Finite Group Databasebeta

← #455470

Subgroups — #455470

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic, complemented
2C2×#2065C212yesmaximal, complemented
3C2×#2025C211yesmaximal, characteristic, complemented
4C2×#1798C211yesmaximal, characteristic, complemented
5#14257C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11227C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#11049C211yesmaximal, characteristic, complemented
11#10969C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2212yes—
20#2629—28no—
21C2×#398—22no—
22C2×#396—22no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29#2065—28no—
30#2025—24no—
31#1798—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1439—22no—
40#1438—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#125—24no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—42no—
52C2×#121—22no—
53C2×#121—42no—
54C2×#121—42no—
55C2×#121Q814yes—
56C2×#117—22no—
57C2×#117—24no—
58C2×#117—22no—
59C2×#117—42no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—42no—
79C2×#106—42no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—42no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—42no—
92#398—216no—
93#396—216no—
94#390—24no—
95#390—22no—
96#390—22no—
97#389—24no—
98#389D418yescomplemented
99#384—24no—
100#380—24no—
101#380—24no—
102#380—42no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#336—22no—
114#336D414yescomplemented
115C25—21no—
116C25C2×D411yescharacteristic
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—82no—
130C22×D4—82no—
131C22×D4C2×Q812yes—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×Q812yes—
149C23×C4#4112yes—
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×D414yes—
165#125—48no—
166#125—48no—
167#121—48no—
168#121—48no—
169#117—48no—
170#117—48no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42C2×D414yes—
230C2×C42C2×D412yes—
231C2×C42#4112yes—
232#106—48no—
233C4⋊C8—48no—
234#102—48no—
235#102—48no—
236C22⋊C8—48no—
237#95—28no—
238#95—28no—
239#95—42no—
240#95—82no—
241#95—82no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24C22⋊Q812yes—
264C24C22×D411yescharacteristic
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—48no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—88no—
283C2×D4—88no—
284C2×D4—84no—
285C2×D4—84no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4#12014yes—
350C22×C4#12114yes—
351C22×C4C22⋊Q812yes—
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)12yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C42−1+411yescharacteristic
361C22×C42−1+412yes—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—48no—
382C4⋊C4—416no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C22⋊C4—84no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—24no—
404C23—24no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—82no—
436C23—82no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—81no—
450C23C2×#12011yescharacteristic
451C23C2×#12111yescharacteristic
452C23C2×C22⋊Q811yescharacteristic
453C23#53411yescharacteristic
454C23#53511yescharacteristic
455C23#53611yescharacteristic
456C23#53711yescharacteristic
457C23#57612yes—
458C23#57712yes—
459C23#57712yes—
460C23#57712yes—
461D4—44no—
462D4—44no—
463D4—816no—
464D4—82no—
465D4—82no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#53512yes—
521C2×C4#53712yes—
522C2×C4#53712yes—
523C2×C4#53812yes—
524C2×C4#54212yes—
525C2×C4#54312yes—
526C2×C4D4212yes—
527C2×C4#54611yescharacteristic
528C2×C4#54612yes—
529C2×C4#54812yes—
530C2×C4D4×Q811yescharacteristic
531C2×C4D4×Q812yes—
532C2×C4#55211yescharacteristic
533C2×C4#55411yescharacteristic
534C2×C4#56212yes—
535C2×C4#56312yes—
536C8—44no—
537C8—82no—
538C22—42no—
539C22—42no—
540C22—44no—
541C22—42no—
542C22—44no—
543C22—82no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—84no—
552C22—82no—
553C22—84no—
554C22—162no—
555C22—161no—
556C22—161no—
557C22—161no—
558C22—161no—
559C22—162no—
560C22#240911yescharacteristic
561C22#241011yescharacteristic
562C22#241711yescharacteristic
563C22#261511yescharacteristic
564C22#262011yescharacteristic
565C22#262611yescharacteristic
566C22#281411yescharacteristic
567C22#301012yes—
568C22#301212yes—
569C22#306012yes—
570C22#306212yes—
571C22#308112yes—
572C22#308412yes—
573C22#319812yes—
574C22#320412yes—
575C22#322011yescharacteristic
576C22#322111yescharacteristic
577C22#322211yescharacteristic
578C22#322211yescharacteristic
579C22#324112yes—
580C22#324412yes—
581C22#325712yes—
582C22#326012yes—
583C4—44no—
584C4—44no—
585C4—42no—
586C4—82no—
587C4—84no—
588C4—84no—
589C4—161no—
590C4—161no—
591C4#267612yes—
592C4#268312yes—
593C4#269212yes—
594C4#270212yes—
595C2—82no—
596C2—162no—
597C2#1626511yescharacteristic
598C2#2430912yes—
599C2#2439712yes—
600C2#2485212yes—
601C2#2967111yescharacteristic
602C2#2967511yescharacteristic
603C2#2985011yescharacteristic
604C2#2985511yescharacteristic
605C2#2996211yescharacteristic
606C2#2996511yescharacteristic
607C2#3275612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2629C2semidirect
2. C2×#2065C2semidirect
3. C2×#2025C2semidirect
4. C2×#1798C2semidirect
6. #11385C2semidirect
7. #11323C2semidirect
8. #11227C2semidirect
9. #11095C2semidirect
10. #11049C2semidirect
98. #389D4semidirect
114. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#20652
C2×#20252
C2×#17982
#142572
#113852
#113232
#112272
#110952
#110492
#109692