Finite Group Databasebeta

← #188513

Subgroups — #188513

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34204C211yesmaximal, characteristic
2#20628C211yesmaximal, characteristic, complemented
3C2×#1519C211yesmaximal, characteristic
4C2×#1507C211yesmaximal, characteristic
5#13951C211yesmaximal, characteristic, complemented
6#8577C212yesmaximal, complemented
7#8575C212yesmaximal, complemented
8#8573C212yesmaximal, complemented
9#8571C212yesmaximal, complemented
10#8016C211yesmaximal, characteristic, complemented
11#8012C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C2×#518C411yescharacteristic, complemented
14C22×#117C2211yescharacteristic
15C2×#424—21no—
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2363—21no—
19#2363—21no—
20#2361—21no—
21#2361—21no—
22#2342—22no—
23D4×C22⋊C4—22no—
24#2324C412yescomplemented
25#2324C412yescomplemented
26#2316C411yescharacteristic, complemented
27#2309C411yescharacteristic, complemented
28C4×#121C2212yes—
29C4×#121C2212yes—
30C2×#389—21no—
31#2280C2211yescharacteristic
32C2×#380C2211yescharacteristic
33#2274C411yescharacteristic, complemented
34C2×C4×C22⋊C4C2211yescharacteristic
35#1798—22no—
36#1797C2212yescomplemented
37#1797C2212yescomplemented
38#1729—21no—
39#1519—24no—
40#1507—24no—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45#1439—22no—
46C4⋊#95—21no—
47C4⋊#95C2212yescomplemented
48#1434C2212yescomplemented
49#1294C2211yescharacteristic, complemented
50#1294C2211yescharacteristic, complemented
51#1294C2211yescharacteristic, complemented
52#1294C2211yescharacteristic, complemented
53#1294C2212yescomplemented
54#1294C2212yescomplemented
55C23×D4—21no—
56C23×D4C2×C411yescharacteristic
57C24×C4C2311yescharacteristic
58C22×M16—21no—
59C23×C8—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121C2×C412yes—
65C2×#121D414yes—
66C2×#121Q814yes—
67C2×#121C2312yes—
68#518—44no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C412yes—
74C2×C4×D4C2×C412yes—
75C2×#117—44no—
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C412yes—
81C2×#117D411yescharacteristic
82C2×#117D412yes—
83C2×#117D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4D412yes—
94C22×C42—21no—
95C22×C42C2312yes—
96#424—44no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99#407—22no—
100#407—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8C2311yescharacteristic
106C2×C22⋊C8C2311yescharacteristic
107C2×C22⋊C8C2312yes—
108#390—21no—
109#390—21no—
110#390—22no—
111#390—22no—
112#390C2×C412yes—
113#390C2×C412yes—
114#389—21no—
115#389—21no—
116#389—44no—
117#389C2×C412yes—
118#389C2×C412yes—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#385—22no—
128#385—22no—
129#381—21no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380C2×C411yescharacteristic
134#380C2×C411yescharacteristic
135#380C2311yescharacteristic
136#380C2311yescharacteristic
137#379—21no—
138#379—22no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2×C412yes—
142C4×C4⋊C4C2×C412yes—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156#336—22no—
157#336—22no—
158#327—24no—
159#327—24no—
160#327—24no—
161#327—24no—
162C25—21no—
163C25C2×D411yescharacteristic
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—81no—
173C22×D4—81no—
174C22×D4—81no—
175C22×D4—81no—
176C22×D4C4⋊C414yescomplemented
177C22×D4C22×C412yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×Q812yes—
195C23×C4#4112yes—
196C23×C4#4112yes—
197C23×C4C2411yescharacteristic
198C2×M16—42no—
199C2×M16—42no—
200C2×M16—42no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207#121—24no—
208#121—24no—
209#121—24no—
210#121—24no—
211#121—24no—
212#121—22no—
213#121—22no—
214#121C4⋊C418yescomplemented
215#121C2×D414yes—
216#121C2×Q814yes—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—82no—
222C4×D4—82no—
223C4×D4—82no—
224C4×D4—82no—
225#117—24no—
226#117—24no—
227#117—44no—
228#117—44no—
229#117—42no—
230#117—42no—
231#117—42no—
232#117—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C412yes—
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C412yes—
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C4⋊C414yescomplemented
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—81no—
280C2×C22⋊C4—81no—
281C2×C22⋊C4C22⋊C412yes—
282C2×C22⋊C4C22⋊C412yes—
283C2×C22⋊C4C22⋊C412yes—
284C2×C22⋊C4C22⋊C412yes—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22×C412yes—
294C2×C42C22×C412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C4⋊C8—44no—
298C4⋊C8—44no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—22no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—22no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—22no—
318#95—21no—
319#95—21no—
320#95—22no—
321#95—44no—
322#95—44no—
323#95—82no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—24no—
330C24—41no—
331C24—41no—
332C24—41no—
333C24—41no—
334C24—42no—
335C24—42no—
336C24—81no—
337C24—81no—
338C24—81no—
339C24—81no—
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24#12112yes—
343C24C22⋊Q812yes—
344C24C22×D411yescharacteristic
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—28no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4C2×C4⋊C414yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C4⋊C412yes—
433C22×C4C2×C4⋊C412yes—
434C22×C4#11712yes—
435C22×C4#11712yes—
436C22×C4#12112yes—
437C22×C4#12112yes—
438C22×C4#12112yes—
439C22×C4C22⋊Q812yes—
440C22×C4C22⋊Q812yes—
441C22×C4C22⋊Q812yes—
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×Q811yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448M16—84no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—42no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—42no—
458C2×C8—84no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—84no—
473C4⋊C4—88no—
474C4⋊C4#9914yes—
475C4⋊C4C2×C4⋊C414yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—28no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—48no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—84no—
499C42—24no—
500C42—24no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—82no—
506C42—82no—
507C42—84no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—81no—
545C23—81no—
546C23—81no—
547C23—81no—
548C23—84no—
549C23—84no—
550C23—162no—
551C23—162no—
552C23—162no—
553C23—162no—
554C23#38012yes—
555C23#38012yes—
556C23C22×C22⋊C411yescharacteristic
557C23C2×#12111yescharacteristic
558C23C2×C22⋊Q811yescharacteristic
559C23#57611yescharacteristic
560C23#57611yescharacteristic
561C23#57711yescharacteristic
562C23#57711yescharacteristic
563D4—44no—
564D4—44no—
565D4—44no—
566D4—44no—
567D4—44no—
568D4—44no—
569D4—48no—
570D4—88no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
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—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—84no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—164no—
644C2×C4—164no—
645C2×C4#38012yes—
646C2×C4#38012yes—
647C2×C4C2×#9914yes—
648C2×C4C2×#9912yes—
649C2×C4#41012yes—
650C2×C4C22×C4⋊C411yescharacteristic
651C2×C4C2×#11711yescharacteristic
652C2×C4C2×#12111yescharacteristic
653C2×C4C2×C22⋊Q811yescharacteristic
654C8—81no—
655C8—81no—
656C8—82no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—168no—
695C22C2×#38011yescharacteristic
696C22C22×#9911yescharacteristic
697C22C2×#41011yescharacteristic
698C22#289711yescharacteristic
699C22#289711yescharacteristic
700C22#290111yescharacteristic
701C22#290111yescharacteristic
702C22#304111yescharacteristic
703C22#304211yescharacteristic
704C22#308111yescharacteristic
705C22#308111yescharacteristic
706C4—21no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—22no—
711C4—24no—
712C4—44no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—82no—
717C4—82no—
718C4—81no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4—162no—
723C4#179914yes—
724C2—21no—
725C2—21no—
726C2—21no—
727C2—21no—
728C2—42no—
729C2—42no—
730C2—84no—
731C2#1760311yescharacteristic
732C2#1760711yescharacteristic
733C2#1761011yescharacteristic
734C2#1761111yescharacteristic
735C2C2×#179911yescharacteristic
736C2#2065011yescharacteristic
737C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20628C2semidirect
5. #13951C2semidirect
6. #8577C2semidirect
7. #8575C2semidirect
8. #8573C2semidirect
9. #8571C2semidirect
10. #8016C2semidirect
11. #8012C2semidirect
13. C2×#518C4semidirect
24. #2324C4semidirect
25. #2324C4semidirect
26. #2316C4semidirect
27. #2309C4semidirect
33. #2274C4semidirect
36. #1797C22semidirect
37. #1797C22semidirect
47. C4⋊#95C22semidirect
48. #1434C22semidirect
49. #1294C22semidirect
50. #1294C22semidirect
51. #1294C22semidirect
52. #1294C22semidirect
53. #1294C22semidirect
54. #1294C22semidirect
176. C22×D4C4⋊C4semidirect
214. #121C4⋊C4semidirect
251. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342042
#206282
C2×#15192
C2×#15072
#139512
#85772
#85752
#85732
#85712
#80162
#80122