Finite Group Databasebeta

← #474847

Subgroups — #474847

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1941C211yesmaximal, characteristic, complemented
3#15181C211yesmaximal, characteristic
4#15129C211yesmaximal, characteristic
5#11228C211yesmaximal, characteristic, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#11199C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#11095C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#394—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26#1954—24no—
27#1941—24no—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36#1442—21no—
37#1441—21no—
38#1440—22no—
39#1439—22no—
40#1438—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×D4D411yescharacteristic
44C24×C4C2312yes—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—42no—
51C2×C4×D4—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D412yes—
66C22×C42C2311yescharacteristic
67C2×#106—22no—
68C2×#106—22no—
69C2×#106—22no—
70C2×#106—22no—
71C2×#106—42no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—42no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80C2×C4×C8—42no—
81C2×C4×C8D414yes—
82#394—44no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#380—24no—
88#380—24no—
89#380—42no—
90#380—42no—
91#380—42no—
92#380—42no—
93#380—42no—
94#380—42no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105#340—22no—
106#340—22no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#340—42no—
113C25C2×D412yes—
114C22×D4—24no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—42no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
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×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8#4114yes—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—48no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4#4112yes—
201C2×C4⋊C4#4112yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4—82no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C2×D412yes—
220C2×C42#4112yes—
221#106—28no—
222#106—44no—
223C4⋊C8—44no—
224C22⋊C8—48no—
225C22⋊C8—48no—
226C4×C8—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—81no—
236#95—81no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24#13514yes—
252C24C22×D411yescharacteristic
253C2×D4—28no—
254C2×D4—48no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—44no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4AGL(1,ℤ/8)12yes—
338C22×C4#13712yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C42−1+412yes—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8#12414yes—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—44no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C22⋊C4—48no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C42—22no—
397C42—22no—
398C42—24no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23C2×#12111yescharacteristic
435C23#53412yes—
436C23#54611yescharacteristic
437C23#54611yescharacteristic
438C23#54712yes—
439C23C2×#13512yes—
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13711yescharacteristic
442C23#57711yescharacteristic
443C23#57711yescharacteristic
444C23#57712yes—
445D4—416no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4#46814yes—
507C2×C4#52511yescharacteristic
508C2×C4C2×#12411yescharacteristic
509C2×C4#53411yescharacteristic
510C2×C4#53511yescharacteristic
511C2×C4#53511yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53812yes—
514C2×C4#54012yes—
515C2×C4#54012yes—
516C2×C4#54212yes—
517C2×C4#54212yes—
518C2×C4#55312yes—
519C2×C4#55312yes—
520C2×C4#55311yescharacteristic
521C2×C4#55311yescharacteristic
522C2×C4#55612yes—
523C2×C4#56212yes—
524C2×C4#56212yes—
525C8—24no—
526C8—42no—
527C8—82no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22#253511yescharacteristic
544C22#254711yescharacteristic
545C22#261711yescharacteristic
546C22#270112yes—
547C22#270112yes—
548C22#305011yescharacteristic
549C22C2×#46811yescharacteristic
550C22#305711yescharacteristic
551C22#306011yescharacteristic
552C22#318812yes—
553C22#319012yes—
554C22#320012yes—
555C22#320412yes—
556C22#329612yes—
557C22#330112yes—
558C22#332411yescharacteristic
559C22#332911yescharacteristic
560C22#333111yescharacteristic
561C22#333211yescharacteristic
562C22#333712yes—
563C22#333712yes—
564C22#335412yes—
565C22#335612yes—
566C4—24no—
567C4—42no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—162no—
572C4—161no—
573C4—162no—
574C4—161no—
575C4#259111yescharacteristic
576C4#263211yescharacteristic
577C4#264111yescharacteristic
578C4#280012yes—
579C4#280012yes—
580C4#281611yescharacteristic
581C2—84no—
582C2#1701111yescharacteristic
583C2#2814411yescharacteristic
584C2#2815011yescharacteristic
585C2#2870711yescharacteristic
586C2#2871311yescharacteristic
587C2#2973511yescharacteristic
588C2#2974111yescharacteristic
589C2#3177812yes—
590C2#3178112yes—
591C2#3178312yes—
592C2#3178412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1941C2semidirect
5. #11228C2semidirect
6. #11227C2semidirect
7. #11200C2semidirect
8. #11199C2semidirect
9. #11095C2semidirect
10. #11095C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#19412
#151812
#151292
#112282
#112272
#112002
#111992
#110952
#108202