Finite Group Databasebeta

← #406675

Subgroups — #406675

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×#1935C212yesmaximal
3#14703C211yesmaximal, characteristic
4#11292C212yesmaximal
5#11207C211yesmaximal, characteristic
6#11207C211yesmaximal, characteristic
7#11067C212yesmaximal
8#11050C211yesmaximal, characteristic
9#11050C211yesmaximal, characteristic
10#10881C212yesmaximal
11#10693C211yesmaximal, characteristic
12C22×#125—21no—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#97C2212yes—
16#2859—28no—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389—21no—
21C2×#389—21no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382—21no—
28C2×C43C2211yescharacteristic
29#1935—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1436—21no—
40#1436—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434—21no—
44#1434—21no—
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#125—28no—
49C2×#125—44no—
50C2×#125—44no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#97—24no—
77C2×#97—24no—
78C2×#97—42no—
79#391—22no—
80#391—22no—
81#391—24no—
82#391—22no—
83#391—22no—
84#389—28no—
85#389—28no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—22no—
90#389—22no—
91#389D418yes—
92#384—24no—
93#384—24no—
94#382—28no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C43—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—42no—
115#336—24no—
116#336—22no—
117#336—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8#4112yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8#4114yes—
154C22×C8#4114yes—
155#125—432no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4#4112yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42C2×D414yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227#103—216no—
228#103—48no—
229#97—48no—
230#95—28no—
231#95—24no—
232#95—24no—
233#95—28no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—42no—
243C24C22×D411yescharacteristic
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
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#12012yes—
335C22×C4#12012yes—
336C22×C4#12412yes—
337C22×C4#12412yes—
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4AGL(1,ℤ/8)12yes—
340C22×C4#13712yes—
341C22×C4#13712yes—
342C22×C4#13712yes—
343C22×C4#13712yes—
344C22×C4C22×D412yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C42−1+412yes—
353C22×C42−1+411yescharacteristic
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—416no—
379C4⋊C4—416no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—84no—
391C42—28no—
392C42—28no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—44no—
398C42—44no—
399C42—42no—
400C42—42no—
401C23C2×#12011yescharacteristic
402C23#52511yescharacteristic
403C23C2×#12411yescharacteristic
404C23#53211yescharacteristic
405C23#53211yescharacteristic
406C23#53611yescharacteristic
407C23#53711yescharacteristic
408C23C2×AGL(1,ℤ/8)11yescharacteristic
409C23C2×AGL(1,ℤ/8)11yescharacteristic
410C23C2×#13711yescharacteristic
411C23C2×#13711yescharacteristic
412C23C2×#13711yescharacteristic
413C23C2×#13711yescharacteristic
414C23#57511yescharacteristic
415C23#57511yescharacteristic
416Q8—44no—
417Q8—44no—
418Q8—82no—
419Q8—82no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4#48914yes—
503C2×C4#49114yes—
504C2×C4#52512yes—
505C2×C4#52912yes—
506C2×C4#52912yes—
507C2×C4#53212yes—
508C2×C4#53912yes—
509C2×C4#54112yes—
510C2×C4#54612yes—
511C2×C4#54712yes—
512C2×C4#54812yes—
513C2×C4#54812yes—
514C2×C4#55212yes—
515C2×C4#55211yescharacteristic
516C2×C4#55211yescharacteristic
517C2×C4#55212yes—
518C2×C4#55311yescharacteristic
519C2×C4#55311yescharacteristic
520C8—44no—
521C8—44no—
522C22#238811yescharacteristic
523C22#243011yescharacteristic
524C22#243611yescharacteristic
525C22#256211yescharacteristic
526C22#261111yescharacteristic
527C22#266411yescharacteristic
528C22#286111yescharacteristic
529C22#300311yescharacteristic
530C22#300411yescharacteristic
531C22#300711yescharacteristic
532C22#300711yescharacteristic
533C22#311211yescharacteristic
534C22#311311yescharacteristic
535C22#311511yescharacteristic
536C22#311611yescharacteristic
537C22C2×#48911yescharacteristic
538C22C2×#49111yescharacteristic
539C22#314111yescharacteristic
540C22#314211yescharacteristic
541C22#317111yescharacteristic
542C22#317211yescharacteristic
543C22#318011yescharacteristic
544C22#318111yescharacteristic
545C22#318211yescharacteristic
546C22#318211yescharacteristic
547C22#318511yescharacteristic
548C22#318511yescharacteristic
549C22#323611yescharacteristic
550C22#323611yescharacteristic
551C22#323811yescharacteristic
552C22#323911yescharacteristic
553C22#325111yescharacteristic
554C22#325211yescharacteristic
555C22#325411yescharacteristic
556C22#325511yescharacteristic
557C4—24no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—84no—
564C4—82no—
565C4—84no—
566C4—82no—
567C4—161no—
568C4—161no—
569C4#246012yes—
570C4#247512yes—
571C4#251912yes—
572C4#255812yes—
573C2#1493311yescharacteristic
574C2#2357611yescharacteristic
575C2#2357711yescharacteristic
576C2#2550111yescharacteristic
577C2#2550411yescharacteristic
578C2#2590811yescharacteristic
579C2#2591211yescharacteristic
580C2#2913111yescharacteristic
581C2#2913211yescharacteristic
582C2#2957911yescharacteristic
583C2#2958111yescharacteristic
584C2#3038211yescharacteristic
585C2#3038511yescharacteristic
586C2#3328211yescharacteristic
587C2#3328711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#19352
#147032
#112922
#112072
#110672
#110502
#108812
#106932