Finite Group Databasebeta

← #432810

Subgroups — #432810

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#15151C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11328C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11066C212yesmaximal
9#10925C211yesmaximal, characteristic
10#10874C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2452—24no—
20#2452—24no—
21C2×#398—21no—
22C2×#398—21no—
23C2×#396—21no—
24C2×#393—21no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#385—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2093—24no—
34#1990—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8D414yes—
56C2×C22⋊Q8—28no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8—44no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—42no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102Q818yescomplemented
93C2×C4×C8—22no—
94C2×C4×C8—42no—
95#398—44no—
96#398—44no—
97#396—28no—
98#393—28no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390D414yes—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—42no—
108#389—42no—
109#385—28no—
110#384—22no—
111#384—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—24no—
129#336—22no—
130#336—42no—
131C25—21no—
132C22×Q8—24no—
133C22×Q8—44no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—82no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C2×D412yes—
144C22×D4C2×Q814yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—44no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—82no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C2×Q814yes—
175C22×C8#4114yes—
176C4⋊Q8—28no—
177C4⋊Q8—44no—
178C4⋊Q8—48no—
179C22⋊Q8—432no—
180C4×D4—48no—
181C4×D4—44no—
182C4×D4—44no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
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—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4D814yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×Q814yes—
229C2×C4⋊C4#4114yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—82no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255#107—48no—
256#102—48no—
257#102—48no—
258C4×C8—44no—
259#95—24no—
260#95—24no—
261#95—22no—
262#95—22no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—44no—
267#95—42no—
268C24—22no—
269C24—24no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24—42no—
279C24C22×D411yescharacteristic
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—84no—
283C2×Q8—88no—
284C2×Q8—88no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4C4⋊Q814yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—88no—
359C22×C4—84no—
360C22×C4—88no—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4C2×D812yes—
364C22×C4C2×D812yes—
365C22×C4C2×D812yes—
366C22×C4#13512yes—
367C22×C4AGL(1,ℤ/8)12yes—
368C22×C4AGL(1,ℤ/8)12yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×Q812yes—
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+412yes—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—28no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—88no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4#12314yes—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—84no—
420C42—22no—
421C42—22no—
422C42—24no—
423C42—41no—
424C42—41no—
425C42—42no—
426C42—42no—
427C42—42no—
428C23—24no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—81no—
442C23—82no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23C2×#12011yescharacteristic
452C23#53211yescharacteristic
453C23#53511yescharacteristic
454C23#54612yes—
455C23D4×Q812yes—
456C23C22×D811yescharacteristic
457C23C2×#13511yescharacteristic
458C23C2×AGL(1,ℤ/8)11yescharacteristic
459C23C2×AGL(1,ℤ/8)11yescharacteristic
460C23#57712yes—
461C23#57812yes—
462Q8—816no—
463D4—28no—
464D4—44no—
465D4—44no—
466D4—44no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—816no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4#44714yes—
530C2×C4#44914yes—
531C2×C4C2×#12311yescharacteristic
532C2×C4C2×C4⋊Q811yescharacteristic
533C2×C4#53312yes—
534C2×C4#53411yescharacteristic
535C2×C4#53611yescharacteristic
536C2×C4#53912yes—
537C2×C4#54212yes—
538C2×C4#54312yes—
539C2×C4#54612yes—
540C2×C4#54612yes—
541C2×C4D4×Q812yes—
542C2×C4#55012yes—
543C2×C4#55212yes—
544C2×C4#55412yes—
545C2×C4#55412yes—
546C2×C4#56212yes—
547C8—44no—
548C8—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—82no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—84no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22#240511yescharacteristic
561C22#243111yescharacteristic
562C22#243411yescharacteristic
563C22#255612yes—
564C22#269212yes—
565C22C2×#44711yescharacteristic
566C22C2×#44911yescharacteristic
567C22#300911yescharacteristic
568C22#301111yescharacteristic
569C22#316911yescharacteristic
570C22#317812yes—
571C22#318411yescharacteristic
572C22#320511yescharacteristic
573C22#320711yescharacteristic
574C22#322512yes—
575C22#329412yes—
576C22#330012yes—
577C22#330812yes—
578C22#331012yes—
579C22Q8×D812yes—
580C22#338112yes—
581C22#338412yes—
582C22#338612yes—
583C4—24no—
584C4—24no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—81no—
589C4—84no—
590C4—82no—
591C4—81no—
592C4—82no—
593C4—162no—
594C4#242312yes—
595C4#254912yes—
596C4#268712yes—
597C4#273212yes—
598C2—84no—
599C2#1546611yescharacteristic
600C2#2403211yescharacteristic
601C2#2403511yescharacteristic
602C2#2559911yescharacteristic
603C2#2561311yescharacteristic
604C2#2583511yescharacteristic
605C2#2583811yescharacteristic
606C2#2894312yes—
607C2#2897412yes—
608C2#3151812yes—
609C2#3151912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1990C2semidirect
4. #15151C2semidirect
5. #11384C2semidirect
6. #11328C2semidirect
7. #11219C2semidirect
10. #10874C2semidirect
92. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
C2×#20932
C2×#19902
#151512
#113842
#113282
#112192
#110662
#109252
#108742
#107882