Finite Group Databasebeta

← #363714

Subgroups — #363714

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4#14686C211yesmaximal, characteristic
5C4×#380C211yesmaximal, characteristic
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10863C212yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10821C212yesmaximal, complemented
12C23×C42C2211yescharacteristic
13C2×#432—24no—
14C2×#432—24no—
15C2×#432C418yescomplemented
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×#97C2212yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#380—21no—
25C2×#380—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32#1948—24no—
33#1947—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45#1434—21no—
46C22⋊#95—21no—
47C4×#95—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2×C414yes—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42Q814yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#432—232no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8C2×C414yes—
88C2×C4⋊C8C2×C414yes—
89C2×C4⋊C8D414yes—
90C2×C4⋊C8D414yes—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8C2×C418yes—
94C2×#97—24no—
95C2×#97—24no—
96C2×#97C2×C418yes—
97#384—22no—
98#384—22no—
99#384—22no—
100#384—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—42no—
106#380—42no—
107#380—44no—
108#380—44no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—44no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—42no—
130#340—42no—
131C25C2×D411yescharacteristic
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4C4⋊C414yes—
163C23×C4C4⋊C414yes—
164C23×C4C22×C412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×Q812yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C22×C414yes—
183C22×C8C22×C414yes—
184C22×C8C2×D414yes—
185C22×C8C2×D414yes—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4#4112yes—
231C2×C4⋊C4#4112yes—
232C2×C4⋊C4#4112yes—
233C2×C4⋊C4#4112yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—28no—
252C2×C42—28no—
253C2×C42—28no—
254C2×C42—28no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C4⋊C414yes—
262C2×C42C4⋊C414yes—
263C2×C42C22×C412yes—
264C2×C42C22×C412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×Q812yes—
268C2×C42C2×Q812yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271#107—48no—
272#106—48no—
273C4⋊C8—28no—
274C4⋊C8C22⋊C4116yescomplemented
275C22⋊C8—216no—
276#97—216no—
277#95—24no—
278#95—24no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—42no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—42no—
287#95—42no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24C2×C4⋊C412yes—
301C24AGL(1,ℤ/8)12yes—
302C24#13712yes—
303C24C22×D411yescharacteristic
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4C2×C4⋊C412yes—
411C22×C4C2×C4⋊C412yes—
412C22×C4C2×C4⋊C412yes—
413C22×C4C2×C4⋊C412yes—
414C22×C4C2×C4⋊C412yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4#12112yes—
418C22×C4#12112yes—
419C22×C4AGL(1,ℤ/8)12yes—
420C22×C4AGL(1,ℤ/8)12yes—
421C22×C4#13712yes—
422C22×C4#13712yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q811yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8C2×C22⋊C418yes—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—88no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—28no—
487C42—28no—
488C42—84no—
489C42—84no—
490C42C2×C22⋊C414yes—
491C42#11714yes—
492C42C22⋊Q814yes—
493C42C22⋊Q814yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23#42814yes—
529C23C22×C4⋊C411yescharacteristic
530C23C2×#12111yescharacteristic
531C23C2×#12111yescharacteristic
532C23#52511yescharacteristic
533C23#52511yescharacteristic
534C23#53511yescharacteristic
535C23#53611yescharacteristic
536C23C2×AGL(1,ℤ/8)11yescharacteristic
537C23C2×AGL(1,ℤ/8)11yescharacteristic
538C23C2×AGL(1,ℤ/8)11yescharacteristic
539C23C2×#13711yescharacteristic
540C23C2×#13711yescharacteristic
541C23C2×#13711yescharacteristic
542C23#57511yescharacteristic
543C23#57511yescharacteristic
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—416no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4#38014yes—
624C2×C4#38014yes—
625C2×C4#42814yes—
626C2×C4#42814yes—
627C2×C4#46814yes—
628C2×C4#46814yes—
629C2×C4#46914yes—
630C2×C4#47014yes—
631C2×C4C22×C22⋊C411yescharacteristic
632C2×C4C2×#11711yescharacteristic
633C2×C4C4×#4111yescharacteristic
634C2×C4C4×#4111yescharacteristic
635C2×C4#52011yescharacteristic
636C2×C4#52011yescharacteristic
637C2×C4C2×C22⋊Q811yescharacteristic
638C2×C4C2×C22⋊Q811yescharacteristic
639C2×C4#53311yescharacteristic
640C2×C4#53311yescharacteristic
641C2×C4#54411yescharacteristic
642C2×C4#54411yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#54611yescharacteristic
645C2×C4#54711yescharacteristic
646C2×C4#54711yescharacteristic
647C2×C4#54811yescharacteristic
648C2×C4#54811yescharacteristic
649C2×C4#55011yescharacteristic
650C2×C4#55011yescharacteristic
651C2×C4#55211yescharacteristic
652C2×C4#55211yescharacteristic
653C2×C4#55311yescharacteristic
654C2×C4#55311yescharacteristic
655C2×C4#55511yescharacteristic
656C2×C4#55511yescharacteristic
657C2×C4#55611yescharacteristic
658C2×C4#55611yescharacteristic
659C8—28no—
660C8—44no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22C2×#38011yescharacteristic
677C22#229611yescharacteristic
678C22#231711yescharacteristic
679C22#241411yescharacteristic
680C22#241411yescharacteristic
681C22#241511yescharacteristic
682C22#241511yescharacteristic
683C22C2×#42811yescharacteristic
684C22C2×#42811yescharacteristic
685C22C2×#42811yescharacteristic
686C22#291111yescharacteristic
687C22C2×#46811yescharacteristic
688C22C2×#46811yescharacteristic
689C22C2×#46911yescharacteristic
690C22C2×#47011yescharacteristic
691C22#305511yescharacteristic
692C22#305511yescharacteristic
693C22#305611yescharacteristic
694C22#305611yescharacteristic
695C22#310911yescharacteristic
696C22#311011yescharacteristic
697C22#311211yescharacteristic
698C22#311311yescharacteristic
699C22#311511yescharacteristic
700C22#311511yescharacteristic
701C22#311611yescharacteristic
702C22#311611yescharacteristic
703C22#321411yescharacteristic
704C22#321611yescharacteristic
705C22#321711yescharacteristic
706C22#321811yescharacteristic
707C22#323511yescharacteristic
708C22#323711yescharacteristic
709C22#323811yescharacteristic
710C22#323911yescharacteristic
711C4—24no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—162no—
720C4—162no—
721C4#180918yes—
722C4#233911yescharacteristic
723C4#234111yescharacteristic
724C4#234611yescharacteristic
725C4#234711yescharacteristic
726C4#243911yescharacteristic
727C4#243911yescharacteristic
728C4#244111yescharacteristic
729C4#244111yescharacteristic
730C2—44no—
731C2#1395111yescharacteristic
732C2C2×#180911yescharacteristic
733C2#2066111yescharacteristic
734C2#2136411yescharacteristic
735C2#2136511yescharacteristic
736C2#2198711yescharacteristic
737C2#2198811yescharacteristic
738C2#2465811yescharacteristic
739C2#2465911yescharacteristic
740C2#2466011yescharacteristic
741C2#2466111yescharacteristic
742C2#2470811yescharacteristic
743C2#2470911yescharacteristic
744C2#2471011yescharacteristic
745C2#2471111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1948C2semidirect
3. C2×#1947C2semidirect
8. #10863C2semidirect
9. #10849C2semidirect
10. #10849C2semidirect
11. #10821C2semidirect
15. C2×#432C4semidirect
274. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×#19482
C2×#19472
#146862
C4×#3802
#110962
#110952
#108632
#108492
#108212