Finite Group Databasebeta

← #192042

Subgroups — #192042

Order 512.

1305 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3#20834C212yesmaximal, complemented
4#20722C211yesmaximal, characteristic
5#8852C214yesmaximal, complemented
6#8851C214yesmaximal, complemented
7#7925C212yesmaximal, complemented
8C24×C8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2975—21no—
11#2974—24no—
12#2973—21no—
13#2973C2212yes—
14#2972C2212yes—
15D4×M16—22no—
16D4×M16—22no—
17#2920—21no—
18C2×#424—22no—
19C2×#407—21no—
20C2×#407C2212yes—
21C22×C22⋊C8—21no—
22C22×C22⋊C8C2211yescharacteristic
23#2280C2212yes—
24#1869—21no—
25#1869—21no—
26#1869C2212yes—
27#1869C2212yes—
28#1850—24no—
29#1850C2218yescomplemented
30#1819C412yes—
31#1799C412yes—
32#1799C412yes—
33#1781C412yes—
34C4×#99C2212yes—
35C4×#99C2212yes—
36#1729—22no—
37C2×#336—21no—
38#1282C2218yescomplemented
39C23×D4C2×C411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—21no—
43C22×M16—21no—
44C22×M16D412yes—
45C22×M16D412yes—
46C23×C8—21no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—21no—
50C23×C8—21no—
51C23×C8—44no—
52C23×C8C2312yes—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4Q811yescharacteristic
62C2×C4×D4Q811yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×#117—22no—
66C22×C4⋊C4Q811yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4Q811yescharacteristic
69C22×C42D411yescharacteristic
70#436—24no—
71#436—24no—
72#435—24no—
73#435—24no—
74#435—24no—
75#435—24no—
76C8×D4—24no—
77C8×D4—24no—
78#432—22no—
79#432—22no—
80#431—22no—
81#431—22no—
82#424—48no—
83#423—21no—
84#423—21no—
85C2×C4⋊C8—28no—
86C2×#99C2×C414yes—
87C2×#99C2×C414yes—
88#407—22no—
89#407—22no—
90#407—22no—
91#407—22no—
92#407—22no—
93#407—22no—
94#407—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—44no—
103C2×C22⋊C8D414yes—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2314yes—
106C4×M16—22no—
107#381—22no—
108#380—22no—
109#380C2×C414yes—
110#379C2×C414yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4C2314yes—
113#342C2×C414yes—
114#342C2×C414yes—
115#336—28no—
116#336—22no—
117#336—22no—
118C25C4⋊C411yescharacteristic
119C25C22×C411yescharacteristic
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—24no—
123C22×D4—42no—
124C22×D4C4⋊C411yescharacteristic
125C22×D4C4⋊C411yescharacteristic
126C22×D4C22×C411yescharacteristic
127C22×D4C22×C411yescharacteristic
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4C4⋊C411yescharacteristic
137C23×C4C4⋊C411yescharacteristic
138C23×C4C4⋊C411yescharacteristic
139C23×C4C4⋊C411yescharacteristic
140C23×C4C4⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×Q811yescharacteristic
149C23×C4C2×Q811yescharacteristic
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—44no—
160C2×M16—44no—
161C2×M16—82no—
162C2×M16—82no—
163C22×C8—24no—
164C22×C8—28no—
165C22×C8—22no—
166C22×C8—24no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—44no—
171C22×C8—44no—
172C22×C8—48no—
173C22×C8—44no—
174C22×C8—42no—
175C22×C8—41no—
176C22×C8—42no—
177C22×C8—41no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8C2×D414yes—
183C22×C8C2×D414yes—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200#117—44no—
201#117—44no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4C4⋊C411yescharacteristic
206C2×C4⋊C4C4⋊C411yescharacteristic
207C2×C4⋊C4C2×Q811yescharacteristic
208C2×C4⋊C4C2×Q811yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4C4⋊C411yescharacteristic
216C2×C22⋊C4C4⋊C411yescharacteristic
217C2×C22⋊C4C4⋊C411yescharacteristic
218C2×C22⋊C4C4⋊C411yescharacteristic
219C2×C22⋊C4C22×C418yes—
220C2×C22⋊C4C22×C411yescharacteristic
221C2×C22⋊C4C22×C411yescharacteristic
222C2×C22⋊C4C2×Q811yescharacteristic
223C2×C22⋊C4C2×Q811yescharacteristic
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—24no—
229C2×C42C4⋊C411yescharacteristic
230C2×C42C4⋊C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—416no—
238#99—28no—
239#99—44no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8—24no—
246C22⋊C8—24no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—42no—
250#97—24no—
251C4×C8—24no—
252#95—24no—
253#95—24no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—42no—
260C24—42no—
261C24—41no—
262C24—42no—
263C24—42no—
264C24C2×C4⋊C411yescharacteristic
265C24C2×C4⋊C411yescharacteristic
266C24C2×C4⋊C411yescharacteristic
267C24C2×C4⋊C411yescharacteristic
268C24C4×D412yes—
269C24C4×D412yes—
270C24C23×C411yescharacteristic
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4#13114yes—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—28no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—82no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—82no—
331C22×C4C2×C4⋊C411yescharacteristic
332C22×C4C2×C4⋊C411yescharacteristic
333C22×C4C2×C4⋊C411yescharacteristic
334C22×C4C2×C4⋊C411yescharacteristic
335C22×C4C2×C4⋊C411yescharacteristic
336C22×C4C2×C4⋊C411yescharacteristic
337C22×C4C2×C4⋊C411yescharacteristic
338C22×C4C2×C4⋊C411yescharacteristic
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4#12011yescharacteristic
342C22×C4#12012yes—
343C22×C4#12011yescharacteristic
344C22×C4C22⋊Q812yes—
345C22×C4C22⋊Q812yes—
346C22×C4C22⋊Q812yes—
347C22×C4C22⋊Q812yes—
348C22×C4#12311yescharacteristic
349C22×C4#12311yescharacteristic
350C22×C4#12312yes—
351C22×C4C22×D412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C2×#4112yes—
355M16—44no—
356M16—44no—
357M16—82no—
358M16—82no—
359M16—82no—
360M16—82no—
361C2×C8—24no—
362C2×C8—28no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—44no—
366C2×C8—42no—
367C2×C8—48no—
368C2×C8—44no—
369C2×C8—42no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—82no—
377C2×C8—82no—
378C2×C8—82no—
379C2×C8—82no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—48no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—44no—
406C42—44no—
407C23—22no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—24no—
412C23—24no—
413C23—21no—
414C23—21no—
415C23—24no—
416C23—24no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—48no—
431C23—48no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—48no—
437C23—42no—
438C23—42no—
439C23—81no—
440C23—81no—
441C23—84no—
442C23#38512yes—
443C23#38511yescharacteristic
444C23#38511yescharacteristic
445C23#42412yescomplemented
446C23C22×C4⋊C411yescharacteristic
447C23C2×C4×D412yes—
448C23C2×#13112yes—
449C23#56812yes—
450D4—28no—
451D4—44no—
452D4—44no—
453D4—44no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4#38511yescharacteristic
508C2×C4#38512yes—
509C2×C4#38511yescharacteristic
510C2×C4#42412yescomplemented
511C2×C4B(4,𝔽2)12yes—
512C2×C4#45812yes—
513C2×C4C2×#12011yescharacteristic
514C2×C4C2×C22⋊Q812yes—
515C2×C4C2×#12311yescharacteristic
516C8—44no—
517C8—44no—
518C8—44no—
519C8—82no—
520C22—24no—
521C22—24no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—48no—
537C22—48no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—81no—
548C22—84no—
549C22—84no—
550C22—81no—
551C22#186712yes—
552C22#186812yes—
553C22C2×#38511yescharacteristic
554C22C2×#42411yescharacteristic
555C22#290511yescharacteristic
556C22#293314yes—
557C4—22no—
558C4—21no—
559C4—21no—
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—88no—
568C4#186712yes—
569C4C2×B(4,𝔽2)11yescharacteristic
570C4C2×#45811yescharacteristic
571C2—24no—
572C2—21no—
573C2—22no—
574C2—22no—
575C2—21no—
576C2—42no—
577C2—42no—
578C2—81no—
579C2#2083611yescharacteristic
580C2C2×#186711yescharacteristic
581C2C2×#186811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20834C2semidirect
5. #8852C2semidirect
6. #8851C2semidirect
7. #7925C2semidirect
29. #1850C22semidirect
38. #1282C22semidirect
445. C23#424semidirect
510. C2×C4#424semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#18692
#208342
#207222
#88522
#88512
#79252