Finite Group Databasebeta

← #208632

Subgroups — #208632

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35315C211yesmaximal, characteristic
2#20779C211yesmaximal, characteristic
3#13805C211yesmaximal, characteristic
4#9913C211yesmaximal, characteristic
5#9910C211yesmaximal, characteristic
6#9714C211yesmaximal, characteristic
7#9711C211yesmaximal, characteristic
8#9152C211yesmaximal, characteristic
9#9152C211yesmaximal, characteristic
10#9145C211yesmaximal, characteristic
11#9145C211yesmaximal, characteristic
12#8221C211yesmaximal, characteristic
13#8219C211yesmaximal, characteristic
14#8217C211yesmaximal, characteristic
15C4×#328C211yesmaximal, characteristic
16#2857C2211yescharacteristic
17#2855C2211yescharacteristic
18#2855C2211yescharacteristic
19#2851C2211yescharacteristic
20#2647C411yescharacteristic, complemented
21#2645C411yescharacteristic, complemented
22#2645C411yescharacteristic, complemented
23#2642C411yescharacteristic, complemented
24#2533—21no—
25#2533—21no—
26#2528—21no—
27#2528—21no—
28#2505—21no—
29#2505—21no—
30#2493—21no—
31#2493—21no—
32#2330C411yescharacteristic, complemented
33#2327C411yescharacteristic, complemented
34#2320C411yescharacteristic, complemented
35#2320C411yescharacteristic, complemented
36C4×C22⋊Q8C2211yescharacteristic
37C4×C22⋊Q8C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#376C2211yescharacteristic
40C2×#376C2211yescharacteristic
41#1840—21no—
42#1832—21no—
43#1817C2211yescharacteristic
44#1817C2211yescharacteristic
45#1797C2211yescharacteristic
46#1797C2211yescharacteristic
47#1768—21no—
48#1749C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1746—21no—
51C2×#328C2211yescharacteristic
52C2×#328C2211yescharacteristic
53C2×#328C2211yescharacteristic
54C2×#328C2211yescharacteristic
55C2×#328C2211yescharacteristic
56C2×#328C2211yescharacteristic
57C2×#328C2211yescharacteristic
58C2×#328C2211yescharacteristic
59#1436—21no—
60#1436—21no—
61#1434—21no—
62#1434C2211yescharacteristic
63#1434C2211yescharacteristic
64C4×#95—21no—
65C4×#95C2211yescharacteristic
66C4×#95C2211yescharacteristic
67#1294C2211yescharacteristic
68#1294C2211yescharacteristic
69#1294C2211yescharacteristic
70#1294C2211yescharacteristic
71#1294C2211yescharacteristic
72#1294C2211yescharacteristic
73#1294C2211yescharacteristic
74#1294C2211yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#125C2×C411yescharacteristic
77C2×C22⋊Q8C2311yescharacteristic
78C2×C22⋊Q8C2311yescharacteristic
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C22×C4⋊C4—21no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C22⋊C8C2311yescharacteristic
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2311yescharacteristic
93C2×C22⋊C8C2311yescharacteristic
94C2×#97—21no—
95C2×C4×C8—21no—
96#401—21no—
97#401—21no—
98#400—21no—
99#400—21no—
100#400—21no—
101#400—21no—
102#400—21no—
103#400—21no—
104#400—21no—
105#400—21no—
106#400—21no—
107#400—21no—
108#400—21no—
109#400—21no—
110#398—21no—
111#398—21no—
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#385—21no—
151#385—21no—
152#385—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#384C2×C411yescharacteristic
157#383—21no—
158#383—21no—
159#382—21no—
160#382—21no—
161#382—22no—
162#382—22no—
163#382C2×C411yescharacteristic
164#381—21no—
165#381—21no—
166#380—21no—
167#380—21no—
168#380C2311yescharacteristic
169#380C2311yescharacteristic
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176#376—22no—
177#376—22no—
178#376C2×C411yescharacteristic
179#376C2×C411yescharacteristic
180#376C2×C411yescharacteristic
181#376C2×C411yescharacteristic
182#376D411yescharacteristic
183#376D411yescharacteristic
184#376D411yescharacteristic
185#376D411yescharacteristic
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95D411yescharacteristic
195C2×#95D411yescharacteristic
196C2×#95D411yescharacteristic
197C2×#95D411yescharacteristic
198C2×#95C2311yescharacteristic
199C2×#95C2311yescharacteristic
200C2×#95C2311yescharacteristic
201C2×#95C2311yescharacteristic
202#336—21no—
203#336—21no—
204#336—21no—
205#336—21no—
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#328—22no—
211#328—22no—
212#328—22no—
213#328—22no—
214#328—22no—
215#328—22no—
216#328—22no—
217#328—22no—
218C22×Q8C22×C411yescharacteristic
219C22×Q8C22×C411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C4⋊Q8—42no—
236#125—42no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C4×Q8—42no—
246C4×Q8—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—81no—
267C2×C4⋊C4—81no—
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—24no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—81no—
296C2×C42—81no—
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C4⋊C8—42no—
310C4⋊C8—42no—
311C22⋊C8—21no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319#97—42no—
320C4×C8—42no—
321#95—24no—
322#95—24no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—42no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—81no—
362#95—81no—
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365#95C22⋊C411yescharacteristic
366#95C22⋊C411yescharacteristic
367#95C22⋊C411yescharacteristic
368#95C22⋊C411yescharacteristic
369#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371#95C22⋊C411yescharacteristic
372#95C22⋊C411yescharacteristic
373#95C22⋊C411yescharacteristic
374#95C22⋊C411yescharacteristic
375#95C22⋊C411yescharacteristic
376#95C22⋊C411yescharacteristic
377#95C22⋊C411yescharacteristic
378#95C22⋊C411yescharacteristic
379#95#4112yes—
380#95#4112yes—
381#95#4112yes—
382#95#4112yes—
383C24C22×D411yescharacteristic
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—41no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473C22×C4C2×#4111yescharacteristic
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—84no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—48no—
540C42—42no—
541C42—42no—
542C42—81no—
543C42—81no—
544C42—84no—
545C42—84no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23C22×C22⋊C411yescharacteristic
551C23#52511yescharacteristic
552C23#52511yescharacteristic
553C23#57711yescharacteristic
554C23#57711yescharacteristic
555C23#57811yescharacteristic
556C23#57811yescharacteristic
557Q8—81no—
558Q8—81no—
559Q8—81no—
560Q8—81no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—81no—
668C2×C4—81no—
669C2×C4#41012yes—
670C2×C4#41012yes—
671C2×C4C4×#4111yescharacteristic
672C2×C4C4×#4111yescharacteristic
673C2×C4#53211yescharacteristic
674C2×C4#53211yescharacteristic
675C8—81no—
676C8—81no—
677C8—81no—
678C8—81no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22#229711yescharacteristic
692C22C2×#41011yescharacteristic
693C22C2×#41011yescharacteristic
694C22#289811yescharacteristic
695C22#289811yescharacteristic
696C22#290111yescharacteristic
697C22#290111yescharacteristic
698C22#312011yescharacteristic
699C22#312011yescharacteristic
700C22#312311yescharacteristic
701C22#312711yescharacteristic
702C4—21no—
703C4—21no—
704C4—21no—
705C4—21no—
706C4—22no—
707C4—22no—
708C4—44no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—81no—
723C4—81no—
724C4—81no—
725C4—81no—
726C4—81no—
727C4—81no—
728C4—81no—
729C4—81no—
730C2—21no—
731C2—21no—
732C2—21no—
733C2—21no—
734C2#1760511yescharacteristic
735C2#1760511yescharacteristic
736C2#1760911yescharacteristic
737C2#1760911yescharacteristic
738C2#2139811yescharacteristic
739C2#2141911yescharacteristic
740C2#2142111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #2647C4semidirect
21. #2645C4semidirect
22. #2645C4semidirect
23. #2642C4semidirect
32. #2330C4semidirect
33. #2327C4semidirect
34. #2320C4semidirect
35. #2320C4semidirect

Maximal subgroups

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

subgroupindex
#353152
#207792
#138052
#99132
#99102
#97142
#97112
#91522
#91452
#82212
#82192
#82172
C4×#3282