Finite Group Databasebeta

← #412264

Subgroups — #412264

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#436C211yesmaximal, characteristic
2C2×#1941C211yesmaximal, characteristic, complemented
3C2×#1809C212yesmaximal, complemented
4C2×#1809C212yesmaximal, complemented
5#15181C211yesmaximal, characteristic
6#15129C211yesmaximal, characteristic
7#11228C211yesmaximal, characteristic, complemented
8#11227C211yesmaximal, characteristic, complemented
9#11200C212yesmaximal, complemented
10#11199C212yesmaximal, complemented
11#10862C211yesmaximal, characteristic, complemented
12C23×M16C2212yes—
13C22×C4×D4C2211yescharacteristic
14C2×#436—24no—
15C2×#436—28no—
16C22×#107C2212yes—
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#394—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29#1941—24no—
30#1809—28no—
31#1809—28no—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1442—21no—
40#1441—21no—
41#1440—22no—
42#1439—22no—
43#1438—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C22×M16—24no—
49C22×M16—24no—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16D418yes—
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42C2311yescharacteristic
77#436—232no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—24no—
84C2×#106—22no—
85C2×#106D414yescomplemented
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C4×C8—24no—
92C2×C4×C8D414yes—
93#394—44no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—24no—
102#380—24no—
103#380—24no—
104#380—42no—
105#380—42no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—24no—
121#340—24no—
122#340—42no—
123C25C2×D412yes—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—42no—
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—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D414yes—
150C23×C4C2×D411yescharacteristic
151C23×C4#4114yes—
152C23×C4C2411yescharacteristic
153C2×M16—216no—
154C2×M16—216no—
155C2×M16—216no—
156C2×M16—48no—
157C2×M16—48no—
158C2×M16—48no—
159C2×M16—48no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8C2×D414yes—
171C22×C8C2×D412yes—
172C22×C8C2×D414yes—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8#4112yes—
176C22×C8#4114yes—
177C4×D4—28no—
178C4×D4—216no—
179C4×D4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
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—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42C2×D412yes—
237C2×C42#4112yes—
238#107—48no—
239#106—28no—
240#106—44no—
241C4⋊C8—28no—
242C22⋊C8—216no—
243C4×C8—28no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—81no—
253#95—81no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24#12114yes—
269C24C22×D411yescharacteristic
270C2×D4—28no—
271C2×D4—48no—
272C2×D4—48no—
273C2×D4—44no—
274C2×D4—44no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
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—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
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—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
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—41no—
332C22×C4—41no—
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—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4#12114yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4AGL(1,ℤ/8)12yes—
355C22×C4#13712yes—
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+412yes—
366M16—432no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8#12014yes—
380C2×C8#12114yes—
381C2×C8#12118yes—
382C2×C8#12318yes—
383C2×C8#12414yes—
384C4⋊C4—28no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C22⋊C4—216no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C42—22no—
415C42—22no—
416C42—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
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—
450C23—42no—
451C23—42no—
452C23C2×#12112yes—
453C23C2×#12111yescharacteristic
454C23D4211yescharacteristic
455C23#54711yescharacteristic
456C23#54712yes—
457C23C2×AGL(1,ℤ/8)11yescharacteristic
458C23C2×#13711yescharacteristic
459C23#57512yes—
460C23#57611yescharacteristic
461C23#57712yes—
462C23#57811yescharacteristic
463D4—416no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
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—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4#46814yes—
525C2×C4C2×#12011yescharacteristic
526C2×C4C2×#12112yes—
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×#12312yes—
529C2×C4#52511yescharacteristic
530C2×C4#52512yes—
531C2×C4C2×#12411yescharacteristic
532C2×C4#53612yes—
533C2×C4#53611yescharacteristic
534C2×C4#53711yescharacteristic
535C2×C4#53712yes—
536C2×C4#54711yescharacteristic
537C2×C4#54812yes—
538C2×C4#55212yes—
539C2×C4#55212yes—
540C2×C4#55312yes—
541C2×C4#55511yescharacteristic
542C2×C4#55612yes—
543C8—42no—
544C8—44no—
545C8#39418yescomplemented
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22#241612yes—
562C22#248612yes—
563C22#253711yescharacteristic
564C22#254711yescharacteristic
565C22#284711yescharacteristic
566C22C2×#46811yescharacteristic
567C22#305411yescharacteristic
568C22#305512yes—
569C22#305612yes—
570C22#305711yescharacteristic
571C22#306111yescharacteristic
572C22#306212yes—
573C22#306312yes—
574C22#327411yescharacteristic
575C22#327512yes—
576C22#327611yescharacteristic
577C22#335112yes—
578C22#335212yes—
579C22#335311yescharacteristic
580C22#335412yes—
581C22#335512yes—
582C22#335612yes—
583C22#335711yescharacteristic
584C4—24no—
585C4—44no—
586C4—42no—
587C4—82no—
588C4—82no—
589C4—161no—
590C4—161no—
591C4—162no—
592C4—162no—
593C4C2×#39411yescharacteristic
594C4#241512yes—
595C4#248512yes—
596C4#253611yescharacteristic
597C4#256211yescharacteristic
598C4#261611yescharacteristic
599C2—84no—
600C2#1512911yescharacteristic
601C2#2479212yes—
602C2#2479312yes—
603C2#2767012yes—
604C2#2767112yes—
605C2#2822911yescharacteristic
606C2#2823011yescharacteristic
607C2#2870911yescharacteristic
608C2#2871311yescharacteristic
609C2#3293511yescharacteristic
610C2#3293811yescharacteristic
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×#1941C2semidirect
3. C2×#1809C2semidirect
4. C2×#1809C2semidirect
7. #11228C2semidirect
8. #11227C2semidirect
9. #11200C2semidirect
10. #11199C2semidirect
11. #10862C2semidirect
85. C2×#106D4semidirect
545. C8#394semidirect

Maximal subgroups

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

subgroupindex
C22×#4362
C2×#19412
C2×#18092
#151812
#151292
#112282
#112272
#112002
#111992
#108622