Finite Group Databasebeta

← #196756

Subgroups — #196756

Order 512.

1323 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2990C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#21851C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic, complemented
5C22×#333C211yesmaximal, characteristic, complemented
6#9136C212yesmaximal, complemented
7#9135C212yesmaximal, complemented
8#9128C212yesmaximal, complemented
9#9127C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#8044C211yesmaximal, characteristic, complemented
12C2×#536C412yes—
13C22×C4⋊Q8C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#2990—24no—
16#2990—24no—
17C2×#446—21no—
18C2×#435—21no—
19C2×#435—21no—
20C2×C8×D4—21no—
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29#2857C2212yes—
30#2857C2212yes—
31#2621C412yes—
32#2621C412yes—
33#2547—21no—
34#2547—21no—
35#2546—21no—
36#2546—21no—
37#2486—21no—
38#2486—21no—
39#2486—21no—
40#2486—21no—
41#2310C412yes—
42#2285C2211yescharacteristic
43#1985—21no—
44#1955—21no—
45#1869—21no—
46#1869—21no—
47#1797C2212yes—
48#1797C2212yes—
49#1558—24no—
50C2×#333—24no—
51C2×#333C2212yes—
52C2×#333C2212yes—
53C2×#333C2212yes—
54C2×#333C2212yes—
55#1296C2212yes—
56#1296C2212yes—
57#1296C2212yes—
58#1296C2212yes—
59C22×#41C2×C411yescharacteristic
60C23×Q8C2×C412yes—
61C22×M16—21no—
62C23×C8—21no—
63C23×C8—21no—
64C23×C8—21no—
65#536—48no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2×C412yes—
74C2×C4⋊Q8C2312yes—
75C2×#124—22no—
76C2×#123—22no—
77C2×#123—22no—
78C2×C22⋊Q8C2×C414yes—
79C2×C22⋊Q8C2×C414yes—
80C4×#41—22no—
81C4×#41—22no—
82C4×#41—22no—
83C4×#41—22no—
84C2×C4×Q8D411yescharacteristic
85C2×C4×Q8D411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×C4×D4D411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C2×C4×D4D411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×#117C2×C412yes—
93C2×#117C2×C411yescharacteristic
94C2×#117C2×C411yescharacteristic
95C2×#117C2×C411yescharacteristic
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101#446—28no—
102#435—28no—
103#435—28no—
104C8×D4—28no—
105#432—22no—
106#432—22no—
107#432—22no—
108#432—22no—
109#432—22no—
110#432D414yes—
111#431—22no—
112#431—22no—
113#423—22no—
114#423—22no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—22no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8D414yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—22no—
136C2×C22⋊C8—22no—
137C2×#97—21no—
138C2×#97—21no—
139C2×#97—21no—
140C2×C4×C8—21no—
141#398—22no—
142#398—22no—
143#398—22no—
144#398—22no—
145#397—22no—
146#397—22no—
147#397—22no—
148#397—22no—
149#391—22no—
150#391—22no—
151#391C2×C414yes—
152#391C2×C414yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#384C2312yes—
158#382C2×C412yes—
159#381—22no—
160#381—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—21no—
166#380—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169#336—22no—
170#336—22no—
171#333—24no—
172#333—24no—
173#333—24no—
174#333—24no—
175C2×#41—24no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—42no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8C22⋊C411yescharacteristic
185C22×Q8C22×C414yes—
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22⋊C411yescharacteristic
188C22×D4C22⋊C411yescharacteristic
189C23×C4—22no—
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—24no—
200C2×M16—82no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—42no—
210C22×C8—82no—
211C22×C8—82no—
212C22×C8—82no—
213C4⋊Q8—24no—
214C4⋊Q8—24no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—48no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223C4⋊Q8—42no—
224#124—48no—
225#123—48no—
226#123—48no—
227C22⋊Q8—48no—
228C22⋊Q8—48no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117—44no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22×C414yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4112yes—
301C2×C42#4112yes—
302C2×C42C2411yescharacteristic
303C4⋊C8—24no—
304C4⋊C8—24no—
305C4⋊C8—24no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—42no—
315C4⋊C8C2×D412yes—
316C4⋊C8C2×D412yes—
317C4⋊C8C2×D412yes—
318C4⋊C8C2×D412yes—
319C22⋊C8—24no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322C22⋊C8—42no—
323C22⋊C8—82no—
324C22⋊C8—82no—
325#97—24no—
326#97—24no—
327#97—42no—
328C4×C8—24no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338#41—48no—
339#41—48no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—41no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—41no—
360C2×D4#10014yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C4×D412yes—
406C22×C4C4×D412yes—
407C22×C4#12012yes—
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4#12312yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414M16—84no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—84no—
430C2×C8—84no—
431C2×C8—84no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—44no—
473C42C4×D412yes—
474C42C4×D412yes—
475C42#12012yes—
476C42#12112yes—
477C42#12112yes—
478C42#12312yes—
479C42C22×D411yescharacteristic
480C42C22×D411yescharacteristic
481C42C2×#4111yescharacteristic
482C42C2×#4111yescharacteristic
483C23—22no—
484C23—22no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—21no—
494C23#38612yes—
495C23C2×#10012yes—
496C23C2×#10012yes—
497C23#41312yes—
498C23C22×C22⋊C411yescharacteristic
499Q8—44no—
500Q8—84no—
501Q8—84no—
502Q8—88no—
503D4—28no—
504D4—44no—
505D4—44no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—88no—
556C2×C4#38612yes—
557C2×C4#38612yes—
558C2×C4#38612yes—
559C2×C4C2×#10012yes—
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×#12011yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12311yescharacteristic
566C2×C4#57712yes—
567C2×C4#57712yes—
568C2×C4#57812yes—
569C2×C4#57812yes—
570C8—42no—
571C8—42no—
572C8—42no—
573C8—42no—
574C8—82no—
575C8—82no—
576C22—24no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—24no—
581C22—22no—
582C22—42no—
583C22—41no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22#190014yes—
591C22C2×#38611yescharacteristic
592C22C22×#10011yescharacteristic
593C22C2×#41311yescharacteristic
594C22#289812yes—
595C22#289812yes—
596C4—22no—
597C4—22no—
598C4—21no—
599C4—21no—
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—84no—
608C4—84no—
609C4#300911yescharacteristic
610C4#301111yescharacteristic
611C4#305711yescharacteristic
612C4#305911yescharacteristic
613C4#306312yes—
614C4#309712yes—
615C2—24no—
616C2—42no—
617C2—81no—
618C2#1777912yes—
619C2#1778012yes—
620C2#2091311yescharacteristic
621C2#2097411yescharacteristic
622C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
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. C22×#333C2semidirect
6. #9136C2semidirect
7. #9135C2semidirect
8. #9128C2semidirect
9. #9127C2semidirect
10. #8044C2semidirect
11. #8044C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29902
#353162
#218512
C2×#15582
C22×#3332
#91362
#91352
#91282
#91272
#80442