Finite Group Databasebeta

← #403630

Subgroups — #403630

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic
4#14699C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11220C212yesmaximal
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#107C2212yes—
15C22×#106C2212yes—
16C4×C4⋊Q8—24no—
17C4×C4⋊Q8—24no—
18C2×#391—21no—
19C2×#389—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C43C2211yescharacteristic
28#2083—24no—
29#2058—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46#1434—21no—
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C2×C4×Q8—28no—
55C2×C4×Q8—44no—
56C2×C4×Q8—44no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42—22no—
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—42no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83#391—28no—
84#389—28no—
85#389—24no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#387—44no—
92#384—22no—
93#384D414yes—
94C4×C4⋊C4—28no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C43—22no—
110C43—22no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—42no—
117#336—22no—
118#336—22no—
119#336—42no—
120#336Q818yescomplemented
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C2×Q814yes—
154C22×C8C2×Q814yes—
155C4⋊Q8—216no—
156C4⋊Q8—216no—
157C4×Q8—432no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4#4112yes—
203C2×C4⋊C4#4112yes—
204C2×C4⋊C4#4114yes—
205C2×C4⋊C4#4114yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—28no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×Q814yes—
235#107—48no—
236#106—48no—
237#95—28no—
238#95—28no—
239#95—24no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249C24C22×D411yescharacteristic
250C2×Q8—216no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—48no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
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—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4#12012yes—
330C22×C4#12012yes—
331C22×C4C4⋊Q812yes—
332C22×C4C4⋊Q812yes—
333C22×C4#13512yes—
334C22×C4#13512yes—
335C22×C4AGL(1,ℤ/8)12yes—
336C22×C4AGL(1,ℤ/8)12yes—
337C22×C4AGL(1,ℤ/8)12yes—
338C22×C4#13712yes—
339C22×C4#13712yes—
340C22×C4#13712yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×Q812yes—
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C42−1+411yescharacteristic
350C22×C42−1+412yes—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
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—
365C4⋊C4—216no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—416no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—84no—
388C4⋊C4#12318yes—
389C42—28no—
390C42—28no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—24no—
395C42—416no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—44no—
401C42—44no—
402C23C2×#12011yescharacteristic
403C23#52511yescharacteristic
404C23#52511yescharacteristic
405C23#52511yescharacteristic
406C23C2×C4⋊Q811yescharacteristic
407C23#53611yescharacteristic
408C23#53711yescharacteristic
409C23C2×#13511yescharacteristic
410C23C2×#13511yescharacteristic
411C23C2×AGL(1,ℤ/8)11yescharacteristic
412C23C2×AGL(1,ℤ/8)11yescharacteristic
413C23C2×AGL(1,ℤ/8)11yescharacteristic
414C23C2×#13711yescharacteristic
415C23C2×#13711yescharacteristic
416C23C2×#13711yescharacteristic
417Q8—432no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4#44914yes—
500C2×C4#45214yes—
501C2×C4#50114yes—
502C2×C4C2×#12312yes—
503C2×C4#52512yes—
504C2×C4#53312yes—
505C2×C4#53312yes—
506C2×C4#54112yes—
507C2×C4#54212yes—
508C2×C4#54611yescharacteristic
509C2×C4#54711yescharacteristic
510C2×C4#54811yescharacteristic
511C2×C4D4×Q812yes—
512C2×C4D4×Q812yes—
513C2×C4#55011yescharacteristic
514C2×C4#55212yes—
515C2×C4#55412yes—
516C2×C4#55612yes—
517C2×C4#55712yes—
518C8—44no—
519C8—44no—
520C22#239911yescharacteristic
521C22#240711yescharacteristic
522C22#243411yescharacteristic
523C22#258411yescharacteristic
524C22#259911yescharacteristic
525C22#263711yescharacteristic
526C22#284911yescharacteristic
527C22C2×#44911yescharacteristic
528C22C2×#45211yescharacteristic
529C22#300311yescharacteristic
530C22#300411yescharacteristic
531C22#310511yescharacteristic
532C22#310511yescharacteristic
533C22#310611yescharacteristic
534C22#310711yescharacteristic
535C22#310811yescharacteristic
536C22#310811yescharacteristic
537C22#310911yescharacteristic
538C22#311011yescharacteristic
539C22#311011yescharacteristic
540C22#311211yescharacteristic
541C22#311211yescharacteristic
542C22#311311yescharacteristic
543C22#316011yescharacteristic
544C22C2×#50111yescharacteristic
545C22#316511yescharacteristic
546C22#316511yescharacteristic
547C22#322811yescharacteristic
548C22#323111yescharacteristic
549C22#323811yescharacteristic
550C22#323911yescharacteristic
551C22#324711yescharacteristic
552C22#324811yescharacteristic
553C22#325311yescharacteristic
554C22#325511yescharacteristic
555C4—24no—
556C4—24no—
557C4—24no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—48no—
562C4—82no—
563C4—84no—
564C4—82no—
565C4—161no—
566C4—161no—
567C4#242312yes—
568C4#244612yes—
569C4#246212yes—
570C4#246512yes—
571C2#1487911yescharacteristic
572C2#2383811yescharacteristic
573C2#2383911yescharacteristic
574C2#2415211yescharacteristic
575C2#2415711yescharacteristic
576C2#2581211yescharacteristic
577C2#2581611yescharacteristic
578C2#2931511yescharacteristic
579C2#2932011yescharacteristic
580C2#2938911yescharacteristic
581C2#2939511yescharacteristic
582C2#3014611yescharacteristic
583C2#3015211yescharacteristic
584C2#3300311yescharacteristic
585C2#3300911yescharacteristic
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
120. #336Q8semidirect

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×#20832
C2×#20582
#146992
#113842
#113572
#113282
#112202
#109082
#109072
#107952