Finite Group Databasebeta

← #468074

Subgroups — #468074

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1862C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic, complemented
5#11380C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11328C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#11189C212yesmaximal
10#11066C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C22⋊Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19#2538—24no—
20#2538—24no—
21C2×#396—22no—
22C2×#395—21no—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29#2099—24no—
30#1862—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42#1441—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8D411yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C22⋊Q8—28no—
52C2×C22⋊Q8—44no—
53C2×C22⋊Q8—44no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—42no—
57C2×C4×Q8—42no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4D414yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×#107—22no—
80C2×#107—42no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102Q814yescomplemented
89#396—216no—
90#395—44no—
91#390—24no—
92#390—24no—
93#389—24no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—42no—
98#389—42no—
99#384—22no—
100#384D414yes—
101#380—28no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—24no—
113#340—24no—
114#340—42no—
115#336—22no—
116#336—22no—
117#336—42no—
118#336—42no—
119C25—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8D814yes—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C2×D412yes—
133C22×D4C2×Q812yes—
134C22×D4#4112yes—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—21no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×Q812yes—
167C22⋊Q8—432no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×Q8—48no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—48no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237#107—44no—
238#102—28no—
239#102—48no—
240#102—44no—
241#95—28no—
242#95—44no—
243#95—42no—
244#95—44no—
245#95—42no—
246#95—81no—
247#95—82no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—42no—
252C24—42no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24—41no—
258C24—41no—
259C24C22×D411yescharacteristic
260C2×Q8—28no—
261C2×Q8—48no—
262C2×Q8—44no—
263C2×Q8—48no—
264C2×Q8—44no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4C22⋊Q814yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—48no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4#12012yes—
339C22×C4#12012yes—
340C22×C4C2×D814yes—
341C22×C4C2×D812yes—
342C22×C4AGL(1,ℤ/8)12yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×Q811yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4H(5,𝔽2)12yes—
349C22×C4H(5,𝔽2)12yes—
350C22×C42−1+411yescharacteristic
351C22×C42−1+412yes—
352C22×C42−1+411yescharacteristic
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—416no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—48no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—84no—
393C4⋊C4—88no—
394C4⋊C4—84no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—88no—
398C42—22no—
399C42—22no—
400C42—24no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—42no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—44no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23C2×#12011yescharacteristic
429C23#53412yes—
430C23#54811yescharacteristic
431C23#54812yes—
432C23D4×Q811yescharacteristic
433C23C22×D811yescharacteristic
434C23C2×AGL(1,ℤ/8)11yescharacteristic
435C23#57512yes—
436C23#57712yes—
437C23#57711yescharacteristic
438C23#57811yescharacteristic
439Q8—416no—
440D4—24no—
441D4—24no—
442D4—42no—
443D4—42no—
444D4—44no—
445D4—44no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—816no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4#44714yes—
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4#53211yescharacteristic
504C2×C4#53411yescharacteristic
505C2×C4#53411yescharacteristic
506C2×C4#53512yes—
507C2×C4#53611yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53912yes—
510C2×C4#53912yes—
511C2×C4#54212yes—
512C2×C4#54412yes—
513C2×C4#54712yes—
514C2×C4#54811yescharacteristic
515C2×C4D4×Q811yescharacteristic
516C2×C4#55112yes—
517C2×C4#55412yes—
518C2×C4#56312yes—
519C2×C4#56312yes—
520C8—42no—
521C8—81no—
522C8—82no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—84no—
527C22—84no—
528C22—82no—
529C22—84no—
530C22—82no—
531C22—84no—
532C22—82no—
533C22—82no—
534C22#240911yescharacteristic
535C22#243211yescharacteristic
536C22#261311yescharacteristic
537C22#272812yes—
538C22#272912yes—
539C22C2×#44711yescharacteristic
540C22#300511yescharacteristic
541C22#300911yescharacteristic
542C22#301311yescharacteristic
543C22#318612yes—
544C22#319412yes—
545C22#320112yes—
546C22#320412yes—
547C22#335811yescharacteristic
548C22#335812yes—
549C22#336711yescharacteristic
550C22#336812yes—
551C22#337612yes—
552C22#337612yes—
553C22#337712yes—
554C22Q8×D811yescharacteristic
555C22#338411yescharacteristic
556C22#338812yes—
557C4—24no—
558C4—44no—
559C4—42no—
560C4—82no—
561C4—84no—
562C4—84no—
563C4—82no—
564C4—82no—
565C4—161no—
566C4#256411yescharacteristic
567C4#262711yescharacteristic
568C4#266311yescharacteristic
569C4#272012yes—
570C4#272312yes—
571C4#281711yescharacteristic
572C2—84no—
573C2#1660111yescharacteristic
574C2#2432611yescharacteristic
575C2#2434311yescharacteristic
576C2#2564511yescharacteristic
577C2#2565611yescharacteristic
578C2#2959211yescharacteristic
579C2#2960911yescharacteristic
580C2#3254412yes—
581C2#3255612yes—
582C2#3256412yes—
583C2#3258512yes—
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×#2099C2semidirect
4. #15158C2semidirect
5. #11380C2semidirect
6. #11357C2semidirect
7. #11328C2semidirect
8. #11292C2semidirect
88. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#20992
C2×#18622
#151582
#113802
#113572
#113282
#112922
#111892
#110662
#107882