Finite Group Databasebeta

← #356160

Subgroups — #356160

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2#20699C211yesmaximal, characteristic
3#20389C211yesmaximal, characteristic
4C2×#1750C211yesmaximal, characteristic
5C2×#1733C211yesmaximal, characteristic
6#10887C211yesmaximal, characteristic, complemented
7#10887C211yesmaximal, characteristic, complemented
8#10883C211yesmaximal, characteristic, complemented
9#10882C211yesmaximal, characteristic, complemented
10#10809C211yesmaximal, characteristic, complemented
11#10809C211yesmaximal, characteristic, complemented
12#10734C211yesmaximal, characteristic
13#10720C211yesmaximal, characteristic
14#10707C211yesmaximal, characteristic, complemented
15#10706C211yesmaximal, characteristic, complemented
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2924—24no—
20#2924C418yescomplemented
21C2×#433—21no—
22C2×#433—21no—
23C2×#432—21no—
24C2×#431—21no—
25C2×#424C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C2×#405C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C22×#97C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2280C2211yescharacteristic
37#1825C412yescomplemented
38#1825C412yescomplemented
39#1824C412yescomplemented
40#1823C412yescomplemented
41#1753—21no—
42#1753—21no—
43#1752—21no—
44C8×C22⋊C4—21no—
45#1750—24no—
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51#1735C2211yescharacteristic
52#1735C2211yescharacteristic
53#1735C2211yescharacteristic
54#1733C418yescomplemented
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62C2×#339C2211yescharacteristic
63#1729C2211yescharacteristic
64#1729C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C24×C4C2311yescharacteristic
68C22×M16—21no—
69C22×M16—22no—
70C22×M16—22no—
71C22×M16—22no—
72C22×M16—41no—
73C22×M16—41no—
74C22×M16C2×C414yes—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#117—21no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—21no—
88C2×#117—22no—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—41no—
92C2×#117—41no—
93C2×#117C2×C414yes—
94C2×#117C2×C414yes—
95C2×#117C2×C411yescharacteristic
96C2×#117C2×C411yescharacteristic
97C2×#117C2×C414yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#433—28no—
111#433—28no—
112#432—28no—
113#431—28no—
114#424—22no—
115#424—22no—
116#424C2×C414yes—
117#424C2×C414yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122#407—22no—
123#407C2×C414yes—
124C2×C22⋊C8—22no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8C2×C412yes—
129C2×C22⋊C8C2×C412yes—
130C2×C22⋊C8C2×C414yes—
131C2×C22⋊C8D411yescharacteristic
132C2×C22⋊C8D411yescharacteristic
133C2×C22⋊C8D411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C2×C22⋊C8D414yes—
136#405—22no—
137#405—22no—
138#405—22no—
139#405—22no—
140#405—41no—
141#405—41no—
142#405C2×C414yes—
143#405C2×C414yes—
144C2×#97—21no—
145C2×#97—21no—
146C2×#97—22no—
147C2×#97—21no—
148C2×#97—21no—
149C2×#97—22no—
150C2×#97—41no—
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8—22no—
154C2×C4×C8—21no—
155C2×C4×C8—21no—
156C2×C4×C8—22no—
157C2×C4×C8—41no—
158#381—21no—
159#380C2×C412yes—
160#380C2×C412yes—
161#379—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164#341—22no—
165#341—22no—
166#341—41no—
167#341—41no—
168#340C2×C412yes—
169#340C2×C412yes—
170#340C2×C412yes—
171#340C2×C412yes—
172#340C2×C412yes—
173#340C2×C412yes—
174#340C2×C412yes—
175#340C2×C412yes—
176#339—22no—
177#339—22no—
178#339—41no—
179#339—41no—
180#339C2×C414yes—
181#339C2×C414yes—
182#339C2×C414yes—
183#339C2×C414yes—
184#336—22no—
185#336—22no—
186#336—21no—
187#336—21no—
188#336—21no—
189#336—21no—
190C25#4111yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4C22×C412yes—
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C412yes—
206C23×C4C22×C411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4#4111yescharacteristic
211C23×C4#4112yes—
212C23×C4C2411yescharacteristic
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—24no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—28no—
233C2×M16—42no—
234C2×M16—42no—
235C2×M16—44no—
236C2×M16—44no—
237C2×M16—44no—
238C2×M16C22×C412yes—
239C2×M16C22×C412yes—
240C2×M16C22×C412yes—
241C2×M16C22×C412yes—
242C2×M16#4112yes—
243C2×M16#4112yes—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—24no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—42no—
270C22×C8—42no—
271C22×C8C22×C412yes—
272C22×C8C22×C412yes—
273C22×C8C22×C412yes—
274C22×C8C22×C412yes—
275C22×C8C2×D412yes—
276C22×C8C2×D412yes—
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D411yescharacteristic
281#117—24no—
282#117—24no—
283#117—24no—
284#117—24no—
285#117—22no—
286#117—22no—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—28no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—21no—
300#117—44no—
301#117—44no—
302#117—44no—
303#117C4214yes—
304#117C4214yes—
305#117C22×C412yes—
306#117C22×C412yes—
307#117C22×C412yes—
308#117C22×C412yes—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4C4214yes—
317C2×C4⋊C4C4214yes—
318C2×C4⋊C4C22×C412yes—
319C2×C4⋊C4C22×C412yes—
320C2×C4⋊C4C22×C411yescharacteristic
321C2×C4⋊C4C22×C411yescharacteristic
322C2×C4⋊C4C22×C411yescharacteristic
323C2×C4⋊C4C22×C411yescharacteristic
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4—42no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42C22×C412yes—
354C2×C42C22×C412yes—
355C2×C42#4112yes—
356C2×C42#4112yes—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364C22⋊C8—24no—
365C22⋊C8C22⋊C414yes—
366C22⋊C8C22⋊C414yes—
367C22⋊C8C22⋊C414yes—
368C22⋊C8C22⋊C414yes—
369#97—24no—
370#97—24no—
371#97—44no—
372#97—42no—
373#97—42no—
374C4×C8—24no—
375C4×C8—24no—
376C4×C8—44no—
377C4×C8—42no—
378C4×C8—42no—
379#95—22no—
380#95—22no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—21no—
385C24—42no—
386C24—42no—
387C24#11712yes—
388C24C4×D411yescharacteristic
389C24C4×D411yescharacteristic
390C24C4×D412yes—
391C24C2×#4111yescharacteristic
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—82no—
448C22×C4C2×C4212yes—
449C22×C4C2×C4212yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C2×C22⋊C412yes—
452C22×C4#11712yes—
453C22×C4C4×D412yes—
454C22×C4C4×D412yes—
455C22×C4C4×D412yes—
456C22×C4C4×D412yes—
457C22×C4C4×D411yescharacteristic
458C22×C4C4×D411yescharacteristic
459C22×C4C4×D412yes—
460C22×C4C23×C411yescharacteristic
461C22×C4C23×C411yescharacteristic
462C22×C4C23×C411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4111yescharacteristic
466C22×C4C2×#4111yescharacteristic
467C22×C4C2×#4111yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4H(5,𝔽2)11yescharacteristic
471C22×C4H(5,𝔽2)11yescharacteristic
472C22×C42−1+411yescharacteristic
473C22×C42−1+411yescharacteristic
474M16—22no—
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479M16—22no—
480M16—22no—
481M16—22no—
482M16—22no—
483M16—48no—
484M16—41no—
485M16—41no—
486M16—41no—
487M16—41no—
488M16—41no—
489M16—41no—
490M16#11714yes—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—48no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C2×C8—41no—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8—41no—
530C2×C8—41no—
531C2×C8—41no—
532C2×C8—41no—
533C2×C8—41no—
534C2×C8C2×C22⋊C412yes—
535C2×C8C2×C22⋊C412yes—
536C2×C8C2×C22⋊C412yes—
537C2×C8C2×C22⋊C412yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—48no—
549C4⋊C4C2×C4212yes—
550C4⋊C4C2×C4212yes—
551C4⋊C4C2×C4212yes—
552C4⋊C4C2×C4212yes—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—82no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—24no—
574C42—24no—
575C42—22no—
576C42—22no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42#11714yes—
584C23—22no—
585C23—22no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—82no—
603C23—82no—
604C23—82no—
605C23—82no—
606C23C4×C22⋊C412yes—
607C23C4×C22⋊C412yes—
608C23C2×#11711yescharacteristic
609C23C2×C4×D411yescharacteristic
610C23C2×C4×D411yescharacteristic
611C23#55111yescharacteristic
612C23#55111yescharacteristic
613C23#55211yescharacteristic
614C23#55211yescharacteristic
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—84no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4C4×C22⋊C412yes—
666C2×C4C4×C22⋊C412yes—
667C2×C4C22×C4211yescharacteristic
668C2×C4C22×C22⋊C411yescharacteristic
669C2×C4C2×#11711yescharacteristic
670C2×C4C2×#11711yescharacteristic
671C2×C4C2×C4×D411yescharacteristic
672C2×C4C2×C4×D411yescharacteristic
673C2×C4C4×#4111yescharacteristic
674C2×C4C4×#4111yescharacteristic
675C2×C4C4×#4111yescharacteristic
676C2×C4C4×#4111yescharacteristic
677C2×C4#51811yescharacteristic
678C2×C4#51911yescharacteristic
679C2×C4#52011yescharacteristic
680C2×C4#52011yescharacteristic
681C2×C4#52011yescharacteristic
682C2×C4#52011yescharacteristic
683C2×C4#54611yescharacteristic
684C2×C4#54611yescharacteristic
685C2×C4#54611yescharacteristic
686C2×C4#54611yescharacteristic
687C2×C4#54711yescharacteristic
688C2×C4#54711yescharacteristic
689C2×C4#54711yescharacteristic
690C2×C4#54711yescharacteristic
691C2×C4#55111yescharacteristic
692C2×C4#55111yescharacteristic
693C2×C4#55211yescharacteristic
694C2×C4#55211yescharacteristic
695C8—22no—
696C8—22no—
697C8—22no—
698C8—22no—
699C8—24no—
700C8—42no—
701C8—42no—
702C8—42no—
703C8—42no—
704C8—42no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—22no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—42no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22C2×C4×C22⋊C411yescharacteristic
728C22#234211yescharacteristic
729C22#234311yescharacteristic
730C22#234311yescharacteristic
731C22#234411yescharacteristic
732C22#235911yescharacteristic
733C22#235911yescharacteristic
734C22#295512yes—
735C22#295712yes—
736C4—22no—
737C4—22no—
738C4—44no—
739C4—42no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4C4×#11711yescharacteristic
747C4#227411yescharacteristic
748C4#233911yescharacteristic
749C4#233911yescharacteristic
750C4#234211yescharacteristic
751C4#234411yescharacteristic
752C4#235911yescharacteristic
753C4#235911yescharacteristic
754C2—21no—
755C2—21no—
756C2—21no—
757C2—21no—
758C2—42no—
759C2—42no—
760C2#1379911yescharacteristic
761C2#2042011yescharacteristic
762C2#2042211yescharacteristic
763C2#2294912yes—
764C2#2295112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10887C2semidirect
7. #10887C2semidirect
8. #10883C2semidirect
9. #10882C2semidirect
10. #10809C2semidirect
11. #10809C2semidirect
14. #10707C2semidirect
15. #10706C2semidirect
20. #2924C4semidirect
37. #1825C4semidirect
38. #1825C4semidirect
39. #1824C4semidirect
40. #1823C4semidirect
54. #1733C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
#206992
#203892
C2×#17502
C2×#17332
#108872
#108832
#108822
#108092
#107342
#107202
#107072
#107062