Finite Group Databasebeta

← #207706

Subgroups — #207706

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4#9879C212yesmaximal, complemented
5#9649C212yesmaximal, complemented
6#9079C214yesmaximal, complemented
7#8155C214yesmaximal, complemented
8C22×#123C412yes—
9C22×#121C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C22×C22⋊C8C2212yes—
13C22×C4×C8—21no—
14#2854—21no—
15#2854—21no—
16#2854—21no—
17#2854—21no—
18#2854—21no—
19#2854—21no—
20#2854—21no—
21#2854—21no—
22C2×#390C411yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C411yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388C412yes—
27C2×#388C2212yes—
28C2×#386C2212yes—
29C2×#380C411yescharacteristic
30#1817—21no—
31#1797—21no—
32#1797C2212yes—
33#1779C2212yes—
34#1749—21no—
35#1749C2212yes—
36C4×C22⋊C8—21no—
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39C4×C22⋊C8C2212yes—
40C2×#336—21no—
41C2×#323C2218yes—
42#1280C2218yes—
43C23×D4C2×C412yes—
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C2×#123—22no—
50C2×#123—22no—
51C2×#123—22no—
52C2×#123—22no—
53C2×#123—48no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—44no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2312yes—
70C22×C42—44no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432—44no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—28no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91C2×#97—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—44no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2312yes—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#323—216no—
135C25C22×C411yescharacteristic
136C25C22×C412yes—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—28no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—24no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—44no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C412yes—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C22×C8—28no—
178C22×C8—24no—
179C22×C8—24no—
180C22×C8—44no—
181C22×C8—44no—
182#123—48no—
183#123—48no—
184#123—48no—
185#123—48no—
186#121—44no—
187#121—44no—
188#121—44no—
189#121—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C4⋊C8—44no—
252C22⋊C8—28no—
253C22⋊C8—24no—
254C22⋊C8—28no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C22⋊C8#4118yes—
258#97—44no—
259C4×C8—44no—
260C4×C8—44no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24C2×C22⋊C411yescharacteristic
282C24C2×C22⋊C411yescharacteristic
283C24#13114yes—
284C24C23×C411yescharacteristic
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—88no—
298C2×D4—816no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—416no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4#9914yes—
365C22×C4#10014yes—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4#13114yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4114yes—
380C2×C8—28no—
381C2×C8—44no—
382C2×C8—416no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—44no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—28no—
439C23—28no—
440C23—21no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—48no—
454C23—84no—
455C23—84no—
456C23—84no—
457C23—84no—
458C23—84no—
459C23—84no—
460C23#40812yes—
461C23#40812yes—
462C23C2×#9912yes—
463C23C2×#10012yes—
464C23C22×C22⋊C411yescharacteristic
465C23C4×#4112yes—
466C23C2×#13112yes—
467C23C2×#13112yes—
468D4—84no—
469D4—88no—
470D4—88no—
471D4—84no—
472C2×C4—28no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—28no—
487C2×C4—21no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4#40812yes—
529C2×C4#40812yes—
530C2×C4C2×#9912yes—
531C2×C4C2×#9912yes—
532C2×C4#41012yes—
533C2×C4C2×#10012yes—
534C2×C4C2×#10012yes—
535C2×C4#41312yes—
536C2×C4#52512yes—
537C2×C4#53212yes—
538C8—48no—
539C8—84no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—416no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—168no—
568C22#147114yes—
569C22#229711yescharacteristic
570C22C2×#40811yescharacteristic
571C22C2×#40811yescharacteristic
572C22C22×#9911yescharacteristic
573C22C2×#41011yescharacteristic
574C22C22×#10011yescharacteristic
575C22C2×#41311yescharacteristic
576C22#296614yes—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4#146814yes—
592C2—24no—
593C2—24no—
594C2—42no—
595C2—162no—
596C2C2×#146811yescharacteristic
597C2C2×#147111yescharacteristic
598C2#1738211yescharacteristic
599C2#1738411yescharacteristic
600C2#2137511yescharacteristic
601C2#2142011yescharacteristic
602C2#2142411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. C2×C4×C22⋊C8C2semidirect
4. #9879C2semidirect
5. #9649C2semidirect
6. #9079C2semidirect
7. #8155C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#207152
C2×C4×C22⋊C82
#98792
#96492
#90792
#81552