Finite Group Databasebeta

← #402840

Subgroups — #402840

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1920C211yesmaximal, characteristic, complemented
4#15181C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11199C212yesmaximal, complemented
7#11095C212yesmaximal, complemented
8#10964C211yesmaximal, characteristic
9#10806C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal, complemented
11#10731C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C22×#106C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17#2354—24no—
18#2354—24no—
19C2×#390—21no—
20C2×#389C2211yescharacteristic
21C2×#388—22no—
22C2×#384C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2100—24no—
29#1920—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1442—21no—
42#1439—22no—
43#1438—21no—
44C4⋊#95C2211yescharacteristic
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—42no—
59C2×#117—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—22no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—22no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87#390—28no—
88#389—22no—
89#389—22no—
90#388—216no—
91#384—22no—
92#384D414yes—
93#380—24no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—42no—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4D414yes—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340—42no—
118#336—22no—
119#336—22no—
120C25C2×D412yes—
121C22×D4—24no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—42no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4#4114yes—
148C23×C4#4114yes—
149C23×C4C2411yescharacteristic
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
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—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8#4112yes—
167C4×D4—28no—
168C4×D4—28no—
169C4×D4—216no—
170#117—48no—
171#117—48no—
172#117—48no—
173#117—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4#4112yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—28no—
233C2×C42—24no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240#106—28no—
241C22⋊C8—48no—
242C22⋊C8—48no—
243#95—28no—
244#95—24no—
245#95—44no—
246#95—42no—
247#95—44no—
248#95—42no—
249#95—81no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24#12314yes—
265C24C22×D411yescharacteristic
266C2×D4—28no—
267C2×D4—48no—
268C2×D4—48no—
269C2×D4—44no—
270C2×D4—44no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
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—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4#12012yes—
348C22×C4#12012yes—
349C22×C4#12314yes—
350C22×C4#13512yes—
351C22×C4#13712yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4C2×#4112yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C42−1+411yescharacteristic
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4#12414yes—
405C22⋊C4—216no—
406C22⋊C4—416no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C42—22no—
411C42—22no—
412C42—24no—
413C42—416no—
414C42—48no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23C2×#12011yescharacteristic
451C23C2×#12312yes—
452C23#54611yescharacteristic
453C23#54612yes—
454C23#54711yescharacteristic
455C23C2×#13511yescharacteristic
456C23C2×#13711yescharacteristic
457C23#57512yes—
458C23#57611yescharacteristic
459C23#57711yescharacteristic
460C23#57712yes—
461D4—416no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
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—81no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4#45214yes—
523C2×C4#52511yescharacteristic
524C2×C4#52511yescharacteristic
525C2×C4#52511yescharacteristic
526C2×C4#52512yes—
527C2×C4C2×#12411yescharacteristic
528C2×C4#52912yes—
529C2×C4#52912yes—
530C2×C4#53511yescharacteristic
531C2×C4#53711yescharacteristic
532C2×C4#53812yes—
533C2×C4#53912yes—
534C2×C4#54612yes—
535C2×C4#54811yescharacteristic
536C2×C4#55011yescharacteristic
537C2×C4#55112yes—
538C2×C4#55212yes—
539C2×C4#55312yes—
540C2×C4#55312yes—
541C8—42no—
542C8—81no—
543C8—82no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22#240711yescharacteristic
560C22#241912yes—
561C22#243511yescharacteristic
562C22#244412yes—
563C22#284911yescharacteristic
564C22C2×#45211yescharacteristic
565C22#300511yescharacteristic
566C22#301011yescharacteristic
567C22#301211yescharacteristic
568C22#309112yes—
569C22#309512yes—
570C22#309612yes—
571C22#309912yes—
572C22#330312yes—
573C22#330511yescharacteristic
574C22#330912yes—
575C22#331112yes—
576C22#331712yes—
577C22#332911yescharacteristic
578C22#333112yes—
579C22#334011yescharacteristic
580C22#335012yes—
581C22#335711yescharacteristic
582C4—24no—
583C4—44no—
584C4—44no—
585C4—42no—
586C4—84no—
587C4—82no—
588C4—82no—
589C4—162no—
590C4—161no—
591C4#239811yescharacteristic
592C4#246012yes—
593C4#246712yes—
594C4#256211yescharacteristic
595C4#259811yescharacteristic
596C4#263611yescharacteristic
597C2—84no—
598C2#1486011yescharacteristic
599C2#2412111yescharacteristic
600C2#2413111yescharacteristic
601C2#2494612yes—
602C2#2496012yes—
603C2#2588611yescharacteristic
604C2#2589311yescharacteristic
605C2#2621312yes—
606C2#2622312yes—
607C2#3298411yescharacteristic
608C2#3298711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
3. C2×#1920C2semidirect
5. #11357C2semidirect
6. #11199C2semidirect
7. #11095C2semidirect
9. #10806C2semidirect
10. #10731C2semidirect
11. #10731C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#21002
C2×#19202
#151812
#113572
#111992
#110952
#109642
#108062
#107312