Finite Group Databasebeta

← #404983

Subgroups — #404983

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C2×#2087C211yesmaximal, characteristic, complemented
3C2×#2075C211yesmaximal, characteristic
4C2×#1944C211yesmaximal, characteristic, complemented
5#14697C212yesmaximal
6#11231C211yesmaximal, characteristic, complemented
7#11228C212yesmaximal, complemented
8#11199C212yesmaximal
9#11095C211yesmaximal, characteristic
10#10881C212yesmaximal, complemented
11#10821C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C2×#432—24no—
14C2×#432—24no—
15C2×#432—24no—
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×#380C2212yes—
28C2×#380C2212yes—
29#2087—24no—
30#2075—24no—
31#1944—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40#1436—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—22no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42—22no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#432—232no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#106—22no—
75C2×#106—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8D414yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8D414yes—
88C2×#97—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97D414yes—
94#389—24no—
95#389—24no—
96#389—24no—
97#389—24no—
98#385—48no—
99#380—24no—
100#380—24no—
101#380—24no—
102#380—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111#340—22no—
112#340—24no—
113#340—22no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119C25C2×D411yescharacteristic
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
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—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
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—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4#4114yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C42—28no—
207C2×C42—28no—
208C2×C42—28no—
209C2×C42—28no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228#107—44no—
229#106—28no—
230C4⋊C8—28no—
231C4⋊C8—28no—
232C22⋊C8—28no—
233C22⋊C8—28no—
234#97—28no—
235#97—28no—
236#95—28no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24#12112yes—
252C24AGL(1,ℤ/8)14yes—
253C24C22×D411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
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—42no—
312C22×C4—42no—
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—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#13714yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C42−1+412yes—
344C22×C42−1+412yes—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8#12014yes—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—88no—
382C4⋊C4—88no—
383C22⋊C4—416no—
384C22⋊C4—88no—
385C42—28no—
386C42—28no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C23—22no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23#46914yes—
416C23C2×#12111yescharacteristic
417C23C2×#12111yescharacteristic
418C23C2×#12111yescharacteristic
419C23#52512yes—
420C23#53612yes—
421C23C2×AGL(1,ℤ/8)12yes—
422C23C2×#13712yes—
423C23#57512yes—
424C23#57512yes—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—416no—
438C2×C4—416no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
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#47014yes—
486C2×C4C2×#12011yescharacteristic
487C2×C4#52511yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4#52912yes—
492C2×C4#53211yescharacteristic
493C2×C4#53611yescharacteristic
494C2×C4#53611yescharacteristic
495C2×C4#54112yes—
496C2×C4#54712yes—
497C2×C4#54712yes—
498C2×C4#54712yes—
499C2×C4#54812yes—
500C2×C4#55212yes—
501C2×C4#55212yes—
502C2×C4#55212yes—
503C2×C4#55612yes—
504C8—24no—
505C8—42no—
506C8—42no—
507C8—42no—
508C22—28no—
509C22—24no—
510C22—44no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—44no—
517C22#239211yescharacteristic
518C22#241312yes—
519C22#241512yes—
520C22#262111yescharacteristic
521C22#284711yescharacteristic
522C22C2×#46911yescharacteristic
523C22C2×#47011yescharacteristic
524C22#305411yescharacteristic
525C22#305411yescharacteristic
526C22#305512yes—
527C22#305512yes—
528C22#305612yes—
529C22#305612yes—
530C22#311112yes—
531C22#311212yes—
532C22#311412yes—
533C22#311612yes—
534C22#323412yes—
535C22#323712yes—
536C22#323812yes—
537C22#324012yes—
538C4—24no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—84no—
544C4—84no—
545C4—162no—
546C4—162no—
547C4#240611yescharacteristic
548C4#243211yescharacteristic
549C4#247512yes—
550C4#247812yes—
551C4#259911yescharacteristic
552C4#261111yescharacteristic
553C2—44no—
554C2#1490711yescharacteristic
555C2#2371012yes—
556C2#2460812yes—
557C2#2461112yes—
558C2#2470612yes—
559C2#2471112yes—
560C2#2988312yes—
561C2#3293611yescharacteristic
562C2#3293711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2087C2semidirect
4. C2×#1944C2semidirect
6. #11231C2semidirect
7. #11228C2semidirect
10. #10881C2semidirect
11. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×#20872
C2×#20752
C2×#19442
#146972
#112312
#112282
#111992
#110952
#108812
#108212