Finite Group Databasebeta

← #465173

Subgroups — #465173

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2C2×#1969C211yesmaximal, characteristic, complemented
3C2×#1923C211yesmaximal, characteristic
4C2×#1835C211yesmaximal, characteristic, complemented
5#15236C212yesmaximal
6#11256C211yesmaximal, characteristic, complemented
7#11096C211yesmaximal, characteristic
8#11077C212yesmaximal, complemented
9#11052C212yesmaximal, complemented
10#11024C211yesmaximal, characteristic, complemented
11#10970C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#382C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2001—24no—
28#1969—24no—
29#1923—24no—
30#1835—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1442—22no—
39#1442—22no—
40#1440—22no—
41#1440—22no—
42C4⋊#95C2212yes—
43C23×Q8C2312yes—
44C24×C4C2311yescharacteristic
45C22×Q16—24no—
46C22×Q16—22no—
47C22×Q16—22no—
48C22×Q16—22no—
49C22×Q16—42no—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—22no—
71C2×#107—42no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—42no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—42no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8D414yes—
91#391—24no—
92#391—24no—
93#391—24no—
94#391—42no—
95#382—24no—
96#382—42no—
97#380—42no—
98#380—42no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—22no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340D414yes—
117#336—22no—
118#336—22no—
119C25C2×D411yescharacteristic
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8#4114yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×Q16—28no—
155C2×Q16—48no—
156C2×Q16—48no—
157C2×Q16—44no—
158C2×Q16—44no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8#4112yes—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182C22⋊Q8—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C2×D414yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42C2×D412yes—
224C2×C42#4112yes—
225#107—44no—
226C4⋊C8—44no—
227#103—48no—
228#103—48no—
229#103—48no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232C4×C8—28no—
233#95—22no—
234#95—22no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244C24—21no—
245C24—22no—
246C24—21no—
247C24—22no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24#13512yes—
255C24C22×D411yescharacteristic
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#13512yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4C2×#4111yescharacteristic
362C22×C4H(5,𝔽2)12yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+411yescharacteristic
367Q16—416no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8#12114yes—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—416no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C22⋊C4—28no—
402C22⋊C4—44no—
403C22⋊C4—416no—
404C22⋊C4—88no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—41no—
410C42—41no—
411C42—42no—
412C23—22no—
413C23—24no—
414C23—22no—
415C23—24no—
416C23—22no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23C2×#12111yescharacteristic
436C23#53512yes—
437C23D4211yescharacteristic
438C23#54612yes—
439C23#54711yescharacteristic
440C23C2×#13511yescharacteristic
441C23C2×#13511yescharacteristic
442C23#57512yes—
443C23#57811yescharacteristic
444C23#57811yescharacteristic
445C23#57812yes—
446Q8—44no—
447Q8—44no—
448Q8—44no—
449Q8—44no—
450Q8—82no—
451Q8—82no—
452Q8—82no—
453Q8—82no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4C2×#12111yescharacteristic
516C2×C4#52511yescharacteristic
517C2×C4#53511yescharacteristic
518C2×C4#53511yescharacteristic
519C2×C4#53511yescharacteristic
520C2×C4#53512yes—
521C2×C4#53711yescharacteristic
522C2×C4#53812yes—
523C2×C4#53912yes—
524C2×C4#54012yes—
525C2×C4#54012yes—
526C2×C4#54612yes—
527C2×C4#54811yescharacteristic
528C2×C4#55112yes—
529C2×C4#55312yes—
530C2×C4#55311yescharacteristic
531C2×C4H(3,𝔽4)12yes—
532C2×C4#56212yes—
533C8—24no—
534C8—42no—
535C8—81no—
536C8—81no—
537C22—44no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—81no—
544C22—81no—
545C22—82no—
546C22—81no—
547C22—81no—
548C22#241611yescharacteristic
549C22#253511yescharacteristic
550C22#261911yescharacteristic
551C22#267012yes—
552C22#270612yes—
553C22#305011yescharacteristic
554C22#305411yescharacteristic
555C22#305911yescharacteristic
556C22#305911yescharacteristic
557C22#321112yes—
558C22#321212yes—
559C22#322312yes—
560C22#322312yes—
561C22#326912yes—
562C22#328311yescharacteristic
563C22#328511yescharacteristic
564C22#330012yes—
565C22#330012yes—
566C22#332712yes—
567C22#332712yes—
568C22#333611yescharacteristic
569C22#334211yescharacteristic
570C22#334912yes—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—84no—
575C4—81no—
576C4—84no—
577C4—81no—
578C4—162no—
579C4—162no—
580C4#241111yescharacteristic
581C4#261911yescharacteristic
582C4#263611yescharacteristic
583C4#271112yes—
584C4#275312yes—
585C4#281911yescharacteristic
586C2—82no—
587C2#1651611yescharacteristic
588C2#2477911yescharacteristic
589C2#2478511yescharacteristic
590C2#2814511yescharacteristic
591C2#2814811yescharacteristic
592C2#2980911yescharacteristic
593C2#2981211yescharacteristic
594C2#3052512yes—
595C2#3053412yes—
596C2#3193912yes—
597C2#3194012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1969C2semidirect
4. C2×#1835C2semidirect
6. #11256C2semidirect
8. #11077C2semidirect
9. #11052C2semidirect
10. #11024C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#19692
C2×#19232
C2×#18352
#152362
#112562
#110962
#110772
#110522
#110242
#109702