Finite Group Databasebeta

← #373769

Subgroups — #373769

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1797C211yesmaximal, characteristic
5C4×#380C211yesmaximal, characteristic
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10849C211yesmaximal, characteristic, complemented
12C22×#117—21no—
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19#2285—24no—
20#2285—24no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382—21no—
24C2×#380—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#1948—24no—
34#1947—24no—
35#1797C418yescomplemented
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45#1434—21no—
46C22⋊#95—21no—
47C4×#95—21no—
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#117—28no—
54C2×#117—44no—
55C2×#117—44no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2×C414yes—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42Q814yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—42no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2×C414yes—
93C2×C4⋊C8C2×C414yes—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8C2×C418yes—
97#384—22no—
98#384—22no—
99#384D414yes—
100#384D414yes—
101#382—28no—
102#380—28no—
103#380—24no—
104#380—24no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C22⋊C4—44no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—42no—
132#340—42no—
133#336C2×C418yes—
134C25C2×D411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
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—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C4⋊C414yes—
166C23×C4C4⋊C414yes—
167C23×C4C22×C412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×Q812yes—
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C22×C414yes—
188C22×C8C22×C414yes—
189#117—432no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
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—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4#4112yes—
235C2×C4⋊C4#4112yes—
236C2×C4⋊C4#4112yes—
237C2×C4⋊C4#4112yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—28no—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—28no—
259C2×C42—28no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42C4⋊C414yes—
269C2×C42C4⋊C414yes—
270C2×C42C22×C412yes—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×Q812yes—
275C2×C42C2×Q812yes—
276C2×C42#4112yes—
277C2×C42#4112yes—
278#107—48no—
279#106—48no—
280C4⋊C8—28no—
281C4⋊C8—28no—
282C22⋊C8—48no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—42no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—42no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24C2×C4⋊C412yes—
306C24AGL(1,ℤ/8)12yes—
307C24#13712yes—
308C24C22×D411yescharacteristic
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
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—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
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—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4C2×C4⋊C412yes—
414C22×C4C2×C4⋊C412yes—
415C22×C4C2×C4⋊C412yes—
416C22×C4C2×C4⋊C412yes—
417C22×C4C2×C4⋊C412yes—
418C22×C4#12012yes—
419C22×C4#12012yes—
420C22×C4#12312yes—
421C22×C4#12312yes—
422C22×C4#13512yes—
423C22×C4#13512yes—
424C22×C4AGL(1,ℤ/8)12yes—
425C22×C4AGL(1,ℤ/8)12yes—
426C22×C4#13712yes—
427C22×C4#13712yes—
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×Q811yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4H(5,𝔽2)11yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C42−1+411yescharacteristic
441C22×C42−1+411yescharacteristic
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—28no—
492C42—28no—
493C42—416no—
494C42—84no—
495C42—84no—
496C42C4×D414yes—
497C42C4×D414yes—
498C42C22⋊Q814yes—
499C42C22⋊Q814yes—
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—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#42814yes—
535C23C22×C4⋊C411yescharacteristic
536C23C2×#12011yescharacteristic
537C23C2×#12311yescharacteristic
538C23#52511yescharacteristic
539C23#52511yescharacteristic
540C23#53711yescharacteristic
541C23#53711yescharacteristic
542C23C2×#13511yescharacteristic
543C23C2×#13511yescharacteristic
544C23C2×AGL(1,ℤ/8)11yescharacteristic
545C23C2×AGL(1,ℤ/8)11yescharacteristic
546C23C2×AGL(1,ℤ/8)11yescharacteristic
547C23C2×#13711yescharacteristic
548C23C2×#13711yescharacteristic
549C23C2×#13711yescharacteristic
550C2×C4—232no—
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—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—416no—
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—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—81no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—82no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4#38514yes—
629C2×C4#38514yes—
630C2×C4#42714yes—
631C2×C4#42814yes—
632C2×C4#42814yes—
633C2×C4#44914yes—
634C2×C4#45214yes—
635C2×C4#48214yes—
636C2×C4#48514yes—
637C2×C4C2×C4×D411yescharacteristic
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C4×#4111yescharacteristic
640C2×C4C4×#4111yescharacteristic
641C2×C4#51811yescharacteristic
642C2×C4#51911yescharacteristic
643C2×C4C2×C22⋊Q811yescharacteristic
644C2×C4C2×C22⋊Q811yescharacteristic
645C2×C4#53311yescharacteristic
646C2×C4#53311yescharacteristic
647C2×C4#54411yescharacteristic
648C2×C4#54411yescharacteristic
649C2×C4#54611yescharacteristic
650C2×C4#54611yescharacteristic
651C2×C4#54711yescharacteristic
652C2×C4#54711yescharacteristic
653C2×C4#54811yescharacteristic
654C2×C4#54811yescharacteristic
655C2×C4#55011yescharacteristic
656C2×C4#55011yescharacteristic
657C2×C4#55211yescharacteristic
658C2×C4#55211yescharacteristic
659C2×C4#55311yescharacteristic
660C2×C4#55311yescharacteristic
661C2×C4#55511yescharacteristic
662C2×C4#55511yescharacteristic
663C2×C4#55611yescharacteristic
664C2×C4#55611yescharacteristic
665C8—44no—
666C8—44no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22C2×#38511yescharacteristic
683C22#229611yescharacteristic
684C22#231611yescharacteristic
685C22#240711yescharacteristic
686C22#240711yescharacteristic
687C22#248511yescharacteristic
688C22#248511yescharacteristic
689C22C2×#42711yescharacteristic
690C22C2×#42811yescharacteristic
691C22C2×#42811yescharacteristic
692C22C2×#42811yescharacteristic
693C22C2×#44911yescharacteristic
694C22C2×#45211yescharacteristic
695C22#300311yescharacteristic
696C22#300411yescharacteristic
697C22C2×#48211yescharacteristic
698C22C2×#48511yescharacteristic
699C22#309211yescharacteristic
700C22#309211yescharacteristic
701C22#310611yescharacteristic
702C22#310611yescharacteristic
703C22#310711yescharacteristic
704C22#310711yescharacteristic
705C22#310911yescharacteristic
706C22#311011yescharacteristic
707C22#311211yescharacteristic
708C22#311311yescharacteristic
709C22#324711yescharacteristic
710C22#324711yescharacteristic
711C22#324811yescharacteristic
712C22#324811yescharacteristic
713C22#325311yescharacteristic
714C22#325411yescharacteristic
715C22#325411yescharacteristic
716C22#325511yescharacteristic
717C4—24no—
718C4—24no—
719C4—42no—
720C4—42no—
721C4—82no—
722C4—84no—
723C4—82no—
724C4—84no—
725C4—82no—
726C4—82no—
727C4#185418yes—
728C4#236211yescharacteristic
729C4#236211yescharacteristic
730C4#236311yescharacteristic
731C4#236311yescharacteristic
732C4#243911yescharacteristic
733C4#243911yescharacteristic
734C4#244111yescharacteristic
735C4#244111yescharacteristic
736C2—44no—
737C2#1412911yescharacteristic
738C2C2×#185411yescharacteristic
739C2#2084411yescharacteristic
740C2#2136211yescharacteristic
741C2#2136511yescharacteristic
742C2#2196211yescharacteristic
743C2#2196411yescharacteristic
744C2#2415211yescharacteristic
745C2#2415311yescharacteristic
746C2#2415611yescharacteristic
747C2#2415711yescharacteristic
748C2#2762611yescharacteristic
749C2#2762711yescharacteristic
750C2#2763011yescharacteristic
751C2#2763111yescharacteristic
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. #10908C2semidirect
9. #10907C2semidirect
10. #10849C2semidirect
11. #10849C2semidirect
35. #1797C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#19482
C2×#19472
C2×#17972
C4×#3802
#110962
#110952
#109082
#109072
#108492