Finite Group Databasebeta

← #414725

Subgroups — #414725

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2538C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1984C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic
7#11328C214yesmaximal, complemented
8#11066C214yesmaximal
9#10849C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#474—24no—
16C2×#474—24no—
17C2×#474—24no—
18C22×#107C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2546—24no—
24#2546—24no—
25#2538—24no—
26#2538—24no—
27C2×#398—22no—
28C2×#396—22no—
29C2×#391—21no—
30C2×#390C2212yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#384C2211yescharacteristic
34C2×#380—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1990—24no—
37#1984—24no—
38C4×#102—24no—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#336C2214yes—
43C23×Q8—21no—
44C23×Q8D411yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—28no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—44no—
56C2×C4⋊Q8—44no—
57C2×C4⋊Q8D414yes—
58C2×C22⋊Q8—28no—
59C2×C22⋊Q8—44no—
60C2×C22⋊Q8—44no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4D414yes—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D412yes—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C42—21no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#474—232no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8D414yes—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102Q818yescomplemented
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106#398—216no—
107#396—216no—
108#391—28no—
109#390—24no—
110#390—24no—
111#389—24no—
112#389—24no—
113#389—28no—
114#389—28no—
115#384—22no—
116#384D414yes—
117#380—28no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4D414yes—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—24no—
123#340—28no—
124#340—24no—
125#340—28no—
126#336—28no—
127C25—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—24no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8D814yes—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C2×D412yes—
151C22×D4C2×Q814yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—24no—
174C22×C8—22no—
175C22×C8—24no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D414yes—
179C22×C8C2×Q814yes—
180C4⋊Q8—28no—
181C4⋊Q8—28no—
182C4⋊Q8—432no—
183C4⋊Q8—44no—
184C4⋊Q8—44no—
185C22⋊Q8—432no—
186C4×Q8—44no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—44no—
193C4×D4—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—216no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×Q814yes—
235C2×C4⋊C4#4114yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—84no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42#4112yes—
259#107—216no—
260C4⋊C8—28no—
261#102—216no—
262#102—216no—
263C4×C8—44no—
264#95—28no—
265#95—216no—
266#95—24no—
267C24—22no—
268C24—24no—
269C24—41no—
270C24—41no—
271C24—42no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—42no—
277C24—42no—
278C24C22×D411yescharacteristic
279C2×Q8—28no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—44no—
283C2×Q8—48no—
284C2×Q8—48no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—48no—
288C2×Q8—44no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4C4⋊Q814yescomplemented
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
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—44no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4#12012yes—
358C22×C4#12012yes—
359C22×C4#12012yes—
360C22×C4#12012yes—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4C2×D814yes—
364C22×C4C2×D812yes—
365C22×C4#13512yes—
366C22×C4AGL(1,ℤ/8)12yes—
367C22×C4AGL(1,ℤ/8)12yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×Q812yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4112yes—
376C22×C42−1+412yes—
377C22×C42−1+412yes—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—28no—
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—44no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—416no—
399C4⋊C4—432no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—84no—
411C4⋊C4C22⋊Q818yescomplemented
412C22⋊C4—416no—
413C22⋊C4—48no—
414C22⋊C4—88no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—48no—
420C42#12014yes—
421C42#12114yes—
422C42#12114yes—
423C42#12714yes—
424C23—24no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—81no—
445C23—81no—
446C23—82no—
447C23C2×#12011yescharacteristic
448C23C2×#12011yescharacteristic
449C23C2×#12111yescharacteristic
450C23#54812yes—
451C23D4×Q812yes—
452C23C22×D811yescharacteristic
453C23C2×#13511yescharacteristic
454C23C2×AGL(1,ℤ/8)11yescharacteristic
455C23C2×AGL(1,ℤ/8)11yescharacteristic
456C23#57712yes—
457C23#57812yes—
458Q8—416no—
459Q8—816no—
460D4—24no—
461D4—24no—
462D4—42no—
463D4—44no—
464D4—42no—
465D4—44no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—816no—
509C2×C4#39214yes—
510C2×C4#39214yes—
511C2×C4#44714yes—
512C2×C4#44714yes—
513C2×C4#44914yes—
514C2×C4#44914yes—
515C2×C4#46614yes—
516C2×C4#46914yes—
517C2×C4C2×#12011yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×C22⋊Q812yes—
521C2×C4C2×#12711yescharacteristic
522C2×C4C2×C4⋊Q811yescharacteristic
523C2×C4#53211yescharacteristic
524C2×C4#53312yes—
525C2×C4#53611yescharacteristic
526C2×C4#53611yescharacteristic
527C2×C4#53612yes—
528C2×C4#54112yes—
529C2×C4#54712yes—
530C2×C4#54712yes—
531C2×C4#54812yes—
532C2×C4D4×Q812yes—
533C2×C4D4×Q812yes—
534C2×C4D4×Q812yes—
535C8—44no—
536C8—44no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—84no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—84no—
547C22—84no—
548C22C2×#39211yescharacteristic
549C22#240912yes—
550C22#242912yes—
551C22#243211yescharacteristic
552C22#243411yescharacteristic
553C22C2×#44711yescharacteristic
554C22C2×#44711yescharacteristic
555C22C2×#44911yescharacteristic
556C22C2×#44911yescharacteristic
557C22#300911yescharacteristic
558C22#301111yescharacteristic
559C22#301312yes—
560C22C2×#46611yescharacteristic
561C22C2×#46911yescharacteristic
562C22#306312yes—
563C22#335812yes—
564C22#336412yes—
565C22#336712yes—
566C22#336812yes—
567C22Q8×D812yes—
568C22#338112yes—
569C22#338412yes—
570C22#338612yes—
571C4—24no—
572C4—48no—
573C4—44no—
574C4—84no—
575C4—82no—
576C4—84no—
577C4—82no—
578C4#199918yes—
579C4#242912yes—
580C4#243211yescharacteristic
581C4#243411yescharacteristic
582C4#254212yes—
583C4#262111yescharacteristic
584C4#285711yescharacteristic
585C2—84no—
586C2#1517811yescharacteristic
587C2C2×#199911yescharacteristic
588C2#2339411yescharacteristic
589C2#2432612yes—
590C2#2432812yes—
591C2#2546912yes—
592C2#2547812yes—
593C2#2564511yescharacteristic
594C2#2565011yescharacteristic
595C2#2583511yescharacteristic
596C2#2583811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
4. C2×#1990C2semidirect
5. C2×#1984C2semidirect
7. #11328C2semidirect
9. #10849C2semidirect
102. C2×#102Q8semidirect
310. C2×D4C4⋊Q8semidirect
411. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#25462
C2×#25382
C2×#19902
C2×#19842
C2×C4×#1022
#113282
#110662
#108492