96#389—22no—
97#389—42no—
98#389—42no—
99#388—28no—
100#384—22no—
101#384—22no—
102#384—22no—
103#384—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4D414yes—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
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#336—24no—
123#336—22no—
124#336—42no—
125C25—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4C2×D412yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8#4114yes—
162#126—432no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4SD1614yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4#4114yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243#106—48no—
244C4⋊C8—28no—
245#102—48no—
246#102—48no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—42no—
252#95—44no—
253#95—42no—
254#95—82no—
255#95—82no—
256C24—22no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—42no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—42no—
266C24—42no—
267C24C22×D411yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
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#12012yes—
343C22×C4#12012yes—
344C22×C4C2×SD1612yes—
345C22×C4C2×SD1612yes—
346C22×C4C2×SD1612yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4#13712yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+412yes—
357C22×C42−1+412yes—
358C22×C42−1+411yescharacteristic
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C4⋊C4—28no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—84no—
404C22⋊C4—84no—
405C42—22no—
406C42—22no—
407C42—24no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—42no—
415C42—48no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—81no—
432C23—81no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—81no—
437C23—81no—
438C23—82no—
439C23C2×#12011yescharacteristic
440C23#53511yescharacteristic
441C23#53711yescharacteristic
442C23#54712yes—
443C23#54812yes—
444C23C22×SD1611yescharacteristic
445C23C2×AGL(1,ℤ/8)11yescharacteristic
446C23C2×#13711yescharacteristic
447C23#57511yescharacteristic
448C23#57612yes—
449C23#57712yes—
450D4—44no—
451D4—44no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#45014yes—
519C2×C4#52511yescharacteristic
520C2×C4#53211yescharacteristic
521C2×C4#53511yescharacteristic
522C2×C4#53511yescharacteristic
523C2×C4#53611yescharacteristic
524C2×C4#53611yescharacteristic
525C2×C4#53711yescharacteristic
526C2×C4#53711yescharacteristic
527C2×C4#53912yes—
528C2×C4#54112yes—
529C2×C4#54112yes—
530C2×C4#54212yes—
531C2×C4#54712yes—
532C2×C4#54712yes—
533C2×C4#55112yes—
534C2×C4#55312yes—
535C2×C4#56312yes—
536C2×C4#56312yes—
537C8—44no—
538C8—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—162no—
547C22—162no—
548C22—162no—
549C22—162no—
550C22#240611yescharacteristic
551C22#243211yescharacteristic
552C22#261511yescharacteristic
553C22#272312yes—
554C22#272512yes—
555C22C2×#45011yescharacteristic
556C22#300711yescharacteristic
557C22#300811yescharacteristic
558C22#301011yescharacteristic
559C22#320611yescharacteristic
560C22#321411yescharacteristic
561C22#322412yes—
562C22#324611yescharacteristic
563C22#325411yescharacteristic
564C22#325712yes—
565C22#333512yes—
566C22#334512yes—
567C22#334612yes—
568C22#335512yes—
569C22#336212yes—
570C22#336612yes—
571C22#336912yes—
572C22#337612yes—
573C4—24no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4—82no—
580C4—84no—
581C4—82no—
582C4—81no—
583C4—162no—
584C4#260311yescharacteristic
585C4#263711yescharacteristic
586C4#266411yescharacteristic
587C4#271812yes—
588C4#272212yes—
589C4#282111yescharacteristic
590C2—162no—
591C2#1667311yescharacteristic
592C2#2408911yescharacteristic
593C2#2410211yescharacteristic
594C2#2564911yescharacteristic
595C2#2566211yescharacteristic
596C2#2966311yescharacteristic
597C2#2966911yescharacteristic
598C2#3243012yes—
599C2#3245212yes—
600C2#3247412yes—
601C2#3248812yes—
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×#1948C2semidirect
4. #15196C2semidirect
5. #11327C2semidirect
6. #11267C2semidirect
7. #11202C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#20712
C2×#19482
#151962
#113272
#112672
#112022
#111212
#110762
#110662
#108062