Finite Group Databasebeta

← #403429

Subgroups — #403429

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4#14703C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11067C212yesmaximal
9#10932C212yesmaximal
10#10907C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C4×C4⋊Q8—24no—
18C4×C4⋊Q8—24no—
19C2×#391—21no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C43C2211yescharacteristic
30#2095—24no—
31#2024—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45#1434—21no—
46#1434—21no—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8D414yes—
55C2×C4×Q8—28no—
56C2×C4×Q8—44no—
57C2×C4×Q8—44no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C42—22no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—42no—
85#391—28no—
86#391—22no—
87#391—22no—
88#391—22no—
89#391—42no—
90#391Q814yescomplemented
91#389—28no—
92#389—24no—
93#389—24no—
94#389—42no—
95#384—22no—
96#384D414yes—
97C4×C4⋊C4—28no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C43—22no—
114C43—22no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—42no—
119#340—42no—
120#336—24no—
121#336—22no—
122#336—22no—
123#336—24no—
124C22×Q8—24no—
125C22×Q8—24no—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8C2×Q812yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
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#4114yes—
168C4⋊Q8—216no—
169C4⋊Q8—216no—
170C4×Q8—432no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
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—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×Q812yes—
208C2×C4⋊C4#4114yes—
209C2×C4⋊C4#4114yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—28no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42Q1614yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×Q812yes—
238C2×C42#4112yes—
239#107—48no—
240#103—48no—
241#103—48no—
242#95—28no—
243#95—28no—
244#95—24no—
245#95—24no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255C24C22×D411yescharacteristic
256C2×Q8—216no—
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—48no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4C22⋊Q812yes—
354C22×C4C22⋊Q812yes—
355C22×C4C2×Q1612yes—
356C22×C4C2×Q1612yes—
357C22×C4C2×Q1612yes—
358C22×C4#13512yes—
359C22×C4AGL(1,ℤ/8)12yes—
360C22×C4AGL(1,ℤ/8)12yes—
361C22×C4#13712yes—
362C22×C4#13712yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+411yescharacteristic
372C22×C42−1+411yescharacteristic
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—216no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—416no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—84no—
405C4⋊C4#12318yes—
406C42—28no—
407C42—24no—
408C42—24no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—28no—
414C42—44no—
415C42—416no—
416C42—42no—
417C42—42no—
418C42—44no—
419C23C2×#12011yescharacteristic
420C23C2×C22⋊Q811yescharacteristic
421C23#52511yescharacteristic
422C23#52511yescharacteristic
423C23#53211yescharacteristic
424C23#53511yescharacteristic
425C23#53611yescharacteristic
426C23C22×Q1611yescharacteristic
427C23C2×#13511yescharacteristic
428C23C2×AGL(1,ℤ/8)11yescharacteristic
429C23C2×AGL(1,ℤ/8)11yescharacteristic
430C23C2×#13711yescharacteristic
431C23C2×#13711yescharacteristic
432C23#57511yescharacteristic
433C23#57511yescharacteristic
434Q8—432no—
435Q8—44no—
436Q8—44no—
437Q8—82no—
438Q8—82no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4#45114yes—
521C2×C4#45214yes—
522C2×C4#47714yes—
523C2×C4C2×#12312yes—
524C2×C4#52512yes—
525C2×C4#52912yes—
526C2×C4#53312yes—
527C2×C4#54212yes—
528C2×C4#54212yes—
529C2×C4#54612yes—
530C2×C4#54811yescharacteristic
531C2×C4#54811yescharacteristic
532C2×C4D4×Q811yescharacteristic
533C2×C4D4×Q811yescharacteristic
534C2×C4#55012yes—
535C2×C4#55212yes—
536C2×C4#55412yes—
537C2×C4#55612yes—
538C2×C4#55912yes—
539C8—44no—
540C8—82no—
541C22#239611yescharacteristic
542C22#240911yescharacteristic
543C22#243111yescharacteristic
544C22#258011yescharacteristic
545C22#260411yescharacteristic
546C22#263211yescharacteristic
547C22#284911yescharacteristic
548C22C2×#45111yescharacteristic
549C22C2×#45211yescharacteristic
550C22#300311yescharacteristic
551C22#300511yescharacteristic
552C22C2×#47711yescharacteristic
553C22#307611yescharacteristic
554C22#307811yescharacteristic
555C22#307911yescharacteristic
556C22#310411yescharacteristic
557C22#310411yescharacteristic
558C22#310511yescharacteristic
559C22#310511yescharacteristic
560C22#311011yescharacteristic
561C22#311011yescharacteristic
562C22#311711yescharacteristic
563C22#311711yescharacteristic
564C22#317011yescharacteristic
565C22#317111yescharacteristic
566C22#318011yescharacteristic
567C22#318511yescharacteristic
568C22#320911yescharacteristic
569C22#321011yescharacteristic
570C22#321311yescharacteristic
571C22#321411yescharacteristic
572C22#323011yescharacteristic
573C22#323311yescharacteristic
574C22#323711yescharacteristic
575C22#323811yescharacteristic
576C4—24no—
577C4—24no—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—48no—
583C4—84no—
584C4—82no—
585C4—82no—
586C4—161no—
587C4—161no—
588C4#242312yes—
589C4#244812yes—
590C4#245812yes—
591C4#246512yes—
592C2#1487511yescharacteristic
593C2#2377711yescharacteristic
594C2#2378211yescharacteristic
595C2#2430211yescharacteristic
596C2#2430311yescharacteristic
597C2#2560011yescharacteristic
598C2#2560211yescharacteristic
599C2#2925911yescharacteristic
600C2#2926011yescharacteristic
601C2#2951511yescharacteristic
602C2#2951911yescharacteristic
603C2#3006111yescharacteristic
604C2#3006311yescharacteristic
605C2#3300911yescharacteristic
606C2#3301011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
90. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20952
C2×#20242
#147032
#113842
#113712
#113282
#110672
#109322
#109072
#107952