Finite Group Databasebeta

← #189920

Subgroups — #189920

Order 512.

951 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34191C211yesmaximal, characteristic
2#21380C211yesmaximal, characteristic
3#20719C211yesmaximal, characteristic
4#8722C211yesmaximal, characteristic, complemented
5#8721C211yesmaximal, characteristic, complemented
6#8720C211yesmaximal, characteristic, complemented
7#8720C211yesmaximal, characteristic, complemented
8#8720C211yesmaximal, characteristic, complemented
9#8720C211yesmaximal, characteristic, complemented
10#8719C211yesmaximal, characteristic, complemented
11#8718C211yesmaximal, characteristic, complemented
12#8035C211yesmaximal, characteristic, complemented
13#8033C211yesmaximal, characteristic, complemented
14#8032C211yesmaximal, characteristic, complemented
15#8030C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2355C411yescharacteristic
22#2355C411yescharacteristic
23#2352C411yescharacteristic
24#2352C411yescharacteristic
25#2352C411yescharacteristic
26#2352C411yescharacteristic
27D4×C4⋊C4C411yescharacteristic
28D4×C4⋊C4C411yescharacteristic
29#2346—21no—
30#2343—21no—
31#2343—21no—
32#2339—21no—
33#2320—21no—
34#2318—21no—
35#2318—21no—
36#2285C2211yescharacteristic
37#2283C2211yescharacteristic
38#2283C2211yescharacteristic
39#2283C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42#2275—21no—
43#1984—21no—
44#1954—21no—
45#1870—21no—
46#1870—21no—
47#1818—21no—
48#1798—21no—
49#1798—21no—
50#1780—21no—
51#1749C2211yescharacteristic
52#1749C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1749C2211yescharacteristic
57#1749C2211yescharacteristic
58#1749C2211yescharacteristic
59#1295C2211yescharacteristic
60#1295C2211yescharacteristic
61#1295C2211yescharacteristic
62#1295C2211yescharacteristic
63#1295C2211yescharacteristic
64#1295C2211yescharacteristic
65#1295C2211yescharacteristic
66#1295C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75C22×#41D411yescharacteristic
76C22×M16—21no—
77C4×#41—22no—
78C4×#41—22no—
79C4×#41—22no—
80C4×#41—22no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8D411yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C2×C4×D4D411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×#117C2×C411yescharacteristic
92C2×#117C2×C411yescharacteristic
93C2×#117D411yescharacteristic
94C2×#117D411yescharacteristic
95C2×#117D411yescharacteristic
96C22×C4⋊C4C2×C411yescharacteristic
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4C2×C411yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103#432—22no—
104#432—22no—
105#432—22no—
106#432—22no—
107#432—22no—
108#432—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊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—
123C4×M16—22no—
124C4×M16—22no—
125C2×#97C2311yescharacteristic
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128C2×#97C2311yescharacteristic
129C2×#97C2311yescharacteristic
130C2×#97C2311yescharacteristic
131C2×#97C2311yescharacteristic
132C2×#97C2311yescharacteristic
133C2×C4×C8—21no—
134#391—21no—
135#391—21no—
136#389—21no—
137#389—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—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#386—21no—
151#386—21no—
152#386—21no—
153#386—21no—
154#385—21no—
155#385—21no—
156#385—21no—
157#385—21no—
158#384C2×C411yescharacteristic
159#384C2×C411yescharacteristic
160#383—21no—
161#383—21no—
162#383—21no—
163#382—21no—
164#382—21no—
165#382—21no—
166#382—21no—
167#382C2×C411yescharacteristic
168#382C2×C411yescharacteristic
169#382C2×C411yescharacteristic
170#382C2×C411yescharacteristic
171#382C2311yescharacteristic
172#382C2311yescharacteristic
173#382C2311yescharacteristic
174#382C2311yescharacteristic
175#381—21no—
176#381—21no—
177#381—21no—
178#380—21no—
179#380—21no—
180#380C2×C411yescharacteristic
181#380C2×C411yescharacteristic
182#380C2×C411yescharacteristic
183#380C2×C411yescharacteristic
184#380C2×C411yescharacteristic
185#380C2×C411yescharacteristic
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2×C411yescharacteristic
191C4×C4⋊C4C2×C411yescharacteristic
192C4×C4⋊C4C2×C411yescharacteristic
193C4×C4⋊C4C2×C411yescharacteristic
194C4×C4⋊C4C2×C411yescharacteristic
195C4×C4⋊C4C2×C411yescharacteristic
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203#376—21no—
204#376—21no—
205#376—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#336—21no—
215#336—21no—
216#336—21no—
217#336—21no—
218C2×#41—22no—
219C2×#41—22no—
220C2×#41—42no—
221C2×#41—42no—
222C22×Q8C22⋊C411yescharacteristic
223C22×D4C22⋊C411yescharacteristic
224C22×D4C22⋊C411yescharacteristic
225C22×D4C22⋊C411yescharacteristic
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4C22×C411yescharacteristic
231C23×C4C22×C411yescharacteristic
232C23×C4C22×C411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C2×M16—42no—
240C2×M16—42no—
241C2×M16—42no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—44no—
267C4×D4—44no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—22no—
274#117—81no—
275#117—81no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—81no—
285C2×C4⋊C4—81no—
286C2×C4⋊C4—81no—
287C2×C4⋊C4—81no—
288C2×C4⋊C4C22⋊C411yescharacteristic
289C2×C4⋊C4C22⋊C411yescharacteristic
290C2×C4⋊C4C22⋊C411yescharacteristic
291C2×C4⋊C4C22⋊C411yescharacteristic
292C2×C4⋊C4C22⋊C411yescharacteristic
293C2×C4⋊C4C22⋊C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C22⋊C4C22⋊C411yescharacteristic
313C2×C22⋊C4C22⋊C411yescharacteristic
314C2×C22⋊C4C22⋊C411yescharacteristic
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42C22×C411yescharacteristic
333C2×C42C22×C411yescharacteristic
334C2×C42C22×C411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42#4112yes—
342C2×C42#4112yes—
343C2×C42#4112yes—
344C2×C42#4112yes—
345C2×C42C2411yescharacteristic
346C4⋊C8—42no—
347C4⋊C8—42no—
348C4⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C22⋊C8—42no—
353C22⋊C8—42no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360#97—21no—
361#97—21no—
362#97—21no—
363#97—21no—
364#97—21no—
365#97—21no—
366#97—21no—
367#97—21no—
368#97—21no—
369#97—21no—
370#97—21no—
371#97—21no—
372#97—21no—
373#97—21no—
374#97—21no—
375#97—21no—
376C4×C8—42no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405C24C2×C22⋊C411yescharacteristic
406C24C2×C22⋊C411yescharacteristic
407C24C2×C22⋊C411yescharacteristic
408#41—48no—
409#41—48no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—41no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
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×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4C2×C22⋊C411yescharacteristic
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×C4#11712yes—
482C22×C4#11712yes—
483C22×C4#12312yes—
484C22×C4#12312yes—
485C22×C4#12312yes—
486C22×C4#12312yes—
487C22×C4C23×C411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490M16—48no—
491C2×C8—22no—
492C2×C8—22no—
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—41no—
513C2×C8—41no—
514C2×C8—41no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—81no—
533C4⋊C4—81no—
534C4⋊C4—81no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C22⋊C4—81no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—42no—
564C42—42no—
565C42—81no—
566C42—81no—
567C42—81no—
568C42—81no—
569C42—81no—
570C42—81no—
571C42#11712yes—
572C42#11712yes—
573C42#12312yes—
574C42#12312yes—
575C42#12312yes—
576C42#12312yes—
577C42C2×#4111yescharacteristic
578C42C2×#4111yescharacteristic
579C42C2×#4111yescharacteristic
580C42C2×#4111yescharacteristic
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23#38112yes—
601C23#42112yes—
602C23C22×C22⋊C411yescharacteristic
603Q8—44no—
604D4—28no—
605D4—44no—
606D4—44no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
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—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
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—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
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—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—81no—
675C2×C4—81no—
676C2×C4—81no—
677C2×C4—81no—
678C2×C4—162no—
679C2×C4—162no—
680C2×C4—162no—
681C2×C4—162no—
682C2×C4#38112yes—
683C2×C4#38112yes—
684C2×C4#38112yes—
685C2×C4#42112yes—
686C2×C4C2×#11711yescharacteristic
687C2×C4C2×#11711yescharacteristic
688C2×C4C2×#12311yescharacteristic
689C2×C4C2×#12311yescharacteristic
690C2×C4C2×#12311yescharacteristic
691C2×C4C2×#12311yescharacteristic
692C8—42no—
693C8—42no—
694C8—81no—
695C8—81no—
696C8—81no—
697C8—81no—
698C8—81no—
699C8—81no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22C2×#38111yescharacteristic
722C22#288811yescharacteristic
723C22#288911yescharacteristic
724C22C2×#42111yescharacteristic
725C22C2×#42111yescharacteristic
726C22#290111yescharacteristic
727C22#290111yescharacteristic
728C4—21no—
729C4—21no—
730C4—22no—
731C4—21no—
732C4—21no—
733C4—22no—
734C4—22no—
735C4—21no—
736C4—21no—
737C4—22no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—81no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4—81no—
748C4—81no—
749C4—81no—
750C4—81no—
751C2—42no—
752C2—42no—
753C2—81no—
754C2#1767911yescharacteristic
755C2#1768011yescharacteristic
756C2#1768111yescharacteristic
757C2#1768211yescharacteristic
758C2#2069511yescharacteristic
759C2#2070511yescharacteristic
760C2#2070611yescharacteristic
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. #8722C2semidirect
5. #8721C2semidirect
6. #8720C2semidirect
7. #8720C2semidirect
8. #8720C2semidirect
9. #8720C2semidirect
10. #8719C2semidirect
11. #8718C2semidirect
12. #8035C2semidirect
13. #8033C2semidirect
14. #8032C2semidirect
15. #8030C2semidirect

Maximal subgroups

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

subgroupindex
#341912
#213802
#207192
#87222
#87212
#87202
#87192
#87182
#80352
#80332
#80322
#80302