Finite Group Databasebeta

← #204184

Subgroups — #204184

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×#1468C214yesmaximal, complemented
5#9651C214yesmaximal, complemented
6#9650C212yesmaximal, complemented
7#9649C212yesmaximal, complemented
8C22×#123C2212yes—
9C22×#121C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C4×C8—21no—
15#2854—21no—
16#2854—21no—
17#2854—21no—
18#2854—21no—
19#2854—21no—
20#2854—21no—
21#2854—21no—
22#2854—21no—
23C2×#390C411yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C411yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388C412yes—
28C2×#388C2212yes—
29C2×#386C412yes—
30C2×#380C411yescharacteristic
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35C4×C22⋊C8—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1728—21no—
40C2×#336—21no—
41#1468—216no—
42C2×#324C2218yes—
43C2×#323C2218yes—
44C23×D4C2×C412yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C23×C8—22no—
51C2×#123—22no—
52C2×#123—22no—
53C2×#123—22no—
54C2×#123—22no—
55C2×#123—28no—
56C2×#123—28no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—44no—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2312yes—
73C22×C42—44no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#423—44no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89#407—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8D418yescomplemented
94C2×C22⋊C8C2314yes—
95C2×C22⋊C8C2314yes—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—44no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#324—216no—
139#323—216no—
140C25C22×C411yescharacteristic
141C25C22×C412yes—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—28no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—24no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—44no—
151C22×D4—44no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
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×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D414yes—
181C23×C4#4114yes—
182C23×C4C2411yescharacteristic
183C2×M16—44no—
184C2×M16—44no—
185C2×M16—44no—
186C22×C8—28no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—44no—
190C22×C8—44no—
191#123—48no—
192#123—48no—
193#123—44no—
194#123—44no—
195#123—44no—
196#123—44no—
197#121—44no—
198#121—44no—
199#121—44no—
200#121—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C4⋊C8—44no—
265C4⋊C8—44no—
266C22⋊C8—28no—
267C22⋊C8—24no—
268C22⋊C8—28no—
269C22⋊C8—44no—
270C22⋊C8C2×D418yes—
271C4×C8—44no—
272C4×C8—44no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—41no—
288C24—41no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C24C4×D414yes—
296C24C23×C411yescharacteristic
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—88no—
310C2×D4—816no—
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—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
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—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—41no—
366C22×C4—416no—
367C22×C4—416no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4#9914yes—
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C4×D414yes—
384C22×C4#12114yes—
385C22×C4#12114yes—
386C22×C4#12412yes—
387C22×C4#12412yes—
388C22×C4#12712yes—
389C22×C4#12712yes—
390C22×C4#13114yes—
391C22×C4#13114yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C2×#4112yes—
396M16—88no—
397C2×C8—28no—
398C2×C8—44no—
399C2×C8—44no—
400C2×C8—416no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—28no—
459C23—28no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—48no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—84no—
478C23—84no—
479C23#39012yes—
480C23#39012yes—
481C23C2×#9912yes—
482C23#41012yes—
483C23C22×C22⋊C411yescharacteristic
484C23C2×C4×D412yes—
485C23C2×#13112yes—
486C23C2×#13112yes—
487D4—84no—
488D4—88no—
489D4—88no—
490D4—84no—
491C2×C4—28no—
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—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4#39012yes—
549C2×C4#39012yes—
550C2×C4#40812yes—
551C2×C4#40812yes—
552C2×C4#40812yes—
553C2×C4#40812yes—
554C2×C4C2×#9912yes—
555C2×C4C2×#9912yes—
556C2×C4#41312yes—
557C2×C4#41312yes—
558C2×C4C2×#12112yes—
559C2×C4C2×#12411yescharacteristic
560C2×C4C2×#12711yescharacteristic
561C8—48no—
562C8—84no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—416no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—168no—
591C22#196414yes—
592C22C2×#39011yescharacteristic
593C22C2×#40811yescharacteristic
594C22C2×#40811yescharacteristic
595C22C22×#9911yescharacteristic
596C22C2×#41011yescharacteristic
597C22C2×#41311yescharacteristic
598C22C2×#41311yescharacteristic
599C22#293114yes—
600C4—22no—
601C4—21no—
602C4—21no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4#146814yescomplemented
616C2—24no—
617C2—24no—
618C2—42no—
619C2—162no—
620C2C2×#146811yescharacteristic
621C2#1737611yescharacteristic
622C2#1738111yescharacteristic
623C2#1738311yescharacteristic
624C2#2118711yescharacteristic
625C2C2×#196411yescharacteristic
626C2#2121811yescharacteristic
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. #20626C2semidirect
3. C2×C4×C22⋊C8C2semidirect
4. C2×#1468C2semidirect
5. #9651C2semidirect
6. #9650C2semidirect
7. #9649C2semidirect
93. C2×C22⋊C8D4semidirect
615. C4#1468semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#206262
C2×C4×C22⋊C82
C2×#14682
#96512
#96502
#96492