Finite Group Databasebeta

← #165277

Subgroups — #165277

Order 512.

993 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36274C211yesmaximal, characteristic
2#22471C211yesmaximal, characteristic
3#20720C211yesmaximal, characteristic
4#7727C211yesmaximal, characteristic, complemented
5#7721C211yesmaximal, characteristic, complemented
6#7701C211yesmaximal, characteristic, complemented
7#7695C211yesmaximal, characteristic, complemented
8#7675C211yesmaximal, characteristic, complemented
9#7667C211yesmaximal, characteristic, complemented
10#7648C211yesmaximal, characteristic, complemented
11#7629C211yesmaximal, characteristic, complemented
12#7609C211yesmaximal, characteristic, complemented
13#7595C211yesmaximal, characteristic, complemented
14#7589C211yesmaximal, characteristic, complemented
15#7575C211yesmaximal, characteristic, complemented
16C2×D4×Q8C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#435—21no—
19C2×#433—21no—
20#2857C2211yescharacteristic
21#2855C2211yescharacteristic
22#2855C2211yescharacteristic
23#2582C411yescharacteristic
24#2568C411yescharacteristic
25#2451—21no—
26#2450—21no—
27#2450—21no—
28#2439—21no—
29#2438—21no—
30#2438—21no—
31#2417C411yescharacteristic
32#2417C411yescharacteristic
33#2409C411yescharacteristic
34#2409C411yescharacteristic
35#2349—21no—
36D4×C4⋊C4C411yescharacteristic
37#2339—21no—
38C4×C22⋊Q8C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41#1985—21no—
42#1870—21no—
43#1869—21no—
44#1817C2211yescharacteristic
45#1817C2211yescharacteristic
46#1817C2211yescharacteristic
47#1817C2211yescharacteristic
48#1798—21no—
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1780—21no—
53#1750—21no—
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57#1728C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1272C2211yescharacteristic
68#1272C2211yescharacteristic
69#1272C2211yescharacteristic
70#1272C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C22×#41D411yescharacteristic
76C23×Q8C2×C411yescharacteristic
77C23×Q8C2×C411yescharacteristic
78C22×M16—21no—
79C23×C8—21no—
80D4×Q8—44no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×C22⋊Q8C2×C411yescharacteristic
88C2×C22⋊Q8C2×C411yescharacteristic
89C2×C22⋊Q8C2×C411yescharacteristic
90C4×#41—44no—
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2×C411yescharacteristic
97C2×C4×D4C2×C411yescharacteristic
98C2×C4×D4C2×C411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4D411yescharacteristic
101C2×C4×D4D411yescharacteristic
102C2×#117D411yescharacteristic
103C2×#117D411yescharacteristic
104C2×#117D411yescharacteristic
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#435—44no—
111#433—44no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×#97—21no—
126C2×#97—21no—
127C2×C4×C8—21no—
128#400—21no—
129#400—21no—
130#398—21no—
131#398—21no—
132#398—21no—
133#398—21no—
134#396—21no—
135#396—21no—
136#395—21no—
137#395—21no—
138#393—21no—
139#393—21no—
140#391—21no—
141#391—21no—
142#391—21no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391C2×C411yescharacteristic
147#391C2×C411yescharacteristic
148#391C2×C411yescharacteristic
149#391C2×C411yescharacteristic
150#391C2×C411yescharacteristic
151#391C2×C411yescharacteristic
152#391C2311yescharacteristic
153#391C2311yescharacteristic
154#391C2311yescharacteristic
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389C2×C411yescharacteristic
160#389C2×C411yescharacteristic
161#389C2311yescharacteristic
162#387—21no—
163#387—21no—
164#387—21no—
165#387—21no—
166#387—21no—
167#386—21no—
168#386—21no—
169#386—21no—
170#386—21no—
171#385—21no—
172#385—21no—
173#385—21no—
174#385—21no—
175#385—21no—
176#385—21no—
177#385—21no—
178#384C2×C411yescharacteristic
179#382—21no—
180#381—21no—
181#381—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380C2×C411yescharacteristic
187#380C2×C411yescharacteristic
188#380C2×C411yescharacteristic
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2×C411yescharacteristic
191C4×C22⋊C4—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336—21no—
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209#336C2311yescharacteristic
210C2×#41—22no—
211C2×#41—22no—
212C2×#41—42no—
213C2×#41—42no—
214C22×Q8—21no—
215C22×Q8—21no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—41no—
219C22×Q8—41no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8—81no—
223C22×Q8—81no—
224C22×Q8C22⋊C411yescharacteristic
225C22×Q8C22×C411yescharacteristic
226C22×Q8C22×C411yescharacteristic
227C22×Q8C22×C411yescharacteristic
228C22×D4C22⋊C411yescharacteristic
229C22×D4C22⋊C411yescharacteristic
230C22×D4C22⋊C411yescharacteristic
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4C22×C411yescharacteristic
236C23×C4C22×C411yescharacteristic
237C23×C4C22×C411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C2×M16—42no—
245C2×M16—42no—
246C2×M16—81no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8—81no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C22⋊Q8—22no—
261C22⋊Q8—44no—
262C22⋊Q8—41no—
263C22⋊Q8—42no—
264C22⋊Q8—44no—
265C22⋊Q8—42no—
266C22⋊Q8—81no—
267C22⋊Q8—81no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×Q8—42no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—44no—
278C4×D4—44no—
279#117—24no—
280#117—42no—
281#117—42no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—81no—
298C2×C4⋊C4—81no—
299C2×C4⋊C4C22⋊C411yescharacteristic
300C2×C4⋊C4C22⋊C411yescharacteristic
301C2×C4⋊C4C22⋊C411yescharacteristic
302C2×C4⋊C4C22⋊C411yescharacteristic
303C2×C4⋊C4C22⋊C411yescharacteristic
304C2×C4⋊C4C22⋊C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C411yescharacteristic
309C2×C4⋊C4C22×C411yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C22⋊C4C22⋊C411yescharacteristic
321C2×C22⋊C4C22⋊C411yescharacteristic
322C2×C22⋊C4C22⋊C411yescharacteristic
323C2×C22⋊C4C22⋊C411yescharacteristic
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C22×C411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2411yescharacteristic
339C4⋊C8—42no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—81no—
352#97—42no—
353#97—42no—
354C4×C8—42no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380C24C2×C22⋊C411yescharacteristic
381C24C2×C22⋊C411yescharacteristic
382C24C2×C22⋊C411yescharacteristic
383#41—48no—
384#41—48no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×Q8—42no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8—84no—
406C2×Q8—84no—
407C2×Q8—84no—
408C2×Q8—84no—
409C2×Q8#13112yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—41no—
418C2×D4—41no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4C2×C22⋊C411yescharacteristic
474C22×C4C2×C22⋊C411yescharacteristic
475C22×C4C2×C22⋊C411yescharacteristic
476C22×C4C2×C22⋊C411yescharacteristic
477C22×C4C2×C22⋊C411yescharacteristic
478C22×C4C2×C22⋊C411yescharacteristic
479C22×C4C2×C22⋊C411yescharacteristic
480C22×C4C2×C22⋊C411yescharacteristic
481C22×C4C2×C22⋊C411yescharacteristic
482C22×C4#13712yes—
483C22×C4#13712yes—
484C22×C4C23×C411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487M16—84no—
488C2×C8—22no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—84no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—42no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C4⋊C4—81no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—81no—
552C22⋊C4—81no—
553C22⋊C4—81no—
554C22⋊C4#9914yes—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—42no—
566C42—42no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23C2×#9912yes—
586C23#41912yes—
587C23#42112yes—
588C23C22×C22⋊C411yescharacteristic
589Q8—42no—
590Q8—42no—
591Q8—44no—
592Q8—84no—
593Q8—81no—
594Q8—81no—
595Q8—81no—
596Q8—81no—
597Q8—84no—
598D4—28no—
599D4—44no—
600D4—44no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—81no—
672C2×C4—81no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—162no—
676C2×C4—162no—
677C2×C4C2×#9912yes—
678C2×C4C2×#9912yes—
679C2×C4#41912yes—
680C2×C4C2×#13111yescharacteristic
681C2×C4#56811yescharacteristic
682C2×C4C2×#13711yescharacteristic
683C2×C4C2×#13711yescharacteristic
684C2×C4#57711yescharacteristic
685C2×C4#57811yescharacteristic
686C8—42no—
687C8—81no—
688C8—81no—
689C8—81no—
690C8—81no—
691C8—81no—
692C8—81no—
693C8—81no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22#288011yescharacteristic
716C22C22×#9911yescharacteristic
717C22#288811yescharacteristic
718C22C2×#41911yescharacteristic
719C22#289811yescharacteristic
720C22C2×#42111yescharacteristic
721C22#290111yescharacteristic
722C4—21no—
723C4—21no—
724C4—22no—
725C4—21no—
726C4—21no—
727C4—22no—
728C4—21no—
729C4—21no—
730C4—22no—
731C4—41no—
732C4—41no—
733C4—42no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—81no—
739C4—81no—
740C4—81no—
741C4—81no—
742C4—81no—
743C4—81no—
744C4—81no—
745C2—42no—
746C2—42no—
747C2—42no—
748C2#1741111yescharacteristic
749C2#1751411yescharacteristic
750C2#1752411yescharacteristic
751C2#1757811yescharacteristic
752C2#1760811yescharacteristic
753C2#1767911yescharacteristic
754C2#1776011yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7727C2semidirect
5. #7721C2semidirect
6. #7701C2semidirect
7. #7695C2semidirect
8. #7675C2semidirect
9. #7667C2semidirect
10. #7648C2semidirect
11. #7629C2semidirect
12. #7609C2semidirect
13. #7595C2semidirect
14. #7589C2semidirect
15. #7575C2semidirect

Maximal subgroups

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

subgroupindex
#362742
#224712
#207202
#77272
#77212
#77012
#76952
#76752
#76672
#76482
#76292
#76092
#75952
#75892
#75752