Finite Group Databasebeta

← #215526

Subgroups — #215526

Order 512.

1228 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
1C4×#535C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#21431C211yesmaximal, characteristic, complemented
4#9678C212yesmaximal, complemented
5#9676C212yesmaximal, complemented
6#9648C211yesmaximal, characteristic, complemented
7#9646C211yesmaximal, characteristic, complemented
8#9645C211yesmaximal, characteristic, complemented
9#9644C211yesmaximal, characteristic, complemented
10#8312C212yesmaximal, complemented
11#8310C212yesmaximal, complemented
12C2×#535C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15#2328C411yescharacteristic
16#2328C411yescharacteristic
17#2326C411yescharacteristic
18#2326C411yescharacteristic
19#2324—22no—
20#2321—22no—
21#2317C411yescharacteristic
22#2311C411yescharacteristic
23#2309C411yescharacteristic
24C4×#127C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C4×#124C2211yescharacteristic
27C4×#124C2211yescharacteristic
28C4×#121—22no—
29C4×#120—22no—
30C4×#117C2211yescharacteristic
31#1924—22no—
32#1924—22no—
33#1917—22no—
34#1917—22no—
35#1842C2211yescharacteristic, complemented
36#1831C2211yescharacteristic, complemented
37#1797C2212yes—
38#1767C2212yescomplemented
39C4×C4⋊C8C2212yescomplemented
40C4×C22⋊C8C2212yes—
41#1744C2211yescharacteristic, complemented
42C4×#97C2211yescharacteristic, complemented
43C2×Q8⋊C8C2212yescomplemented
44C2×Q8⋊C8C2212yescomplemented
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yescomplemented
50C2×D4⋊C8C2212yescomplemented
51C22×#41C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C24×C4—21no—
54#535—24no—
55#535—24no—
56C2×#127C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#124C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#121—22no—
61C2×#120—22no—
62C4×#41—42no—
63C4×#41—42no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—42no—
73C2×C4×D4—42no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C2×#117C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×#97—22no—
95C2×#97—22no—
96C2×C4×C8C2312yes—
97C2×C4×C8C2312yes—
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#390—22no—
101#390—22no—
102#390—22no—
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#389—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#380—22no—
127#379—22no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D412yes—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4D412yes—
133C4×C22⋊C4D412yes—
134#376C2×C411yescharacteristic
135#376C2×C411yescharacteristic
136C43C2311yescharacteristic
137C43C2311yescharacteristic
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144Q8⋊C8—42no—
145Q8⋊C8—42no—
146D4⋊C8—42no—
147D4⋊C8—42no—
148D4⋊C8—42no—
149D4⋊C8—42no—
150D4⋊C8—42no—
151D4⋊C8—42no—
152C25—21no—
153C2×#41—22no—
154C2×#41—42no—
155C2×#41—42no—
156C2×#41M1614yes—
157C22×Q8C22×C411yescharacteristic
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4M1614yes—
166C22×D4C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—82no—
180C23×C4—82no—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188#127—22no—
189#127—22no—
190#127—22no—
191#127—22no—
192#124—22no—
193#124—22no—
194#124—22no—
195#124—22no—
196#121—216no—
197#120—216no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—42no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—82no—
217C4×D4—82no—
218C4×D4—82no—
219C4×D4—82no—
220#117—22no—
221#117—22no—
222#117—42no—
223#117—44no—
224#117—42no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C22⋊C412yes—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C22⋊C412yes—
254C2×C22⋊C4C22⋊C412yes—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2411yescharacteristic
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—44no—
279C4⋊C8—82no—
280C22⋊C8—44no—
281C22⋊C8—82no—
282#97—44no—
283#97—82no—
284C4×C8—22no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95C22⋊C412yes—
293#95C22⋊C412yes—
294#95C22⋊C412yes—
295#95C22⋊C412yes—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—24no—
300C24—42no—
301C24—42no—
302C24—82no—
303C24—82no—
304C24C2×C22⋊C411yescharacteristic
305#41—48no—
306#41—48no—
307C2×Q8—21no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8C2×M1612yes—
311C2×D4—24no—
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—48no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—48no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4C2×M1612yes—
335C2×D4C2×M1612yes—
336C2×D4C2×M1612yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×M1612yes—
384C22×C4C2×M1612yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—24no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C42—22no—
429C42—21no—
430C42—21no—
431C42—22no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—41no—
447C42—41no—
448C42H(5,𝔽2)12yes—
449C42H(5,𝔽2)12yes—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—48no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—88no—
471C23—84no—
472C23—84no—
473C23—88no—
474C23#40712yes—
475C23C22×C22⋊C411yescharacteristic
476Q8—24no—
477Q8—42no—
478D4—24no—
479D4—24no—
480D4—24no—
481D4—48no—
482D4—48no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—88no—
487D4—88no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—48no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#40712yes—
539C2×C4#51812yes—
540C2×C4#53512yes—
541C2×C4#53511yescharacteristic
542C2×C4#53511yescharacteristic
543C2×C4C22×M1611yescharacteristic
544C2×C4C22×M1611yescharacteristic
545C2×C4#56812yes—
546C2×C4#57612yes—
547C2×C4#57712yes—
548C8—44no—
549C8—82no—
550C8—82no—
551C8—82no—
552C22—22no—
553C22—24no—
554C22—41no—
555C22—41no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—816no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22#230911yescharacteristic
574C22C2×#40711yescharacteristic
575C22#287911yescharacteristic
576C22#289712yes—
577C22#290112yes—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4#297312yes—
592C4#297312yes—
593C4#322011yescharacteristic
594C4#322211yescharacteristic
595C4#322412yes—
596C2—42no—
597C2—82no—
598C2—82no—
599C2—82no—
600C2#1745312yes—
601C2#1750512yes—
602C2#2179011yescharacteristic
603C2#2181311yescharacteristic
604C2#2181711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #21431C2semidirect
4. #9678C2semidirect
5. #9676C2semidirect
6. #9648C2semidirect
7. #9646C2semidirect
8. #9645C2semidirect
9. #9644C2semidirect
10. #8312C2semidirect
11. #8310C2semidirect
35. #1842C22semidirect
36. #1831C22semidirect
38. #1767C22semidirect
39. C4×C4⋊C8C22semidirect
41. #1744C22semidirect
42. C4×#97C22semidirect
43. C2×Q8⋊C8C22semidirect
44. C2×Q8⋊C8C22semidirect
49. C2×D4⋊C8C22semidirect
50. C2×D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#219262
#214312
#96782
#96762
#96482
#96462
#96452
#96442
#83122
#83102