Finite Group Databasebeta

← #397970

Subgroups — #397970

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#1864C211yesmaximal, characteristic
3C2×#1802C212yesmaximal
4#15143C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11323C212yesmaximal, complemented
8#11322C211yesmaximal, characteristic, complemented
9#11231C212yesmaximal, complemented
10#11228C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2212yes—
15C2×#476—28no—
16C2×#476—24no—
17C22×#106C2212yes—
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C2×#395—22no—
25C2×#389C2211yescharacteristic
26C2×#384C2212yes—
27C22×#95—21no—
28#1864—24no—
29#1802—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1444—21no—
38#1444—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C22⋊#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4D414yes—
65C2×C4×D4Q818yescomplemented
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79#476—232no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—22no—
84C2×#106—42no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—42no—
90C2×#102—22no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—22no—
94C2×#102—24no—
95C2×#102—22no—
96C2×#102D418yes—
97C2×#102Q814yescomplemented
98#395—48no—
99#389—22no—
100#389—22no—
101#384—24no—
102#384—42no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—44no—
111C2×#95D412yes—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340—22no—
117#340—42no—
118#340—42no—
119#336—22no—
120#336—24no—
121C25—21no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—42no—
125C22×Q8—81no—
126C22×Q8—81no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C2×D414yes—
133C22×D4C2×D412yes—
134C22×D4C2×Q814yes—
135C22×D4C2×Q812yes—
136C22×D4#4112yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—28no—
143C23×C4—21no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2411yescharacteristic
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D414yes—
175C22×C8C2×Q812yes—
176C4⋊Q8—28no—
177C4⋊Q8—48no—
178C4⋊Q8—44no—
179C4×D4—28no—
180C4×D4—216no—
181C4×D4—44no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4—48no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2×Q814yes—
221C2×C4⋊C4C2×Q812yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C2×D412yes—
234C2×C42C2×Q814yes—
235C2×C42#4112yes—
236C2×C42#4114yes—
237#106—216no—
238#106—44no—
239C4⋊C8—28no—
240C4⋊C8—48no—
241#102—216no—
242#102—28no—
243#102—44no—
244#95—24no—
245#95—22no—
246#95—44no—
247#95—42no—
248#95—42no—
249#95—44no—
250#95—44no—
251#95—81no—
252#95—82no—
253#95SD1614yes—
254C24—22no—
255C24—24no—
256C24—24no—
257C24—22no—
258C24—22no—
259C24—41no—
260C24—42no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—41no—
265C24C22×D411yescharacteristic
266C2×Q8—48no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4#12014yes—
285C2×D4C22⋊Q814yes—
286C2×D4C22⋊Q818yes—
287C2×D4C4⋊Q814yes—
288C2×D4C4⋊Q818yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—48no—
310C22×C4—416no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—48no—
332C22×C4—416no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—88no—
339C22×C4—84no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q814yes—
350C22×C4C2×SD1612yes—
351C22×C4C2×SD1614yes—
352C22×C4AGL(1,ℤ/8)14yes—
353C22×C4#13712yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C22×Q812yes—
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—42no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—88no—
401C4⋊C4—84no—
402C4⋊C4—88no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C42—22no—
406C42—24no—
407C42—22no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—28no—
416C23—28no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—44no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23C2×#12111yescharacteristic
436C23C2×C22⋊Q812yes—
437C23D4212yes—
438C23#54611yescharacteristic
439C23D4×Q811yescharacteristic
440C23C22×SD1611yescharacteristic
441C23C2×AGL(1,ℤ/8)12yes—
442C23C2×#13711yescharacteristic
443C23#57611yescharacteristic
444C23#57711yescharacteristic
445C23#57712yes—
446Q8—816no—
447D4—24no—
448D4—24no—
449D4—24no—
450D4—28no—
451D4—28no—
452D4#39518yescomplemented
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—816no—
487C2×C4—832no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4#46114yes—
507C2×C4#47618yescomplemented
508C2×C4C2×#12011yescharacteristic
509C2×C4C2×C22⋊Q811yescharacteristic
510C2×C4C2×C22⋊Q812yes—
511C2×C4C2×C4⋊Q811yescharacteristic
512C2×C4C2×C4⋊Q812yes—
513C2×C4#53211yescharacteristic
514C2×C4#53312yes—
515C2×C4#53411yescharacteristic
516C2×C4#53711yescharacteristic
517C2×C4#54312yes—
518C2×C4#54412yes—
519C2×C4#54612yes—
520C2×C4#54612yes—
521C2×C4#54712yes—
522C2×C4#55112yes—
523C2×C4#55211yescharacteristic
524C2×C4#55412yes—
525C2×C4#55411yescharacteristic
526C8—42no—
527C8—44no—
528C8—81no—
529C22—28no—
530C22—44no—
531C22—44no—
532C22—48no—
533C22—48no—
534C22—44no—
535C22—44no—
536C22—48no—
537C22—48no—
538C22—44no—
539C22—44no—
540C22#241011yescharacteristic
541C22#241711yescharacteristic
542C22#243712yes—
543C22#244012yes—
544C22#285711yescharacteristic
545C22C2×#46111yescharacteristic
546C22#303811yescharacteristic
547C22#304211yescharacteristic
548C22#304511yescharacteristic
549C22C2×#47612yes—
550C22#307912yes—
551C22#308112yes—
552C22#308412yes—
553C22#326412yes—
554C22#326612yes—
555C22#327212yes—
556C22D4×SD1612yes—
557C22#329511yescharacteristic
558C22#330612yes—
559C22#333111yescharacteristic
560C22Q8×SD1611yescharacteristic
561C22#338412yes—
562C22#338511yescharacteristic
563C4—24no—
564C4—28no—
565C4—42no—
566C4—44no—
567C4—82no—
568C4—84no—
569C4—84no—
570C4—82no—
571C4—161no—
572C4C2×#39511yescharacteristic
573C4#243011yescharacteristic
574C4#243812yes—
575C4#254912yes—
576C4#256411yescharacteristic
577C4#265911yescharacteristic
578C2—48no—
579C2#1475811yescharacteristic
580C2#2435811yescharacteristic
581C2#2437211yescharacteristic
582C2#2481511yescharacteristic
583C2#2482911yescharacteristic
584C2#2595812yes—
585C2#2597812yes—
586C2#2604412yes—
587C2#2605812yes—
588C2#3319711yescharacteristic
589C2#3320511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #15143C2semidirect
6. #11384C2semidirect
7. #11323C2semidirect
8. #11322C2semidirect
9. #11231C2semidirect
10. #11228C2semidirect
65. C2×C4×D4Q8semidirect
97. C2×#102Q8semidirect
452. D4#395semidirect
507. C2×C4#476semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#18642
C2×#18022
#151432
D4×#952
#113842
#113232
#113222
#112312
#112282
#107882