Finite Group Databasebeta

← #223868

Subgroups — #223868

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36164C211yesmaximal, characteristic
2#21271C211yesmaximal, characteristic, complemented
3#21108C211yesmaximal, characteristic, complemented
4#9895C211yesmaximal, characteristic, complemented
5#9890C211yesmaximal, characteristic, complemented
6#9124C211yesmaximal, characteristic, complemented
7#9124C211yesmaximal, characteristic, complemented
8#9123C211yesmaximal, characteristic, complemented
9#9120C211yesmaximal, characteristic, complemented
10#8719C211yesmaximal, characteristic, complemented
11#8718C211yesmaximal, characteristic, complemented
12#8581C211yesmaximal, characteristic, complemented
13#8226C211yesmaximal, characteristic, complemented
14#8033C211yesmaximal, characteristic, complemented
15#8030C211yesmaximal, characteristic, complemented
16C2×#554C411yescharacteristic
17C2×#547C411yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2862C2211yescharacteristic
23#2850C2211yescharacteristic
24#2850C2211yescharacteristic
25#2568C411yescharacteristic
26#2563C411yescharacteristic
27#2504C411yescharacteristic
28#2504C411yescharacteristic
29#2503C411yescharacteristic
30#2503C411yescharacteristic
31#2489—21no—
32#2489—21no—
33#2488—21no—
34#2488—21no—
35#2488—21no—
36#2488—21no—
37#2339—21no—
38#2339—21no—
39C4×#124C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C42×D4C2211yescharacteristic
43#1955—21no—
44#1940—21no—
45#1923—21no—
46#1923—21no—
47#1923—21no—
48#1916—21no—
49#1870—21no—
50#1851—21no—
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1749C2211yescharacteristic
57#1745C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#329C2211yescharacteristic
60C2×#329C2211yescharacteristic
61C2×#329C2211yescharacteristic
62C2×#329C2211yescharacteristic
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×#41—21no—
76C22×M16—21no—
77#554—44no—
78#547—44no—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#125—21no—
82C2×#124C2311yescharacteristic
83C2×#123—21no—
84C2×#123—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8C2×C411yescharacteristic
89C2×C22⋊Q8C2×C411yescharacteristic
90C2×#121C2×C411yescharacteristic
91C2×#121C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4C2×C411yescharacteristic
95C2×C4×D4C2×C411yescharacteristic
96C2×C4×D4D411yescharacteristic
97C2×C4×D4D411yescharacteristic
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4C2×C411yescharacteristic
100C22×C4⋊C4C2×C411yescharacteristic
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#432—22no—
105#432—22no—
106#432—22no—
107#432—41no—
108#432—41no—
109#432—41no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
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—
122C4×M16—22no—
123C4×M16—41no—
124C2×#97—21no—
125C2×#97C2311yescharacteristic
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128C2×#97C2311yescharacteristic
129C2×#97C2311yescharacteristic
130C2×#97C2311yescharacteristic
131C2×#97C2311yescharacteristic
132C2×#97C2311yescharacteristic
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135#400—21no—
136#400—21no—
137#400—21no—
138#400—21no—
139#400—21no—
140#400—21no—
141#397—21no—
142#397—21no—
143#397—21no—
144#397—21no—
145#393—21no—
146#393—21no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391C2×C411yescharacteristic
154#391C2×C411yescharacteristic
155#391C2×C411yescharacteristic
156#390C2×C411yescharacteristic
157#390C2×C411yescharacteristic
158#390C2×C411yescharacteristic
159#388—21no—
160#388—21no—
161#388—21no—
162#388—21no—
163#388—21no—
164#388—21no—
165#386—21no—
166#386—21no—
167#386—21no—
168#386—21no—
169#386—21no—
170#386—21no—
171#386—21no—
172#386—21no—
173#382—21no—
174#382—21no—
175#382—21no—
176#382—21no—
177#382C2×C411yescharacteristic
178#382C2×C411yescharacteristic
179#382C2×C411yescharacteristic
180#382C2×C411yescharacteristic
181#382C2×C411yescharacteristic
182#382C2×C411yescharacteristic
183#382C2311yescharacteristic
184#382C2311yescharacteristic
185#382C2311yescharacteristic
186#381—21no—
187#381—21no—
188#381—21no—
189#381—21no—
190#380—21no—
191#380—21no—
192C4×C4⋊C4C2×C411yescharacteristic
193C4×C4⋊C4D411yescharacteristic
194C4×C4⋊C4D411yescharacteristic
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4C2×C411yescharacteristic
198C4×C22⋊C4C2×C411yescharacteristic
199C4×C22⋊C4D411yescharacteristic
200C4×C22⋊C4D411yescharacteristic
201C4×C22⋊C4D411yescharacteristic
202C4×C22⋊C4D411yescharacteristic
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C43C2311yescharacteristic
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#336—21no—
215#336—21no—
216#329—22no—
217#329—22no—
218#329—41no—
219#329—41no—
220C2×#41—44no—
221C2×#41—44no—
222C2×#41—44no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8C22×C411yescharacteristic
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C22⋊C411yescharacteristic
229C22×D4C22×C411yescharacteristic
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4C22×C411yescharacteristic
236C23×C4C22×C411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C2×M16—42no—
242C2×M16—42no—
243C2×M16—42no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C4⋊Q8—42no—
253C4⋊Q8—42no—
254#125—44no—
255#124—21no—
256#124—21no—
257#124—21no—
258#124—21no—
259#123—44no—
260#123—44no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C22⋊Q8—42no—
264C22⋊Q8—44no—
265C22⋊Q8—42no—
266#121—42no—
267#121—42no—
268C4×Q8—42no—
269C4×D4—24no—
270C4×D4—42no—
271C4×D4—42no—
272C4×D4—42no—
273C4×D4—44no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
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—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
310C2×C22⋊C4C22⋊C411yescharacteristic
311C2×C22⋊C4C22⋊C411yescharacteristic
312C2×C22⋊C4C22⋊C411yescharacteristic
313C2×C22⋊C4C22⋊C411yescharacteristic
314C2×C22⋊C4C22⋊C411yescharacteristic
315C2×C42—21no—
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×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2411yescharacteristic
337C4⋊C8—42no—
338C4⋊C8—42no—
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—
349#97—21no—
350#97—21no—
351#97—21no—
352#97—21no—
353#97—21no—
354#97—21no—
355#97—21no—
356#97—21no—
357#97—21no—
358#97—21no—
359#97—21no—
360#97—21no—
361#97—21no—
362#97—21no—
363#97—42no—
364#97#4112yescomplemented
365#97#4112yescomplemented
366C4×C8—42no—
367C4×C8—42no—
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—
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#95C22⋊C411yescharacteristic
393#95C22⋊C411yescharacteristic
394#95C22⋊C411yescharacteristic
395#95C22⋊C411yescharacteristic
396#95C22⋊C411yescharacteristic
397#95C22⋊C411yescharacteristic
398C24—21no—
399C24C2×C22⋊C411yescharacteristic
400C24C2×C22⋊C411yescharacteristic
401#41—816no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
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—21no—
445C22×C4—21no—
446C22×C4—21no—
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—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484M16—48no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
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—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—81no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—81no—
546C22⋊C4—81no—
547C22⋊C4#10014yes—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—22no—
561C42—22no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—42no—
567C42C2×#4111yescharacteristic
568C42C2×#4111yescharacteristic
569C42H(5,𝔽2)11yescharacteristic
570C422−1+411yescharacteristic
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—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23C2×#10012yes—
591C23#42112yes—
592C23C22×C22⋊C411yescharacteristic
593Q8—81no—
594Q8—81no—
595Q8—84no—
596Q8—162no—
597D4—28no—
598D4—81no—
599D4—81no—
600D4—84no—
601D4—84no—
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—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—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
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—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—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—81no—
666C2×C4—81no—
667C2×C4—81no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—162no—
672C2×C4—162no—
673C2×C4C2×#10012yes—
674C2×C4C2×#10012yes—
675C2×C4#42112yes—
676C2×C4C4×#4111yescharacteristic
677C2×C4#52011yescharacteristic
678C2×C4#54611yescharacteristic
679C2×C4#54611yescharacteristic
680C2×C4#54711yescharacteristic
681C2×C4#54711yescharacteristic
682C8—42no—
683C8—42no—
684C8—42no—
685C8—42no—
686C8—81no—
687C8—81no—
688C8—81no—
689C8—81no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22#233911yescharacteristic
712C22C22×#10011yescharacteristic
713C22#288911yescharacteristic
714C22C2×#42111yescharacteristic
715C22C2×#42111yescharacteristic
716C22#290111yescharacteristic
717C22#290111yescharacteristic
718C4—22no—
719C4—22no—
720C4—21no—
721C4—21no—
722C4—22no—
723C4—21no—
724C4—21no—
725C4—21no—
726C4—21no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—81no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—81no—
739C4—81no—
740C4—81no—
741C2—42no—
742C2—42no—
743C2—161no—
744C2#1766611yescharacteristic
745C2#1768011yescharacteristic
746C2#1768211yescharacteristic
747C2#1768511yescharacteristic
748C2#2247911yescharacteristic
749C2#2250611yescharacteristic
750C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21271C2semidirect
3. #21108C2semidirect
4. #9895C2semidirect
5. #9890C2semidirect
6. #9124C2semidirect
7. #9124C2semidirect
8. #9123C2semidirect
9. #9120C2semidirect
10. #8719C2semidirect
11. #8718C2semidirect
12. #8581C2semidirect
13. #8226C2semidirect
14. #8033C2semidirect
15. #8030C2semidirect
364. #97#41semidirect
365. #97#41semidirect

Maximal subgroups

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

subgroupindex
#361642
#212712
#211082
#98952
#98902
#91242
#91232
#91202
#87192
#87182
#85812
#82262
#80332
#80302