Finite Group Databasebeta

← #106835

Subgroups — #106835

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C22×#324C211yesmaximal, characteristic
3C2×#1441C211yesmaximal, characteristic
4#10730C212yesmaximal
5C2×#1294C212yesmaximal
6C22×#117—21no—
7C23×C4⋊C4C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2316—24no—
11#2316—22no—
12#2316C414yescomplemented
13C2×#389—24no—
14C2×#380—21no—
15C2×#380C2212yes—
16C22×#95—21no—
17C22×#95—21no—
18C22×#95C2211yescharacteristic
19C2×#340—24no—
20C2×#324—22no—
21C2×#324—22no—
22C2×#324C414yescomplemented
23#1441—22no—
24#1441C414yescomplemented
25#1294—24no—
26C24×C4—21no—
27C24×C4—21no—
28C24×C4D411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4C2311yescharacteristic
31C23×C8—22no—
32C2×#117—24no—
33C2×#117—24no—
34C2×#117—42no—
35C2×#117—42no—
36C2×#117—42no—
37C2×#117—42no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—44no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4Q812yes—
56C22×C22⋊C4—22no—
57C22×C42—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8C2×C414yes—
64#389—216no—
65#389—216no—
66#380—24no—
67#380—24no—
68#380—22no—
69#380—22no—
70#380C2×C414yes—
71#380D414yes—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—24no—
77C2×#95—24no—
78C2×#95—21no—
79C2×#95—44no—
80C2×#95—42no—
81C2×#95—42no—
82C2×#95C2×C412yes—
83#340—48no—
84#324—28no—
85C25C2×D411yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—41no—
114C23×C4—41no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C4⋊C412yes—
130C23×C4C4⋊C412yes—
131C23×C4D812yes—
132C23×C4SD1612yes—
133C23×C4SD1612yes—
134C23×C4Q1612yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×Q811yescharacteristic
143C23×C4#4112yes—
144C23×C4#4112yes—
145C22×C8—24no—
146C22×C8—44no—
147C22×C8—44no—
148C22×C8—42no—
149C22×C8—42no—
150#117—48no—
151#117—48no—
152#117—48no—
153#117—48no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C4214yes—
183C2×C4⋊C4C22⋊C414yes—
184C2×C4⋊C4C22⋊C418yes—
185C2×C4⋊C4C4⋊C414yes—
186C2×C4⋊C4C4⋊C414yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C42—28no—
194C2×C42—44no—
195C2×C42—44no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200#95—28no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—48no—
205#95—48no—
206#95—42no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24C2×C22⋊C411yescharacteristic
212C24C2×C22⋊C411yescharacteristic
213C24C2×C4⋊C411yescharacteristic
214C24#12011yescharacteristic
215C24#12011yescharacteristic
216C24#12311yescharacteristic
217C24#12311yescharacteristic
218C24C2×D811yescharacteristic
219C24C2×SD1611yescharacteristic
220C24C2×SD1611yescharacteristic
221C24C2×Q1611yescharacteristic
222C24AGL(1,ℤ/8)11yescharacteristic
223C24AGL(1,ℤ/8)11yescharacteristic
224C24#13711yescharacteristic
225C24#13711yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—82no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4—82no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4#9514yes—
338C22×C4#9514yes—
339C22×C4#10214yes—
340C22×C4#10214yes—
341C22×C4#10314yes—
342C22×C4#10314yes—
343C22×C4#10614yes—
344C22×C4#10714yes—
345C22×C4C2×C4211yescharacteristic
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C4⋊C411yescharacteristic
349C22×C4C2×C4⋊C411yescharacteristic
350C22×C4#11712yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4C4×D412yes—
354C22×C4C4×D412yes—
355C22×C4#12012yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4C22⋊Q812yes—
359C22×C4C22⋊Q812yes—
360C22×C4#12312yes—
361C2×C8—48no—
362C2×C8—44no—
363C2×C8—82no—
364C2×C8—82no—
365C2×C8—84no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—48no—
372C4⋊C4—44no—
373C4⋊C4—48no—
374C4⋊C4—416no—
375C4⋊C4—84no—
376C4⋊C4—88no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C42—416no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—22no—
413C23—22no—
414C23C2×#9511yescharacteristic
415C23#37911yescharacteristic
416C23#38111yescharacteristic
417C23#38511yescharacteristic
418C23#38511yescharacteristic
419C23#38611yescharacteristic
420C23#38611yescharacteristic
421C23C2×#10211yescharacteristic
422C23C2×#10211yescharacteristic
423C23C2×#10311yescharacteristic
424C23C2×#10311yescharacteristic
425C23#41711yescharacteristic
426C23#41711yescharacteristic
427C23#41811yescharacteristic
428C23#41911yescharacteristic
429C23C2×#10611yescharacteristic
430C23C2×#10711yescharacteristic
431C23#42811yescharacteristic
432C23#42811yescharacteristic
433C23#44711yescharacteristic
434C23#44711yescharacteristic
435C23#44811yescharacteristic
436C23#44811yescharacteristic
437C23#45011yescharacteristic
438C23#45011yescharacteristic
439C23#45111yescharacteristic
440C23#45111yescharacteristic
441C23#45412yes—
442C23#45612yes—
443C23B(4,𝔽2)12yes—
444C23#45812yes—
445C23#48011yescharacteristic
446C23#48011yescharacteristic
447C23#48111yescharacteristic
448C23#48111yescharacteristic
449C23#48311yescharacteristic
450C23#48311yescharacteristic
451C23#48411yescharacteristic
452C23#48411yescharacteristic
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#34018yescomplemented
524C2×C4C4×C22⋊C412yes—
525C2×C4#38012yes—
526C2×C4#38512yes—
527C2×C4#38612yes—
528C8—84no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22C22⋊#9511yescharacteristic
570C22#159812yes—
571C22#159912yes—
572C22#160412yes—
573C22#160512yes—
574C22#161012yes—
575C22#161112yes—
576C22#161612yes—
577C22#161712yes—
578C22C2×#34011yescharacteristic
579C22#173611yescharacteristic
580C22#178611yescharacteristic
581C22#178711yescharacteristic
582C22#179012yes—
583C22#179312yes—
584C22#182311yescharacteristic
585C22#182411yescharacteristic
586C22#185211yescharacteristic
587C22#185211yescharacteristic
588C22#185311yescharacteristic
589C22#185311yescharacteristic
590C22#185712yes—
591C22#186712yes—
592C22#187211yescharacteristic
593C22#187211yescharacteristic
594C22#187411yescharacteristic
595C22#187411yescharacteristic
596C22#188611yescharacteristic
597C22#188811yescharacteristic
598C22#188911yescharacteristic
599C22#189011yescharacteristic
600C4—28no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—44no—
606C4—84no—
607C4—84no—
608C4—88no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—21no—
617C2#840412yes—
618C2#840612yes—
619C2#885612yes—
620C2#885712yes—
621C2#901511yescharacteristic
622C2#901911yescharacteristic
623C2#902311yescharacteristic
624C2#902711yescharacteristic
625C2#1073011yescharacteristic
626C2#1074511yescharacteristic
627C2#1075011yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2316C4semidirect
22. C2×#324C4semidirect
24. #1441C4semidirect
523. C2×C4#340semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C22×#3242
C2×#14412
#107302
C2×#12942