Finite Group Databasebeta

← #493876

Subgroups — #493876

Order 512.

1687 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34259C211yesmaximal, characteristic
2#20593C212yesmaximal
3#12863C212yesmaximal, complemented
4#12862C212yesmaximal, complemented
5#12861C212yesmaximal, complemented
6#12860C212yesmaximal, complemented
7#12586C212yesmaximal, complemented
8C2×#1394C212yesmaximal, complemented
9C22×2−1+4C2211yescharacteristic
10C2×#520C411yescharacteristic
11C2×C4×#41C411yescharacteristic
12C2×#413—22no—
13C22×#101C2214yes—
14#2879—22no—
15Q8×C22⋊C4—22no—
16#2339—22no—
17#2339—22no—
18#2339—22no—
19#2311C411yescharacteristic
20#2311C411yescharacteristic
21#2311C2211yescharacteristic
22#2310C411yescharacteristic
23#2310C2211yescharacteristic
24#2310C2211yescharacteristic
25#2297C411yescharacteristic
26#2297C411yescharacteristic
27#2297C2211yescharacteristic
28C2×#391C411yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31#1785—22no—
32#1785C2214yes—
33#1784—22no—
34#1784C2214yes—
35#1394C2214yes—
36#1394C2214yes—
37#1394C2214yes—
38#1394C2214yes—
39C2×2−1+4—24no—
40C2×2−1+4—24no—
41C2×2−1+4—44no—
42C2×2−1+4—44no—
43C22×#41C2×C411yescharacteristic
44C22×#41C2×C411yescharacteristic
45C22×#41C2×C411yescharacteristic
46C22×#41D412yes—
47C22×#41D412yes—
48C22×#41D412yes—
49C22×#41C2311yescharacteristic
50C23×Q8C2×C411yescharacteristic
51C23×Q8D412yes—
52C23×Q8C2311yescharacteristic
53C23×Q8C2311yescharacteristic
54C22×M16—24no—
55#520—44no—
56C4×#41—44no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77#413—48no—
78C2×#101—28no—
79C2×#101—28no—
80C2×#101—24no—
81C2×#101—24no—
82C2×#101C2318yes—
83C2×#100—24no—
84#408—48no—
85#407—24no—
86#407—24no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391—22no—
91#391—22no—
92#391—22no—
93#391C2×C411yescharacteristic
94#391C2×C411yescharacteristic
95#391C2×C411yescharacteristic
96#391C2×C411yescharacteristic
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#391C2311yescharacteristic
102#391C2311yescharacteristic
103#391C2311yescharacteristic
104#391C2311yescharacteristic
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#381—22no—
116#381—22no—
117#381—22no—
118#380—22no—
119#380—22no—
120#380—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
1272−1+4—416no—
1282−1+4—432no—
1292−1+4—416no—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41—28no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—28no—
138C2×#41—28no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—28no—
144C2×#41—28no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—44no—
153C2×#41—44no—
154C2×#41—42no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—28no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—42no—
168C22×Q8C22⋊C411yescharacteristic
169C22×Q8C22⋊C412yes—
170C22×Q8C22×C411yescharacteristic
171C22×Q8C22×C411yescharacteristic
172C22×Q8C2×D412yes—
173C22×Q8C2×D412yes—
174C22×Q8#4114yes—
175C22×Q8#4114yes—
176C22×Q8C2411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C412yes—
179C22×D4C22⋊C412yes—
180C22×D4C22⋊C411yescharacteristic
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C22×D4C2×D412yes—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C22⋊C412yes—
191C23×C4C22⋊C412yes—
192C23×C4C22⋊C412yes—
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C2×M16—28no—
200C2×M16—44no—
201C2×M16—48no—
202C2×M16—48no—
203C22×C8—44no—
204C4×Q8—44no—
205C4×Q8—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—44no—
212#117—42no—
213#117—42no—
214#117—42no—
215#117—42no—
216#117—42no—
217#117—42no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254#101—28no—
255#101—28no—
256#101—48no—
257#101—48no—
258#100—48no—
259C22⋊C8—48no—
260C22⋊C8—48no—
261#95—22no—
262#95—22no—
263#95—22no—
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—
273C24C2×C22⋊C412yes—
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277#41—216no—
278#41—216no—
279#41—416no—
280#41—416no—
281#41—416no—
282#41—416no—
283#41—48no—
284#41—48no—
285#41—48no—
286#41—48no—
287#41—48no—
288#41—48no—
289#41—432no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—42no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—44no—
310C2×Q8—48no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—432no—
317C2×Q8#11712yes—
318C2×Q8#11712yes—
319C2×Q8#12318yes—
320C2×Q8C2×#4112yes—
321C2×Q8C2×#4112yes—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4#9914yes—
343C2×D4#9914yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4#11712yes—
400C22×C4#11712yes—
401C22×C4#12318yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D412yes—
404M16—48no—
405M16—48no—
406M16—88no—
407C2×C8—48no—
408C2×C8—88no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23#38112yes—
455C23C2×#9912yes—
456C23C2×#9912yes—
457C23C2×#9912yes—
458C23C2×#9912yes—
459C23C22×C22⋊C411yescharacteristic
460Q8—28no—
461Q8—42no—
462Q8—42no—
463Q8—48no—
464Q8—42no—
465Q8—42no—
466Q8—48no—
467Q8—48no—
468Q8—42no—
469Q8—42no—
470D4—28no—
471D4—28no—
472D4—28no—
473D4—48no—
474D4—48no—
475D4—42no—
476D4—42no—
477D4—48no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—162no—
530C2×C4—162no—
531C2×C4—162no—
532C2×C4#38112yes—
533C2×C4#38114yes—
534C2×C4C2×#9912yes—
535C2×C4C2×#9912yes—
536C2×C4C2×#11711yescharacteristic
537C2×C4C2×#11711yescharacteristic
538C2×C4C2×#12314yes—
539C8—88no—
540C22—21no—
541C22—22no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—82no—
561C22#181914yes—
562C22C2×#38111yescharacteristic
563C22C22×#9911yescharacteristic
564C22C22×#9911yescharacteristic
565C4—24no—
566C4—21no—
567C4—21no—
568C4—22no—
569C4—24no—
570C4—22no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—81no—
581C2—21no—
582C2—21no—
583C2—24no—
584C2—42no—
585C2—42no—
586C2—82no—
587C2C2×#181911yescharacteristic
588C2#3337011yescharacteristic
589C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C22×2−1+4 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12863C2semidirect
4. #12862C2semidirect
5. #12861C2semidirect
6. #12860C2semidirect
7. #12586C2semidirect
8. C2×#1394C2semidirect

Maximal subgroups

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

subgroupindex
#342592
#205932
#128632
#128622
#128612
#128602
#125862
C2×#13942