Finite Group Databasebeta

← #208382

Subgroups — #208382

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic, complemented
3#13806C211yesmaximal, characteristic, complemented
4#9911C211yesmaximal, characteristic, complemented
5#9908C211yesmaximal, characteristic, complemented
6#9715C211yesmaximal, characteristic
7#9712C211yesmaximal, characteristic
8#9157C212yesmaximal, complemented
9#9149C212yesmaximal, complemented
10#8220C212yesmaximal, complemented
11C4×#327C212yesmaximal, complemented
12C2×#539—21no—
13#2857C2211yescharacteristic
14#2856C2211yescharacteristic
15#2854C2212yes—
16#2639—21no—
17#2618—21no—
18#2560—22no—
19#2558—22no—
20#2512C412yescomplemented
21#2506C412yescomplemented
22#2452C412yescomplemented
23#2445C412yescomplemented
24#2325—21no—
25C4×#121C2212yes—
26C2×#383C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1916—22no—
30#1916—22no—
31#1817C2212yes—
32#1797C2211yescharacteristic
33#1797C2211yescharacteristic
34C4×C22⋊C8C2212yes—
35C2×#327C2212yes—
36C2×#327C2212yes—
37C2×#327C2212yes—
38C2×#327C2212yes—
39#1436—21no—
40#1436—21no—
41#1434—21no—
42#1434C2211yescharacteristic
43#1434C2211yescharacteristic
44C4×#95—21no—
45C4×#95C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#539—44no—
52#539—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#126—22no—
56C2×#125—22no—
57C2×#123—22no—
58C2×#123—22no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2312yes—
64C2×C4×D4—22no—
65C2×C4×D4C2×C412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C22⋊C8C2312yes—
77C2×C22⋊C8C2312yes—
78C2×C4×C8—22no—
79#400—22no—
80#398—22no—
81#397—22no—
82#396—22no—
83#396—22no—
84#394—22no—
85#394—22no—
86#393—22no—
87#391—21no—
88#390—21no—
89#390—21no—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#390C2×C412yes—
94#389—21no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#387—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#383—22no—
111#383—22no—
112#380—21no—
113#380C2312yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D412yes—
119C4×C22⋊C4—22no—
120#376—22no—
121#376—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95D411yescharacteristic
132C2×#95D411yescharacteristic
133C2×#95C2311yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137#336—21no—
138#336—21no—
139#336—24no—
140#336—22no—
141#327—24no—
142#327—24no—
143#327—24no—
144#327—24no—
145C22×Q8—21no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C4⋊Q8—81no—
165#127—81no—
166#126—48no—
167#125—48no—
168#123—48no—
169#123—44no—
170#123—44no—
171C22⋊Q8—48no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—42no—
179#121—42no—
180#121—42no—
181#121—42no—
182C4×D4—44no—
183C4×D4—44no—
184C4×D4—82no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—81no—
229C2×C42—81no—
230C2×C42—81no—
231C2×C42—81no—
232C2×C42C22×C412yes—
233C2×C42C22×C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C4⋊C8—81no—
241C4⋊C8—81no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C4×C8—82no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—81no—
265#95—81no—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95#4112yes—
271#95#4112yes—
272#95#4112yes—
273#95#4112yes—
274C24—22no—
275C24C22×D411yescharacteristic
276C2×Q8—42no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4—81no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
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—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
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—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
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—44no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C412yes—
361C22×C4#13512yes—
362C22×C4#13512yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4AGL(1,ℤ/8)12yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—81no—
393C4⋊C4—81no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—42no—
412C42—41no—
413C42—41no—
414C42—42no—
415C42—41no—
416C42—41no—
417C42—48no—
418C42—44no—
419C42—84no—
420C42—84no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23C22×C22⋊C411yescharacteristic
433C23#52511yescharacteristic
434C23#52511yescharacteristic
435C23C2×#13511yescharacteristic
436C23C2×#13511yescharacteristic
437C23C2×AGL(1,ℤ/8)11yescharacteristic
438C23C2×AGL(1,ℤ/8)11yescharacteristic
439Q8—162no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—82no—
444D4—82no—
445D4—82no—
446D4—162no—
447D4—162no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
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—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—164no—
529C2×C4#41612yes—
530C2×C4#41612yes—
531C2×C4#41814yes—
532C2×C4C4×#4111yescharacteristic
533C2×C4C4×#4111yescharacteristic
534C2×C4#53211yescharacteristic
535C2×C4#53211yescharacteristic
536C8—82no—
537C8—82no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22#229711yescharacteristic
558C22#288311yescharacteristic
559C22#288411yescharacteristic
560C22C2×#41611yescharacteristic
561C22C2×#41811yescharacteristic
562C22#290111yescharacteristic
563C22#290111yescharacteristic
564C22#310611yescharacteristic
565C22#310611yescharacteristic
566C22#310611yescharacteristic
567C22#310611yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—22no—
573C4—22no—
574C4—24no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—162no—
592C2#1761511yescharacteristic
593C2#1761611yescharacteristic
594C2#1762011yescharacteristic
595C2#1763111yescharacteristic
596C2#2139011yescharacteristic
597C2#2141911yescharacteristic
598C2#2142311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21912C2semidirect
3. #13806C2semidirect
4. #9911C2semidirect
5. #9908C2semidirect
8. #9157C2semidirect
9. #9149C2semidirect
10. #8220C2semidirect
11. C4×#327C2semidirect
20. #2512C4semidirect
21. #2506C4semidirect
22. #2452C4semidirect
23. #2445C4semidirect

Maximal subgroups

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

subgroupindex
#350062
#219122
#138062
#99112
#99082
#97152
#97122
#91572
#91492
#82202
C4×#3272