Finite Group Databasebeta

← #398650

Subgroups — #398650

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1862C211yesmaximal, characteristic
2C2×#1802C212yesmaximal
3C2×#1802C211yesmaximal, characteristic
4#15145C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic, complemented
7#11322C211yesmaximal, characteristic, complemented
8#11267C212yesmaximal, complemented
9#11232C212yesmaximal, complemented
10#11227C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C2×#398—21no—
21C2×#398—21no—
22C2×#389C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2212yes—
25C22×#95—21no—
26#1862—24no—
27#1802—28no—
28#1802—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1444—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C22⋊#95—22no—
43C4×#95C2211yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4—22no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×C4×D4—42no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4D414yes—
59C2×C4×D4Q818yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2312yes—
74C2×#107—22no—
75C2×#107—42no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8—42no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102D414yes—
89#398—44no—
90#398—44no—
91#389—22no—
92#389—22no—
93#384—24no—
94#384—22no—
95#384—22no—
96#384—42no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—44no—
105C2×#95D411yescharacteristic
106C2×#95D412yes—
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—42no—
113#340—42no—
114#336—22no—
115#336—24no—
116C25—21no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4C2×D412yes—
123C22×D4C2×D412yes—
124C22×D4C2×Q814yes—
125C22×D4#4112yes—
126C22×D4#4114yes—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—28no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—42no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—44no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8C2×D412yes—
166C4×D4—216no—
167C4×D4—28no—
168C4×D4—44no—
169C4×D4—48no—
170C4×D4—48no—
171C4×D4—48no—
172C4×D4—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42C2×D412yes—
226C2×C42C2×Q814yes—
227C2×C42#4112yes—
228C2×C42#4114yes—
229#107—44no—
230C4⋊C8—48no—
231C4⋊C8—44no—
232#102—28no—
233#102—48no—
234#102—44no—
235#95—22no—
236#95—24no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—44no—
241#95—44no—
242#95—81no—
243#95—82no—
244#95D814yes—
245C24—22no—
246C24—24no—
247C24—22no—
248C24—24no—
249C24—22no—
250C24—41no—
251C24—41no—
252C24—42no—
253C24—41no—
254C24—42no—
255C24—41no—
256C24C22×D411yescharacteristic
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4#12114yes—
271C2×D4C22⋊Q818yes—
272C2×D4#12314yes—
273C2×D4#12518yes—
274C2×D4C4⋊Q814yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—48no—
295C22×C4—416no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—48no—
313C22×C4—416no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—88no—
327C22×C4—84no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4C22⋊Q814yes—
337C22×C4C22⋊Q814yes—
338C22×C4C2×D812yes—
339C22×C4C2×D814yes—
340C22×C4AGL(1,ℤ/8)12yes—
341C22×C4#13714yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×Q812yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4H(5,𝔽2)12yes—
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C42−1+411yescharacteristic
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—88no—
388C4⋊C4—84no—
389C4⋊C4—88no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C22⋊C4—48no—
393C22⋊C4—416no—
394C42—24no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—28no—
405C23—28no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—42no—
412C23—44no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23C2×#12111yescharacteristic
425C23C2×C22⋊Q812yes—
426C23D4211yescharacteristic
427C23#54612yes—
428C23#54811yescharacteristic
429C23C22×D811yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×#13712yes—
432C23#57611yescharacteristic
433C23#57711yescharacteristic
434C23#57712yes—
435D4—28no—
436D4—24no—
437D4—28no—
438D4—24no—
439D4—24no—
440D4#39818yescomplemented
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—832no—
473C2×C4—816no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#45914yes—
497C2×C4#47418yes—
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×C22⋊Q812yes—
500C2×C4C2×#12311yescharacteristic
501C2×C4C2×#12512yes—
502C2×C4C2×C4⋊Q811yescharacteristic
503C2×C4#53211yescharacteristic
504C2×C4#53312yes—
505C2×C4#53411yescharacteristic
506C2×C4#53711yescharacteristic
507C2×C4#54312yes—
508C2×C4#54412yes—
509C2×C4#54612yes—
510C2×C4#54711yescharacteristic
511C2×C4#55112yes—
512C2×C4#55112yes—
513C2×C4#55212yes—
514C2×C4#55311yescharacteristic
515C2×C4#55312yes—
516C8—42no—
517C8—81no—
518C8—82no—
519C22—28no—
520C22—48no—
521C22—48no—
522C22—48no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—48no—
530C22#241211yescharacteristic
531C22#241611yescharacteristic
532C22#243712yes—
533C22#244212yes—
534C22#285711yescharacteristic
535C22C2×#45911yescharacteristic
536C22#303911yescharacteristic
537C22#304211yescharacteristic
538C22#304511yescharacteristic
539C22C2×#47412yes—
540C22#307912yes—
541C22#308112yes—
542C22#308412yes—
543C22#326612yes—
544C22#327211yescharacteristic
545C22D4×D811yescharacteristic
546C22#329412yes—
547C22#330612yes—
548C22#331412yes—
549C22#333112yes—
550C22#335811yescharacteristic
551C22#336611yescharacteristic
552C22#337212yes—
553C4—28no—
554C4—24no—
555C4—44no—
556C4—42no—
557C4—84no—
558C4—82no—
559C4—82no—
560C4—162no—
561C4—161no—
562C4C2×#39811yescharacteristic
563C4#243812yes—
564C4#249811yescharacteristic
565C4#254311yescharacteristic
566C4#256912yes—
567C4#265911yescharacteristic
568C2—48no—
569C2#1476611yescharacteristic
570C2#2452611yescharacteristic
571C2#2454811yescharacteristic
572C2#2472611yescharacteristic
573C2#2474811yescharacteristic
574C2#2595012yes—
575C2#2597812yes—
576C2#2610612yes—
577C2#2612412yes—
578C2#3319511yescharacteristic
579C2#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. #15145C2semidirect
6. #11373C2semidirect
7. #11322C2semidirect
8. #11267C2semidirect
9. #11232C2semidirect
10. #11227C2semidirect
440. D4#398semidirect

Maximal subgroups

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

subgroupindex
C2×#18622
C2×#18022
#151452
D4×#952
#113732
#113222
#112672
#112322
#112272
#107882