Finite Group Databasebeta

← #196747

Subgroups — #196747

Order 512.

1221 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2989C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic, complemented
5C2×#1548C211yesmaximal, characteristic, complemented
6#9136C212yesmaximal, complemented
7#9135C212yesmaximal, complemented
8#9128C212yesmaximal, complemented
9#9127C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#8042C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#2989—24no—
15#2989—24no—
16C2×#446—21no—
17C2×#436—21no—
18C2×#435—21no—
19C2×#435—21no—
20C2×#432C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28#2857C2212yes—
29#2857C2212yes—
30#2628—21no—
31#2628—21no—
32#2568C412yes—
33#2567—21no—
34#2563—21no—
35#2542C412yes—
36#2487—21no—
37#2487—21no—
38#2409C412yes—
39#2409C412yes—
40#2317—21no—
41#2311—21no—
42#2285C2211yescharacteristic
43#1818—21no—
44#1797C2212yes—
45#1797C2212yes—
46#1780—21no—
47#1750—21no—
48#1750—21no—
49#1558—24no—
50#1548—24no—
51C2×#333C2212yes—
52C2×#333C2212yes—
53C2×#333C2212yes—
54C2×#333C2212yes—
55#1296C2212yes—
56#1296C2212yes—
57#1296C2212yes—
58#1296C2212yes—
59C22×#41D411yescharacteristic
60C23×Q8C2×C412yes—
61C22×M16—21no—
62C22×M16—21no—
63C23×C8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8C2×C412yes—
71C2×C4⋊Q8C2312yes—
72C2×C22⋊Q8C2×C414yes—
73C2×C22⋊Q8C2×C414yes—
74C4×#41—22no—
75C4×#41—22no—
76C4×#41—22no—
77C4×#41—22no—
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C2×#117D411yescharacteristic
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#446—28no—
96#436—28no—
97#435—28no—
98#435—28no—
99#432—22no—
100#432—22no—
101#432—22no—
102#432—22no—
103#432—22no—
104#432D414yes—
105#431—22no—
106#431—22no—
107#431—22no—
108#431—22no—
109#423—22no—
110#423—22no—
111#423—22no—
112#423D414yes—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×#97—21no—
132C2×#97—21no—
133C2×#97—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136#400—22no—
137#400—22no—
138#399—22no—
139#399—22no—
140#395—22no—
141#395—22no—
142#393—22no—
143#393—22no—
144#391—22no—
145#391C2×C414yes—
146#391C2×C414yes—
147#390—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#386—22no—
153#386—22no—
154#385—22no—
155#385—22no—
156#384—22no—
157#384C2312yes—
158#382—22no—
159#382—21no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380—21no—
165#380C2×C412yes—
166C4×C4⋊C4—21no—
167C4×C4⋊C4C2×C412yes—
168#336—22no—
169#336—22no—
170#333—24no—
171#333—24no—
172#333—24no—
173#333—24no—
174C2×#41—22no—
175C2×#41—22no—
176C2×#41—24no—
177C2×#41—42no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—84no—
182C22×Q8C22⋊C411yescharacteristic
183C22×Q8C22×C414yes—
184C22×D4C22⋊C411yescharacteristic
185C22×D4C22⋊C411yescharacteristic
186C22×D4C22⋊C411yescharacteristic
187C23×C4—22no—
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—82no—
200C2×M16—82no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—24no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—82no—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218C4⋊Q8—42no—
219C22⋊Q8—84no—
220C22⋊Q8—84no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237#117—22no—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—44no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C22⋊C411yescharacteristic
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22×C414yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42#4112yes—
293C2×C42#4112yes—
294C2×C42C2411yescharacteristic
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8C2×D412yes—
307C4⋊C8C2×D412yes—
308C4⋊C8C2×D412yes—
309C4⋊C8C2×D412yes—
310C22⋊C8—24no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—44no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—44no—
317#97—24no—
318#97—24no—
319#97—42no—
320C4×C8—24no—
321C4×C8—42no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328C24C2×C22⋊C411yescharacteristic
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331#41—48no—
332#41—48no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—41no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—84no—
342C2×Q8—88no—
343C2×Q8—88no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—84no—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4#12012yes—
395C22×C4#12112yes—
396C22×C4#12112yes—
397C22×C4#12312yes—
398C22×C4#13714yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402M16—84no—
403M16—84no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—84no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—44no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—44no—
459C42C4×D412yes—
460C42C4×D412yes—
461C42#12012yes—
462C42#12112yes—
463C42#12112yes—
464C42#12312yes—
465C42C22×D411yescharacteristic
466C42C22×D411yescharacteristic
467C42C2×#4111yescharacteristic
468C42C2×#4111yescharacteristic
469C23—22no—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—21no—
482C23#38612yes—
483C23#41912yes—
484C23C22×C22⋊C411yescharacteristic
485Q8—44no—
486Q8—84no—
487Q8—84no—
488Q8—88no—
489D4—44no—
490D4—44no—
491D4—44no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—88no—
538C2×C4#38612yes—
539C2×C4#38612yes—
540C2×C4#38612yes—
541C2×C4#41912yes—
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×#12011yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12311yescharacteristic
548C2×C4C2×#13712yes—
549C2×C4#57512yes—
550C2×C4#57712yes—
551C2×C4#57812yes—
552C8—42no—
553C8—42no—
554C8—42no—
555C8—42no—
556C8—82no—
557C8—82no—
558C22—24no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#38611yescharacteristic
574C22#288812yes—
575C22C2×#41911yescharacteristic
576C22#289611yescharacteristic
577C22#289812yes—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—21no—
582C4—21no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—84no—
590C4—84no—
591C4#300911yescharacteristic
592C4#301111yescharacteristic
593C4#305612yes—
594C4#305612yes—
595C4#309712yes—
596C2—42no—
597C2—42no—
598C2—81no—
599C2#1776012yes—
600C2#1776812yes—
601C2#2091311yescharacteristic
602C2#2097211yescharacteristic
603C2#2099011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1558C2semidirect
5. C2×#1548C2semidirect
6. #9136C2semidirect
7. #9135C2semidirect
8. #9128C2semidirect
9. #9127C2semidirect
10. #8046C2semidirect
11. #8042C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29892
#352792
#207212
C2×#15582
C2×#15482
#91362
#91352
#91282
#91272
#80462
#80422