Finite Group Databasebeta

← #192305

Subgroups — #192305

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34083C211yesmaximal, characteristic
2#21388C211yesmaximal, characteristic
3#21361C211yesmaximal, characteristic, complemented
4C2×#1500C211yesmaximal, characteristic
5#13854C211yesmaximal, characteristic, complemented
6#8872C212yesmaximal, complemented
7#8870C212yesmaximal
8#8869C212yesmaximal
9#8867C212yesmaximal, complemented
10#8013C211yesmaximal, characteristic, complemented
11#7908C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2337—21no—
16#2336—21no—
17#2334—22no—
18#2329C2212yes—
19#2325C2212yes—
20#2322—22no—
21#2298—21no—
22#2296—21no—
23#2295—21no—
24#2290—21no—
25#2290—21no—
26#2283C2211yescharacteristic
27C42×D4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1989—21no—
32#1983—21no—
33#1949—22no—
34#1874C2212yes—
35#1872C2212yes—
36#1853C2212yes—
37#1852C2212yes—
38C4×#107—21no—
39C4×#106—21no—
40C4×#103—21no—
41C4×#102—21no—
42#1500C418yescomplemented
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#327C2211yescharacteristic
46C2×#327C2211yescharacteristic
47#1442—22no—
48#1442C2212yes—
49#1442C2212yes—
50C4×#95—21no—
51C4×#95—21no—
52#1294C2212yes—
53#1294C2212yes—
54#1280C2212yes—
55#1280C2212yes—
56#529—22no—
57#529—22no—
58#529C2×C414yes—
59#529Q814yes—
60C2×#126—21no—
61C2×#125D411yescharacteristic
62C2×#124D411yescharacteristic
63C2×#123—21no—
64C2×C22⋊Q8C2311yescharacteristic
65C2×#121C2311yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4D411yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2312yes—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80#431—22no—
81#431—22no—
82#431—22no—
83#431C2×C414yescomplemented
84C2×#107—22no—
85C2×#106—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×#103—22no—
89C2×#102—22no—
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2311yescharacteristic
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95#391—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#386—22no—
110#386—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#383—21no—
116#382D412yes—
117#381—21no—
118#380C2312yes—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137#376—21no—
138#376—21no—
139#376—22no—
140#376C2×C414yescomplemented
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C43—21no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#328C2×C414yescomplemented
156#328C2×C414yescomplemented
157#327C2×C414yes—
158#327C2×C414yes—
159C22×Q8C2×D411yescharacteristic
160C22×D4—21no—
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175#126—24no—
176#126—24no—
177#125—22no—
178#125C4⋊C414yes—
179#124—22no—
180#124C4⋊C414yes—
181#123—24no—
182#123—24no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8C22×C412yes—
186C22⋊Q8C2×Q812yes—
187#121—21no—
188#121—21no—
189#121C22×C412yes—
190#121C2×Q812yes—
191C4×Q8—22no—
192C4×Q8C4⋊C414yes—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4C4⋊C414yes—
199#117—24no—
200#117—24no—
201#117—82no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—82no—
249C2×C42C22×C412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×Q812yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257#107—44no—
258#106—44no—
259C4⋊C8—24no—
260C4⋊C8—42no—
261#103—44no—
262#102—44no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8C22×C412yes—
268C22⋊C8C22×C412yes—
269C4×C8—24no—
270C4×C8—42no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—22no—
279#95—22no—
280#95—21no—
281#95—21no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—21no—
286#95—21no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—41no—
291#95—41no—
292#95—44no—
293#95C22×C412yes—
294#95C22×C412yes—
295C24—21no—
296C24C22×D411yescharacteristic
297C2×Q8—21no—
298C2×Q8—42no—
299C2×Q8C2×C4⋊C412yes—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4C2×C4⋊C412yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4C22⋊Q812yes—
356C22×C4C22⋊Q812yes—
357C22×C4#13514yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4C2×C4⋊C412yes—
390C4⋊C4C2×C4⋊C412yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C42—24no—
404C42—48no—
405C42—44no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—81no—
411C42—81no—
412C42—82no—
413C42—82no—
414C42—82no—
415C42—82no—
416C42C2×C4⋊C412yes—
417C42C2×C4⋊C412yes—
418C42C4×D412yes—
419C42C4×D412yes—
420C42C22⋊Q812yes—
421C42C22⋊Q812yes—
422C42#12312yes—
423C42#12312yes—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23C2×C4×D411yescharacteristic
433C23C2×#12011yescharacteristic
434C23C2×C22⋊Q811yescharacteristic
435C23C2×#13512yes—
436C23#57512yes—
437Q8—42no—
438Q8—81no—
439D4—42no—
440D4—44no—
441D4—42no—
442D4—42no—
443D4—81no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—88no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4#38512yes—
502C2×C4#38512yes—
503C2×C4#38512yes—
504C2×C4#38512yes—
505C2×C4#45412yes—
506C2×C4#45512yes—
507C2×C4C22×C4⋊C411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×C22⋊Q811yescharacteristic
510C2×C4C2×#12311yescharacteristic
511C8—42no—
512C8—42no—
513C8—82no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22C2×#38511yescharacteristic
528C22C4×#13511yescharacteristic
529C22#294311yescharacteristic
530C22#300212yes—
531C22C2×#45511yescharacteristic
532C22C2×#45411yescharacteristic
533C22#302512yes—
534C22#307612yes—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—44no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—84no—
552C4#186014yes—
553C2—21no—
554C2—21no—
555C2—22no—
556C2—82no—
557C2#1812912yes—
558C2#1813012yes—
559C2C2×#186011yescharacteristic
560C2#2087211yescharacteristic
561C2#2088811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21361C2semidirect
5. #13854C2semidirect
6. #8872C2semidirect
9. #8867C2semidirect
10. #8013C2semidirect
11. #7908C2semidirect
42. #1500C4semidirect
83. #431C2×C4semidirect
140. #376C2×C4semidirect
155. #328C2×C4semidirect
156. #328C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340832
#213882
#213612
C2×#15002
#138542
#88722
#88702
#88692
#88672
#80132
#79082