Finite Group Databasebeta

← #482063

Subgroups — #482063

Order 512.

1063 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic, complemented
3C2×#1461C211yesmaximal, characteristic, complemented
4#11883C212yesmaximal, complemented
5#11882C212yesmaximal, complemented
6#11879C214yesmaximal, complemented
7#11698C214yesmaximal, complemented
8C2×#536—21no—
9C2×#532C411yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#431—21no—
12C2×#423—21no—
13C22×#101—21no—
14#2854—21no—
15#2854—21no—
16#2618C412yes—
17#2618C2212yes—
18#2548C411yescharacteristic
19#2548C2211yescharacteristic
20#2546C411yescharacteristic
21#2546C2211yescharacteristic
22#2545—21no—
23#2545—21no—
24#2538—21no—
25#2538—21no—
26#2536C412yes—
27#2536C2212yes—
28#2311—21no—
29#2285C411yescharacteristic
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34C2×#343C2212yes—
35#1461—21no—
36#1461—21no—
37#1461C2212yes—
38#1461C2212yes—
39C2×#324—21no—
40C2×#324C2212yes—
41#1346C2218yes—
42#1345C2218yes—
43C22×#41—21no—
44C23×Q8D411yescharacteristic
45C22×M16—22no—
46#536—44no—
47#536—44no—
48#532—41no—
49#532—41no—
50#532—41no—
51#532—41no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#124—21no—
58C2×#124—21no—
59C2×#123—22no—
60C2×#123—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×#121—21no—
64C2×#121C2×C412yes—
65C2×C4×Q8—41no—
66C2×C4×Q8—41no—
67C2×C4×Q8—41no—
68C2×C4×Q8—41no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4C2×C412yes—
79C2×#117—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86#431—22no—
87#431—22no—
88#431—41no—
89#431—41no—
90#423—22no—
91#423—22no—
92#423—41no—
93#423—41no—
94C2×C4⋊C8—41no—
95C2×C4⋊C8—41no—
96C2×C4⋊C8—41no—
97C2×#101—22no—
98C2×#101—22no—
99C2×#101—44no—
100C2×C22⋊C8—22no—
101C2×C4×C8—41no—
102#398C2×C412yes—
103#398C2×C412yes—
104#398C2×C412yes—
105#398C2312yes—
106#396—22no—
107#396—22no—
108#396—22no—
109#396—22no—
110#394C2×C412yes—
111#394C2×C412yes—
112#394C2×C412yes—
113#394C2312yes—
114#391—21no—
115#391—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#386—22no—
131#386—22no—
132#384—21no—
133#382C2×C412yes—
134#380—21no—
135#380—22no—
136#380—22no—
137#380C2×C412yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141#343—22no—
142#343C2314yes—
143#324—22no—
144#324—22no—
145#324—22no—
146#324C2314yes—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—44no—
150C2×#41—44no—
151C22×Q8—21no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8C22⋊C411yescharacteristic
156C22×Q8C22⋊C411yescharacteristic
157C22×Q8D814yes—
158C22×Q8SD1614yes—
159C22×Q8C2×D411yescharacteristic
160C22×Q8C2×D411yescharacteristic
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22×C412yes—
165C23×C4—22no—
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C2×M16—24no—
174C2×M16—42no—
175C2×M16—44no—
176C2×M16—44no—
177C22×C8—42no—
178C22×C8—42no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—44no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184#127—41no—
185#127—41no—
186#124—42no—
187#124—42no—
188#124—44no—
189#124—42no—
190#124—42no—
191#123—48no—
192#123—44no—
193#123—44no—
194C22⋊Q8—48no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197#121—42no—
198#121—42no—
199#121—41no—
200#121—41no—
201#121—41no—
202#121—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223#117—44no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2411yescharacteristic
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C4⋊C8—42no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295#101—44no—
296#101—44no—
297C22⋊C8—24no—
298C22⋊C8—42no—
299C4×C8—42no—
300C4×C8—42no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95C22×C412yes—
307#95C22×C412yes—
308C24—22no—
309C24C2×C22⋊C411yescharacteristic
310#41—88no—
311#41—88no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8#10214yes—
328C2×Q8#10214yes—
329C2×Q8C2×D812yes—
330C2×Q8C2×D812yes—
331C2×Q8C2×SD1612yes—
332C2×Q8C2×SD1612yes—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—82no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4#9911yescharacteristic
372C22×C4#9911yescharacteristic
373C22×C4#9911yescharacteristic
374C22×C4#9911yescharacteristic
375C22×C4#10214yes—
376C22×C4#10214yes—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×D812yes—
389C22×C4C2×SD1612yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393M16—44no—
394M16—44no—
395M16—84no—
396C2×C8—44no—
397C2×C8—44no—
398C2×C8—84no—
399C2×C8—84no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—82no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—42no—
462C42—42no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C2×#9911yescharacteristic
473C23C2×#9911yescharacteristic
474C23C2×#10212yes—
475C23#42112yes—
476C23C22×C22⋊C411yescharacteristic
477Q8—44no—
478Q8—44no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—84no—
484D4—82no—
485D4—84no—
486D4—84no—
487D4—84no—
488D4—84no—
489D4—82no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—88no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4C2×#9911yescharacteristic
545C2×C4C2×#9911yescharacteristic
546C2×C4C2×#9911yescharacteristic
547C2×C4C2×#9911yescharacteristic
548C2×C4C2×#10212yes—
549C2×C4C2×#10212yes—
550C2×C4C2×#10212yes—
551C2×C4C2×#10212yes—
552C2×C4C2×#10212yes—
553C2×C4C22×D811yescharacteristic
554C2×C4C22×SD1611yescharacteristic
555C8—84no—
556C8—84no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—42no—
564C22—42no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22#151912yes—
573C22#151912yes—
574C22C22×#9911yescharacteristic
575C22C22×#10211yescharacteristic
576C22C2×#42111yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—21no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4#213312yes—
597C2—21no—
598C2—21no—
599C2—22no—
600C2—22no—
601C2—162no—
602C2C2×#151911yescharacteristic
603C2#3334711yescharacteristic
604C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1801C2semidirect
3. C2×#1461C2semidirect
4. #11883C2semidirect
5. #11882C2semidirect
6. #11879C2semidirect
7. #11698C2semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×#18012
C2×#14612
#118832
#118822
#118792
#116982