Finite Group Databasebeta

← #403407

Subgroups — #403407

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2093C211yesmaximal, characteristic
3C2×#1925C211yesmaximal, characteristic
4#15157C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11324C212yesmaximal, complemented
8#11049C212yesmaximal
9#10931C212yesmaximal
10#10907C212yesmaximal, complemented
11#10806C211yesmaximal, characteristic, complemented
12C22×C4×Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C2×#393—21no—
20Q8×C4⋊C4—24no—
21Q8×C4⋊C4—24no—
22C2×#390C2212yes—
23C2×#390C2211yescharacteristic
24C2×#389—22no—
25C2×#389—21no—
26C2×#384C2212yes—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2093—24no—
31#1925—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41#1439—21no—
42#1439—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4×Q8—28no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—24no—
54C2×C4×Q8—24no—
55C2×C4×Q8—44no—
56C2×C4×Q8—44no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42—22no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—22no—
79C2×#107—42no—
80C2×#107—42no—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—22no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102Q814yescomplemented
89#393—44no—
90#390—22no—
91#390—22no—
92#390—24no—
93#390—24no—
94#389—28no—
95#389—216no—
96#384—24no—
97#384—24no—
98#384—22no—
99#384D414yes—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D414yes—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110#340—24no—
111#340—22no—
112#340—24no—
113#340—42no—
114#340—42no—
115#336—22no—
116#336—22no—
117#336—24no—
118#336—24no—
119C25—21no—
120C22×Q8—22no—
121C22×Q8—24no—
122C22×Q8—22no—
123C22×Q8—24no—
124C22×Q8D814yes—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C2×Q812yes—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2411yescharacteristic
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×Q812yes—
161C22×C8#4114yes—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8—216no—
165C4×Q8—432no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
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—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4#4112yes—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—28no—
226C2×C42—24no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42#4114yes—
234C2×C42#4114yes—
235#107—48no—
236#107—44no—
237#102—28no—
238#102—48no—
239#102—44no—
240#95—28no—
241#95—24no—
242#95—42no—
243#95—44no—
244#95—82no—
245#95—81no—
246C24—22no—
247C24—41no—
248C24—42no—
249C24—41no—
250C24—42no—
251C24—41no—
252C24—41no—
253C24—42no—
254C24—42no—
255C24—41no—
256C24—41no—
257C24C22×D411yescharacteristic
258C2×Q8—28no—
259C2×Q8—216no—
260C2×Q8—28no—
261C2×Q8—28no—
262C2×Q8—216no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C22×C4—22no—
282C22×C4—22no—
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—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4#12012yes—
329C22×C4#12012yes—
330C22×C4#12314yes—
331C22×C4#12314yes—
332C22×C4C2×D812yes—
333C22×C4C2×D814yes—
334C22×C4AGL(1,ℤ/8)12yes—
335C22×C4#13714yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q811yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4H(5,𝔽2)12yes—
344C22×C42−1+411yescharacteristic
345C22×C42−1+411yescharacteristic
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—28no—
364C4⋊C4—216no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—416no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—48no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—84no—
385C4⋊C4—88no—
386C4⋊C4C22⋊Q814yes—
387C22⋊C4—88no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C22⋊C4—88no—
391C42—22no—
392C42—22no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—416no—
399C42—48no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—42no—
404C23—44no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—81no—
418C23—82no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23C2×#12011yescharacteristic
424C23C2×#12312yes—
425C23#54811yescharacteristic
426C23D4×Q811yescharacteristic
427C23#55012yes—
428C23C22×D811yescharacteristic
429C23C2×AGL(1,ℤ/8)11yescharacteristic
430C23C2×#13712yes—
431C23#57612yes—
432C23#57711yescharacteristic
433C23#57811yescharacteristic
434Q8—232no—
435D4—42no—
436D4—42no—
437D4—44no—
438D4—81no—
439D4—81no—
440D4—82no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4#44714yes—
495C2×C4#48018yes—
496C2×C4C2×C22⋊Q811yescharacteristic
497C2×C4#52511yescharacteristic
498C2×C4#52511yescharacteristic
499C2×C4#52512yes—
500C2×C4#52912yes—
501C2×C4#53211yescharacteristic
502C2×C4#53312yes—
503C2×C4#53511yescharacteristic
504C2×C4#53611yescharacteristic
505C2×C4#54212yes—
506C2×C4#54212yes—
507C2×C4#54612yes—
508C2×C4#54811yescharacteristic
509C2×C4D4×Q811yescharacteristic
510C2×C4#55212yes—
511C2×C4#55412yes—
512C2×C4#55612yes—
513C2×C4#55912yes—
514C8—42no—
515C8—44no—
516C8—81no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—84no—
521C22—84no—
522C22—161no—
523C22—161no—
524C22—162no—
525C22—161no—
526C22—162no—
527C22—161no—
528C22#240911yescharacteristic
529C22#242312yes—
530C22#243111yescharacteristic
531C22#244812yes—
532C22#284911yescharacteristic
533C22C2×#44711yescharacteristic
534C22#300411yescharacteristic
535C22#300811yescharacteristic
536C22#301311yescharacteristic
537C22C2×#48012yes—
538C22#309212yes—
539C22#309612yes—
540C22#309812yes—
541C22#335811yescharacteristic
542C22#336711yescharacteristic
543C22#337512yes—
544C22Q8×D811yescharacteristic
545C22#338411yescharacteristic
546C22#338512yes—
547C22#339112yes—
548C22#339812yes—
549C22#339912yes—
550C22#340012yes—
551C4—24no—
552C4—28no—
553C4—44no—
554C4—44no—
555C4—42no—
556C4—82no—
557C4—84no—
558C4—161no—
559C4—162no—
560C4#239611yescharacteristic
561C4#245812yes—
562C4#246512yes—
563C4#258011yescharacteristic
564C4#260411yescharacteristic
565C4#263211yescharacteristic
566C2—162no—
567C2#1487511yescharacteristic
568C2#2432611yescharacteristic
569C2#2434111yescharacteristic
570C2#2505612yes—
571C2#2506812yes—
572C2#2563111yescharacteristic
573C2#2564011yescharacteristic
574C2#2631612yes—
575C2#2633012yes—
576C2#3297811yescharacteristic
577C2#3298611yescharacteristic
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
1. C2×Q8×C4⋊C4C2semidirect
5. #11380C2semidirect
6. #11357C2semidirect
7. #11324C2semidirect
10. #10907C2semidirect
11. #10806C2semidirect
88. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#20932
C2×#19252
#151572
#113802
#113572
#113242
#110492
#109312
#109072
#108062