Finite Group Databasebeta

← #404978

Subgroups — #404978

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic
2C2×#2087C211yesmaximal, characteristic, complemented
3C2×#2076C211yesmaximal, characteristic
4C2×#1942C211yesmaximal, characteristic, complemented
5#14697C212yesmaximal
6#11232C211yesmaximal, characteristic, complemented
7#11228C212yesmaximal, complemented
8#11200C212yesmaximal
9#11096C211yesmaximal, characteristic
10#10880C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C2×#431—24no—
14C2×#431—24no—
15C2×#431—24no—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×#380C2212yes—
28C2×#380C2212yes—
29#2087—24no—
30#2076—24no—
31#1942—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#431—232no—
71C2×#107—22no—
72C2×#107—24no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
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×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8D414yes—
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—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4Q1614yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
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—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4#4114yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C42—28no—
209C2×C42—28no—
210C2×C42—28no—
211C2×C42—28no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#107—28no—
231#107—44no—
232C4⋊C8—28no—
233C4⋊C8—28no—
234C22⋊C8—28no—
235C22⋊C8—28no—
236C4×C8—28no—
237C4×C8—28no—
238#95—28no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24#12112yes—
255C24C2×Q1612yes—
256C24#13512yes—
257C24C22×D411yescharacteristic
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
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—22no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—44no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
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—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4C2×Q1612yes—
341C22×C4C2×Q1612yes—
342C22×C4#13512yes—
343C22×C4#13512yes—
344C22×C4AGL(1,ℤ/8)14yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C42−1+412yes—
354C22×C42−1+412yes—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8#12014yes—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
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—42no—
390C4⋊C4—42no—
391C4⋊C4—88no—
392C4⋊C4—88no—
393C22⋊C4—416no—
394C22⋊C4—88no—
395C42—28no—
396C42—28no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C23—22no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23#46714yes—
428C23C2×#12111yescharacteristic
429C23C2×#12111yescharacteristic
430C23C2×#12111yescharacteristic
431C23#52512yes—
432C23#53612yes—
433C23C22×Q1611yescharacteristic
434C23C2×#13511yescharacteristic
435C23C2×#13511yescharacteristic
436C23C2×#13511yescharacteristic
437C23C2×AGL(1,ℤ/8)12yes—
438C23#57512yes—
439C2×C4—232no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—416no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
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—82no—
488C2×C4—82no—
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#46714yes—
502C2×C4#46714yes—
503C2×C4#46914yes—
504C2×C4C2×#12011yescharacteristic
505C2×C4#52511yescharacteristic
506C2×C4#52511yescharacteristic
507C2×C4#52511yescharacteristic
508C2×C4#52511yescharacteristic
509C2×C4#52912yes—
510C2×C4#53211yescharacteristic
511C2×C4#53611yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#54112yes—
514C2×C4#54712yes—
515C2×C4#54712yes—
516C2×C4#54712yes—
517C2×C4#54812yes—
518C2×C4#55212yes—
519C2×C4#55212yes—
520C2×C4#55212yes—
521C2×C4#55612yes—
522C8—24no—
523C8—42no—
524C8—42no—
525C8—42no—
526C22—28no—
527C22—24no—
528C22—44no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22#239211yescharacteristic
538C22#241312yes—
539C22#241512yes—
540C22#262111yescharacteristic
541C22#284711yescharacteristic
542C22C2×#46711yescharacteristic
543C22C2×#46711yescharacteristic
544C22C2×#46711yescharacteristic
545C22#305011yescharacteristic
546C22#305011yescharacteristic
547C22#305011yescharacteristic
548C22C2×#46911yescharacteristic
549C22#305411yescharacteristic
550C22#305512yes—
551C22#305512yes—
552C22#310212yes—
553C22#310412yes—
554C22#310512yes—
555C22#310612yes—
556C22#322812yes—
557C22#323012yes—
558C22#323212yes—
559C22#323312yes—
560C4—24no—
561C4—24no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—84no—
566C4—84no—
567C4—162no—
568C4—162no—
569C4#240611yescharacteristic
570C4#243211yescharacteristic
571C4#247512yes—
572C4#247812yes—
573C4#259911yescharacteristic
574C4#261111yescharacteristic
575C2—44no—
576C2#1490711yescharacteristic
577C2#2369711yescharacteristic
578C2#2369811yescharacteristic
579C2#2460212yes—
580C2#2460512yes—
581C2#2470212yes—
582C2#2470412yes—
583C2#2987811yescharacteristic
584C2#2987911yescharacteristic
585C2#3293111yescharacteristic
586C2#3293611yescharacteristic
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×#1942C2semidirect
6. #11232C2semidirect
7. #11228C2semidirect
10. #10880C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
C2×#20872
C2×#20762
C2×#19422
#146972
#112322
#112282
#112002
#110962
#108802
#108202