Finite Group Databasebeta

← #182620

Subgroups — #182620

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3C2×#1508C212yesmaximal
4#13753C211yesmaximal, characteristic
5#8221C214yesmaximal, complemented
6#8217C214yesmaximal, complemented
7#8019C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C2×#407C2212yes—
12C2×C4×M16—21no—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8C412yescomplemented
18C4×C22⋊Q8C412yescomplemented
19C4×C22⋊Q8C2212yes—
20C4×C22⋊Q8C2212yes—
21C2×#391C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#387—21no—
24C2×#385—21no—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C411yescharacteristic, complemented
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4C2212yes—
30C2×#376—21no—
31#1749—21no—
32#1749C2212yes—
33C4×C22⋊C8—21no—
34C4×C22⋊C8C2212yes—
35#1728—21no—
36#1508C4116yescomplemented
37C2×#328C2218yes—
38#1434C2212yes—
39#1434C2212yes—
40C22⋊#95—21no—
41C22⋊#95—21no—
42C4×#95—21no—
43#1294C2218yes—
44C23×Q8C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8C2×C418yes—
53C2×C22⋊Q8C2×C412yes—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—44no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42—44no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#407—24no—
86#407C2×C418yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8C2314yes—
89C4×M16—44no—
90C2×#97—22no—
91C2×C4×C8—22no—
92#391—21no—
93#391—21no—
94#391C2×C412yes—
95#391C2×C412yes—
96#389—21no—
97#389—21no—
98#389C2×C412yes—
99#389C2×C412yes—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#380—21no—
109#380—21no—
110#380C2×C412yes—
111#380C2312yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—24no—
125C4×C22⋊C4C2×C418yes—
126#376—44no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95C2314yes—
137#336—22no—
138#336—22no—
139#328C2×C4132yescomplemented
140C25C2×D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8C4214yes—
146C22×Q8C22×C412yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—42no—
154C23×C4—42no—
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×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C414yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4114yes—
179C23×C4C2411yescharacteristic
180C2×M16—28no—
181C2×M16—44no—
182C2×M16—44no—
183C22×C8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—24no—
188C22⋊Q8C4218yes—
189C22⋊Q8C22×C418yes—
190C4×Q8—44no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×Q8—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C4214yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—28no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C22⋊C8—28no—
260C22⋊C8—24no—
261C22⋊C8C22×C418yes—
262#97—44no—
263C4×C8—44no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—24no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95C22×C418yes—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—41no—
285C24—41no—
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288C24C4×D414yes—
289C24C22×D411yescharacteristic
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8C2×C4214yes—
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—24no—
314C22×C4—24no—
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—28no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
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—42no—
357C22×C4—42no—
358C22×C4—416no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4C2×C4212yes—
374C22×C4C2×C4212yes—
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×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4#11712yes—
386C22×C4#11712yes—
387C22×C4C4×D414yes—
388C22×C4C4×D414yes—
389C22×C4C4×D414yes—
390C22×C4#13714yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C23×C412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4112yes—
395M16—416no—
396C2×C8—28no—
397C2×C8—44no—
398C2×C8—44no—
399C4⋊C4—22no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—24no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4C2×C4214yes—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—416no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—44no—
433C42—44no—
434C42—416no—
435C42—44no—
436C42—44no—
437C42#9914yes—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—28no—
453C23—28no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23C4×C22⋊C412yes—
460C23C4×C22⋊C412yes—
461C23#42112yes—
462C23#42112yes—
463C23C22×C22⋊C411yescharacteristic
464C23C2×C4×D412yes—
465C23C2×#13712yes—
466C23#57512yes—
467Q8—44no—
468Q8—44no—
469Q8—48no—
470C2×C4—28no—
471C2×C4—21no—
472C2×C4—21no—
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—28no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
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—24no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4C4×C22⋊C412yes—
541C2×C4C4×C22⋊C412yes—
542C2×C4C2×#9912yes—
543C2×C4C2×#9912yes—
544C2×C4C2×#9912yes—
545C2×C4#41012yes—
546C2×C4#41912yes—
547C2×C4#41912yes—
548C2×C4C22×C4211yescharacteristic
549C2×C4C2×#11711yescharacteristic
550C2×C4C2×C4×D412yes—
551C8—48no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—416no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22#175914yes—
573C22C2×C4×C22⋊C411yescharacteristic
574C22C22×#9911yescharacteristic
575C22C2×#41011yescharacteristic
576C22C2×#41911yescharacteristic
577C22#289611yescharacteristic
578C22C2×#42111yescharacteristic
579C22C2×#42111yescharacteristic
580C22C4×#13714yes—
581C4—22no—
582C4—21no—
583C4—22no—
584C4—21no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—44no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—44no—
595C4—88no—
596C4—82no—
597C4—82no—
598C4C4×#9914yes—
599C2—24no—
600C2—24no—
601C2—42no—
602C2#1757811yescharacteristic
603C2#1757911yescharacteristic
604C2#1758211yescharacteristic
605C2#1758411yescharacteristic
606C2C2×C4×#9911yescharacteristic
607C2C2×#175911yescharacteristic
608C2C4×#41911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ 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
5. #8221C2semidirect
6. #8217C2semidirect
9. C22×C4×Q8C4semidirect
17. C4×C22⋊Q8C4semidirect
18. C4×C22⋊Q8C4semidirect
21. C2×#391C4semidirect
22. C2×#389C4semidirect
27. C2×C4×C4⋊C4C4semidirect
36. #1508C4semidirect
139. #328C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C4×#4072
C2×#15082
#137532
#82212
#82172
#80192