Finite Group Databasebeta

← #480381

Subgroups — #480381

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#17366C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#11528C214yesmaximal, complemented
5#11520C212yesmaximal, complemented
6#11504C212yesmaximal, complemented
7#11472C214yesmaximal, complemented
8C2×#536C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic
11#2920—21no—
12C2×#432—21no—
13#2327—21no—
14#2326—21no—
15#2323—21no—
16#2323—21no—
17#2322C412yes—
18#2322C2212yes—
19#2317C2211yescharacteristic
20#2311C411yescharacteristic
21#2310C2211yescharacteristic
22C4×C4⋊Q8—21no—
23C4×#124—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×#123C412yes—
27C4×#123C2212yes—
28C4×#117C411yescharacteristic
29#1467—21no—
30#1466C2212yes—
31#1465—21no—
32#1465C2212yes—
33#1464—21no—
34#1464—21no—
35#1464C2212yes—
36#1464C2212yes—
37#1461—21no—
38#1460C2212yes—
39#1459—21no—
40#1459C2212yes—
41#1326C2218yes—
42#1317C2218yes—
43C22×#41C2×C411yescharacteristic
44C23×Q8C2×C411yescharacteristic
45#536—28no—
46C2×C4⋊Q8—21no—
47C2×#124—21no—
48C2×#123C2×C412yes—
49C2×#123C2×C412yes—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—41no—
61C2×C4×Q8—41no—
62C2×C4×Q8—41no—
63C2×C4×Q8—41no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C412yes—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C412yes—
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83#432—22no—
84#432—22no—
85#432—22no—
86#432—41no—
87#432—41no—
88#432—41no—
89#431—22no—
90#431—41no—
91#423—41no—
92C2×C4⋊C8—41no—
93#407—22no—
94C2×C22⋊C8—22no—
95C4×M16—41no—
96C2×#97—41no—
97#391—21no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391—21no—
102#390—21no—
103#389—22no—
104#389—22no—
105#389—21no—
106#389—21no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#386C2×C412yes—
116#386C2×C412yes—
117#385—22no—
118#385—22no—
119#385C2×C412yes—
120#385C2312yes—
121#384—21no—
122#382—21no—
123#381C2×C412yes—
124#381C2312yes—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4C2×C412yes—
134C4×C22⋊C4C2×C412yes—
135#376—21no—
136C43—21no—
137#324—22no—
138#324—22no—
139#324—22no—
140#324—22no—
141#323—22no—
142#323—22no—
143#323C2314yes—
144#323C2314yes—
145C2×#41—24no—
146C2×#41—41no—
147C2×#41—41no—
148C2×#41—42no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—24no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8C22⋊C411yescharacteristic
156C22×D4C22⋊C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—42no—
171C22×C8—42no—
172C4⋊Q8—28no—
173#124—28no—
174#123—24no—
175#123—24no—
176#123—24no—
177#123M1618yescomplemented
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206#117—24no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22×C412yes—
241C2×C22⋊C4C22×C412yes—
242C2×C22⋊C4C22×C411yescharacteristic
243C2×C22⋊C4C2×D411yescharacteristic
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C22⋊C4C2411yescharacteristic
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280C22⋊C8—24no—
281C22⋊C8—24no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284#97—42no—
285#97—42no—
286#97—42no—
287C4×C8—42no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298C24C2×C22⋊C411yescharacteristic
299C24C2×C22⋊C411yescharacteristic
300C24C23×C411yescharacteristic
301#41—48no—
302#41—48no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—42no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4C2×M1614yes—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—82no—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
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×M1614yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376M16—84no—
377C2×C8—44no—
378C2×C8—44no—
379C2×C8—84no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C22⋊C4—28no—
399C22⋊C4—22no—
400C22⋊C4—28no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—42no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4C2×M1614yes—
411C42—24no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—44no—
431C42—44no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—82no—
436C42#9911yescharacteristic
437C42#9911yescharacteristic
438C42#9911yescharacteristic
439C42#9911yescharacteristic
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23#40712yes—
454C23C22×C22⋊C411yescharacteristic
455C23C22×M1612yes—
456Q8—44no—
457Q8—44no—
458Q8—44no—
459Q8—48no—
460D4—44no—
461D4—44no—
462D4—44no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
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—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—84no—
519C2×C4#40712yes—
520C2×C4C2×#9911yescharacteristic
521C2×C4C2×#9911yescharacteristic
522C2×C4C2×#9911yescharacteristic
523C2×C4C2×#9911yescharacteristic
524C2×C4C2×#9911yescharacteristic
525C2×C4C2×#9911yescharacteristic
526C2×C4#41312yes—
527C8—84no—
528C8—84no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22C2×#40711yescharacteristic
547C22C22×#9911yescharacteristic
548C22C2×#41311yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—22no—
556C4—44no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—44no—
565C4—82no—
566C4—82no—
567C4#212312yes—
568C2—21no—
569C2—21no—
570C2—42no—
571C2—42no—
572C2—81no—
573C2#1736211yescharacteristic
574C2C2×#212311yescharacteristic
575C2#3334711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11528C2semidirect
5. #11520C2semidirect
6. #11504C2semidirect
7. #11472C2semidirect
177. #123M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#173662
#173632
#115282
#115202
#115042
#114722