Finite Group Databasebeta

← #195852

Subgroups — #195852

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic
2#36625C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4C2×#1481C212yesmaximal, complemented
5#9075C214yesmaximal, complemented
6#9071C214yesmaximal, complemented
7#7893C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#431—21no—
11C2×#431—21no—
12C2×#431—28no—
13C2×#431C2212yes—
14C2×#431C2212yes—
15C22×C4⋊C8—21no—
16C22×C4⋊C8C2212yes—
17C22×C22⋊C8—21no—
18C22×C22⋊C8—21no—
19C22×C4×C8C2212yes—
20#2857C411yescharacteristic
21#2857C411yescharacteristic
22#2857C411yescharacteristic
23#2857C411yescharacteristic
24#2855—21no—
25#2855C412yes—
26#2851—21no—
27#2851—21no—
28#2848—21no—
29#2848C412yes—
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8—21no—
32C4×C22⋊Q8C2212yes—
33C4×C22⋊Q8C2212yes—
34#2296—21no—
35#2285C2211yescharacteristic
36#2280—21no—
37#1797—21no—
38#1797—21no—
39#1797C2212yes—
40#1797C2212yes—
41C2×#336—21no—
42#1481—28no—
43C2×Q8⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C23×C8—22no—
49C23×C8—22no—
50#533—24no—
51#533—24no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×Q8C2312yes—
63C2×#117—21no—
64C2×#117—21no—
65C22×C42—24no—
66C22×C42—24no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#431—28no—
80#431—28no—
81#431—24no—
82#431—28no—
83#431D418yes—
84C2×C4⋊C8—28no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2314yes—
91C2×C22⋊C8—28no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—44no—
97C2×C4×C8—22no—
98C2×C4×C8—28no—
99C2×C4×C8C2314yes—
100#391—21no—
101#391—21no—
102#391C2×C412yes—
103#391C2×C412yes—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389C2×C412yes—
111#389C2×C412yes—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#382—21no—
125#381—22no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380C2×C412yes—
134#380C2×C412yes—
135#380C2×C412yes—
136#379—22no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4C2312yes—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143#336—22no—
144#336—22no—
145Q8⋊C8—216no—
146C25C22⋊C411yescharacteristic
147C22×Q8—21no—
148C22×Q8C22×C412yes—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
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×C4SD1614yes—
168C23×C4Q1614yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C22×C8—28no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—44no—
182C22×C8—44no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8—44no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188#125—42no—
189#125—42no—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—44no—
200#117—44no—
201#117—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D414yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4114yes—
259C2×C42C2411yescharacteristic
260C4⋊C8—28no—
261C4⋊C8—28no—
262C4⋊C8—24no—
263C4⋊C8—24no—
264C4⋊C8—44no—
265C22⋊C8—28no—
266C22⋊C8—28no—
267C22⋊C8—44no—
268C22⋊C8—44no—
269C4×C8—28no—
270C4×C8—24no—
271C4×C8C2×D418yes—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—41no—
287C24—41no—
288C24#10314yes—
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
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—216no—
317C22×C4—216no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
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—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—416no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4#10314yes—
353C22×C4#10314yes—
354C22×C4#10314yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C4×D414yes—
365C22×C4#12012yes—
366C22×C4#12114yes—
367C22×C4#12312yes—
368C22×C4#13114yes—
369C22×C4C2×SD1612yes—
370C22×C4C2×SD1612yes—
371C22×C4C2×SD1612yes—
372C22×C4C2×Q1612yes—
373C22×C4C2×Q1612yes—
374C22×C4C2×Q1612yes—
375C22×C4#13514yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C2×C8—28no—
380C2×C8—416no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—88no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C42—22no—
411C42—22no—
412C42—21no—
413C42—21no—
414C42—22no—
415C42—22no—
416C42—21no—
417C42—21no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—44no—
425C42—44no—
426C42C4×D414yes—
427C42#12012yes—
428C42#12114yes—
429C42#12312yes—
430C42C22×D412yes—
431C42C2×#4112yes—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—28no—
446C23—28no—
447C23—21no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23#38612yes—
454C23#40812yes—
455C23C2×#10312yes—
456C23C2×#10312yes—
457C23C2×#10312yes—
458C23#41612yes—
459C23#42112yes—
460C23C22×C22⋊C411yescharacteristic
461Q8—48no—
462Q8—84no—
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—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—28no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
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—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4#38612yes—
518C2×C4#38612yes—
519C2×C4#38612yes—
520C2×C4#40812yes—
521C2×C4C2×#10312yes—
522C2×C4C2×#10312yes—
523C2×C4C2×#10312yes—
524C2×C4#41612yes—
525C2×C4#42112yes—
526C2×C4#46518yes—
527C2×C4#46718yes—
528C2×C4C2×C4×D412yes—
529C2×C4C2×#12011yescharacteristic
530C2×C4C2×#12112yes—
531C2×C4C2×#12311yescharacteristic
532C2×C4C2×#13112yes—
533C2×C4#56812yes—
534C2×C4C22×SD1611yescharacteristic
535C2×C4C22×Q1611yescharacteristic
536C2×C4C2×#13512yes—
537C8—48no—
538C8—84no—
539C22—22no—
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—416no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22#148114yescomplemented
560C22#189214yes—
561C22C2×#38611yescharacteristic
562C22C2×#40811yescharacteristic
563C22#287911yescharacteristic
564C22C22×#10311yescharacteristic
565C22C2×#41611yescharacteristic
566C22C2×#42111yescharacteristic
567C22C2×#42111yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#189214yes—
583C4#293314yes—
584C4C2×#46512yes—
585C4C2×#46712yes—
586C2—24no—
587C2—24no—
588C2—42no—
589C2C2×#148111yescharacteristic
590C2#1745811yescharacteristic
591C2#1748611yescharacteristic
592C2#1748811yescharacteristic
593C2#2089711yescharacteristic
594C2C2×#189211yescharacteristic
595C2#2097911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1481C2semidirect
5. #9075C2semidirect
6. #9071C2semidirect
7. #7893C2semidirect
43. C2×Q8⋊C8C22semidirect
44. #1279C22semidirect
559. C22#1481semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
#366252
C2×#17972
C2×#14812
#90752
#90712
#78932