Finite Group Databasebeta

← #480330

Subgroups — #480330

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#17361C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#11528C214yesmaximal, complemented
5#11526C214yesmaximal, complemented
6#11454C212yesmaximal, complemented
7#11428C212yesmaximal, complemented
8C2×#536C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic
11#2920—21no—
12#2920—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#1467—21no—
31#1466C2212yes—
32#1466C2212yes—
33#1465—21no—
34#1465—21no—
35#1465C2212yes—
36#1465C2212yes—
37#1458—21no—
38#1458—21no—
39#1458C2212yes—
40#1458C2212yes—
41#1326C2218yes—
42#1317C2218yes—
43C22×#41C2×C411yescharacteristic
44C23×Q8C2×C411yescharacteristic
45#536—24no—
46#536—24no—
47C2×C4⋊Q8—21no—
48C2×#124—21no—
49C2×#123C2×C412yes—
50C2×#123C2×C412yes—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C4×#41—41no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—41no—
62C2×C4×Q8—41no—
63C2×C4×Q8—41no—
64C2×C4×Q8—41no—
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C412yes—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C412yes—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84#432—22no—
85#432—22no—
86#432—41no—
87#432—41no—
88#431—22no—
89#431—22no—
90#431—41no—
91#431—41no—
92#423—41no—
93#423—41no—
94#407—22no—
95#407—22no—
96C4×M16—41no—
97C4×M16—41no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391—21no—
103#390—21no—
104#389—22no—
105#389—22no—
106#389—21no—
107#389—21no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#386C2×C412yes—
117#386C2×C412yes—
118#385—22no—
119#385—22no—
120#385C2×C412yes—
121#385C2312yes—
122#384—21no—
123#382—21no—
124#381C2×C412yes—
125#381C2312yes—
126#380—21no—
127#380—21no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4C2×C412yes—
135C4×C22⋊C4C2×C412yes—
136#376—21no—
137C43—21no—
138#324—22no—
139#324—22no—
140#324—22no—
141#324—22no—
142#323—22no—
143#323—22no—
144#323C2314yes—
145#323C2314yes—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—42no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—42no—
157C22×Q8C22⋊C411yescharacteristic
158C22×Q8M1614yescomplemented
159C22×D4C22⋊C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C22×D4C22×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×C412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C2×M16—42no—
174C2×M16—42no—
175C4⋊Q8—28no—
176#124—28no—
177#123—24no—
178#123—24no—
179#123—24no—
180#123—24no—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—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—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209#117—22no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—41no—
214#117—41no—
215#117—41no—
216#117—42no—
217#117M1614yescomplemented
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4C22⋊C411yescharacteristic
243C2×C22⋊C4C22⋊C411yescharacteristic
244C2×C22⋊C4C22×C412yes—
245C2×C22⋊C4C22×C412yes—
246C2×C22⋊C4C22×C411yescharacteristic
247C2×C22⋊C4C2×D411yescharacteristic
248C2×C22⋊C4C2×D411yescharacteristic
249C2×C22⋊C4C2411yescharacteristic
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—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C4⋊C8—42no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—42no—
289#97—42no—
290C4×C8—42no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304C24C23×C411yescharacteristic
305#41—44no—
306#41—44no—
307#41—44no—
308#41—44no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—82no—
372C22×C4C2×C22⋊C411yescharacteristic
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×M1612yes—
383C22×C4C2×M1612yes—
384C22×C4C2×M1614yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387M16—84no—
388M16—84no—
389C2×C8—44no—
390C2×C8—44no—
391C4⋊C4—28no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—22no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—42no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4C2×M1614yes—
422C42—24no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—44no—
442C42—44no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—82no—
447C42#9911yescharacteristic
448C42#9911yescharacteristic
449C42#9911yescharacteristic
450C42#9911yescharacteristic
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23C22×C22⋊C411yescharacteristic
466C23C22×M1611yescharacteristic
467C23C22×M1611yescharacteristic
468Q8—48no—
469Q8—44no—
470Q8—44no—
471Q8—44no—
472D4—44no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
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—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—84no—
534C2×C4#40712yes—
535C2×C4#40712yes—
536C2×C4C2×#9911yescharacteristic
537C2×C4C2×#9911yescharacteristic
538C2×C4C2×#9911yescharacteristic
539C2×C4C2×#9911yescharacteristic
540C2×C4C2×#9911yescharacteristic
541C2×C4C2×#9911yescharacteristic
542C2×C4#41312yes—
543C8—84no—
544C8—84no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—81no—
563C22—81no—
564C22C2×#40711yescharacteristic
565C22C22×#9911yescharacteristic
566C22C2×#41311yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—22no—
571C4—44no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—44no—
581C4—82no—
582C4—82no—
583C4#146512yes—
584C4#146512yes—
585C4#212312yes—
586C2—21no—
587C2—21no—
588C2—41no—
589C2—41no—
590C2—81no—
591C2—81no—
592C2C2×#146511yescharacteristic
593C2C2×#212311yescharacteristic
594C2#3334711yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #11526C2semidirect
6. #11454C2semidirect
7. #11428C2semidirect
158. C22×Q8M16semidirect
217. #117M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#173612
#115282
#115262
#114542
#114282