Finite Group Databasebeta

← #404545

Subgroups — #404545

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2319C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#2093C211yesmaximal, characteristic
4#14277C211yesmaximal, characteristic, complemented
5#11328C212yesmaximal, complemented
6#11208C211yesmaximal, characteristic, complemented
7#11066C212yesmaximal
8#11051C211yesmaximal, characteristic
9#10907C212yesmaximal, complemented
10#10796C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#107C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16#2319—24no—
17#2319—24no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#388—21no—
24C2×#388—21no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#381—21no—
28C2×#380—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#2100—24no—
32#2093—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1444—21no—
43#1444—21no—
44#1442—21no—
45#1442—21no—
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390D414yes—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—42no—
93#389—42no—
94#389D414yes—
95#388—28no—
96#388—28no—
97#384—22no—
98#384—22no—
99#382—24no—
100#381—28no—
101#380—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104#376—22no—
105#376—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—42no—
123#336—24no—
124#336—22no—
125#336—22no—
126#336—24no—
127C25—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C2×D412yes—
136C22×D4C2×Q812yes—
137C22×D4#4112yes—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
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—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8#4114yes—
169C4×D4—28no—
170C4×D4—28no—
171C4×D4—28no—
172C4×D4—28no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
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—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×Q812yes—
208C2×C4⋊C4#4114yes—
209C2×C4⋊C4#4112yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—82no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×Q812yes—
236C2×C42#4112yes—
237#107—48no—
238#102—48no—
239#102—48no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—44no—
250#95—82no—
251#95—82no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24C22×D411yescharacteristic
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4C22⋊Q814yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—216no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4#12012yes—
341C22×C4#12012yes—
342C22×C4C22⋊Q812yes—
343C22×C4C22⋊Q812yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+412yes—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4C22⋊Q814yes—
389C22⋊C4—216no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—24no—
399C42—24no—
400C42—42no—
401C42—42no—
402C42—84no—
403C23—24no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—44no—
408C23—42no—
409C23—44no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23C2×#12011yescharacteristic
427C23C2×C22⋊Q811yescharacteristic
428C23#52511yescharacteristic
429C23#52511yescharacteristic
430C23#53211yescharacteristic
431C23#53511yescharacteristic
432C23#53611yescharacteristic
433C23#57612yes—
434C23#57612yes—
435C23#57712yes—
436C23#57812yes—
437D4—28no—
438D4—44no—
439D4—44no—
440D4—44no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—816no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4#52511yescharacteristic
504C2×C4#52511yescharacteristic
505C2×C4#52912yes—
506C2×C4#53312yes—
507C2×C4#54212yes—
508C2×C4#54412yes—
509C2×C4#54612yes—
510C2×C4#54712yes—
511C2×C4#54712yes—
512C2×C4#55012yes—
513C2×C4#55112yes—
514C2×C4#55212yes—
515C2×C4#55212yes—
516C2×C4#55312yes—
517C8—44no—
518C8—82no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—84no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22#239611yescharacteristic
531C22#240611yescharacteristic
532C22#243111yescharacteristic
533C22#259811yescharacteristic
534C22#261111yescharacteristic
535C22#262811yescharacteristic
536C22#284811yescharacteristic
537C22#300811yescharacteristic
538C22#300811yescharacteristic
539C22#300911yescharacteristic
540C22#301111yescharacteristic
541C22#308112yes—
542C22#308312yes—
543C22#311812yes—
544C22#311912yes—
545C22#312412yes—
546C22#312412yes—
547C22#317612yes—
548C22#317812yes—
549C22#322012yes—
550C22#322512yes—
551C22#324112yes—
552C22#324312yes—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—82no—
559C4—81no—
560C4—82no—
561C4—81no—
562C4—84no—
563C4—162no—
564C4#243812yes—
565C4#244112yes—
566C4#247012yes—
567C4#247912yes—
568C2—84no—
569C2#1489311yescharacteristic
570C2#2379112yes—
571C2#2406911yescharacteristic
572C2#2407311yescharacteristic
573C2#2560811yescharacteristic
574C2#2561311yescharacteristic
575C2#2938112yes—
576C2#2958412yes—
577C2#2999112yes—
578C2#3296211yescharacteristic
579C2#3296311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
4. #14277C2semidirect
5. #11328C2semidirect
6. #11208C2semidirect
9. #10907C2semidirect
10. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23192
C2×#21002
C2×#20932
#142772
#113282
#112082
#110662
#110512
#109072
#107962
#107882