Finite Group Databasebeta

← #192724

Subgroups — #192724

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21884C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic, complemented
4#20817C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic
6#8897C212yesmaximal, complemented
7#8894C212yesmaximal, complemented
8#8885C212yesmaximal, complemented
9#8882C212yesmaximal, complemented
10#8045C211yesmaximal, characteristic, complemented
11C2×#1279C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520—21no—
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2367—22no—
19#2366—21no—
20#2365—21no—
21#2354C2212yes—
22#2352—22no—
23D4×C4⋊C4C2212yes—
24#2325—22no—
25#2311—21no—
26C4×#121—22no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1988—21no—
31#1982—21no—
32#1949—22no—
33#1948C2212yes—
34#1947C2212yes—
35#1893—22no—
36#1845C2211yescharacteristic
37#1845C2211yescharacteristic
38#1834C2211yescharacteristic
39#1834C2211yescharacteristic
40#1808—22no—
41#1802C2212yes—
42#1802C2212yes—
43C4×#107—22no—
44#1736—21no—
45#1492C418yescomplemented
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×#41—21no—
53C22×M16—21no—
54C23×C8—21no—
55C23×C8—21no—
56#532—22no—
57#532—22no—
58#532C2×C414yes—
59#532Q814yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×#127D411yescharacteristic
62C2×#124—21no—
63C2×#121—21no—
64C2×#121D412yes—
65#520—44no—
66#520—44no—
67C2×C4×Q8—22no—
68C2×C4×D4—24no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4C2312yes—
73C2×#117—21no—
74C2×#117—22no—
75C2×#117—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#428—44no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#106—22no—
86#423—22no—
87#423C2×C414yescomplemented
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×#102—22no—
95C2×#102—22no—
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98C2×C4×C8C2×C414yescomplemented
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#391—22no—
102#390—22no—
103#390—22no—
104#389—22no—
105#388—24no—
106#388—24no—
107#387—24no—
108#386—24no—
109#386—24no—
110#385—24no—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#382—21no—
117#380—21no—
118#380—22no—
119#380—22no—
120#380—21no—
121#380—21no—
122#380D412yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—24no—
133C2×#95—21no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140D4⋊C8C2×C418yescomplemented
141D4⋊C8C2×C418yescomplemented
142C2×#41—24no—
143C2×#41—24no—
144C2×#41—42no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4C2×D412yes—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—21no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×M16—24no—
161C2×M16—42no—
162C2×M16—42no—
163C22×C8—21no—
164C22×C8—24no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—81no—
172C22×C8—81no—
173C4⋊Q8—22no—
174C4⋊Q8C4⋊C414yes—
175#127—22no—
176#127C4⋊C414yes—
177#124—24no—
178#124—24no—
179#121—24no—
180#121—24no—
181#121—24no—
182#121C4⋊C418yes—
183C4×Q8—48no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—48no—
189C4×D4—416no—
190C4×D4—82no—
191C4×D4—82no—
192C4×D4C22×C414yes—
193C4×D4C2×Q814yes—
194#117—44no—
195#117—42no—
196#117—42no—
197#117—44no—
198#117—42no—
199#117—42no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4D814yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—41no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—41no—
237C2×C42—81no—
238C2×C42—81no—
239C2×C42C22×C412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×Q812yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42C2411yescharacteristic
247#107—44no—
248#107—44no—
249#107—44no—
250#106—44no—
251C4⋊C8—24no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—42no—
256C4⋊C8C22×C414yes—
257#102—82no—
258#102—82no—
259C4×C8—24no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8C22×C412yes—
263C4×C8C22×C412yes—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—41no—
271#95—41no—
272C24—22no—
273C24#12011yescharacteristic
274#41—48no—
275#41—48no—
276#41—48no—
277#41—48no—
278C2×Q8—24no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×D4—28no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4C2×C4⋊C414yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—48no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4C2×C4⋊C412yes—
325C22×C4C4×D412yes—
326C22×C4C4×D412yes—
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12011yescharacteristic
330C22×C4C22⋊Q812yes—
331C22×C4C22⋊Q812yes—
332C22×C4#12312yes—
333C22×C4C2×D812yes—
334C22×C4#13512yes—
335C22×C4AGL(1,ℤ/8)12yes—
336C22×C4#13712yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340M16—48no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—48no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—84no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4C2×C4⋊C414yes—
384C4⋊C4C2×D814yes—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—28no—
388C22⋊C4—48no—
389C22⋊C4—44no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C22⋊C4—82no—
393C42—48no—
394C42—42no—
395C42—42no—
396C42—44no—
397C42—82no—
398C42—82no—
399C42C2×C4⋊C412yes—
400C42C4×D412yes—
401C42C4×D412yes—
402C42C22⋊Q812yes—
403C42C22⋊Q812yes—
404C42#12312yes—
405C42C23×C411yescharacteristic
406C42C22×Q811yescharacteristic
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—24no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23#38512yes—
421C23#44712yes—
422C23#45212yes—
423C23C2×#12011yescharacteristic
424Q8—48no—
425D4—24no—
426D4—24no—
427D4—416no—
428D4—42no—
429D4—42no—
430D4—48no—
431D4—84no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—168no—
472C2×C4#38512yes—
473C2×C4#38512yes—
474C2×C4#38512yes—
475C2×C4C4×D814yes—
476C2×C4#44014yes—
477C2×C4#44712yes—
478C2×C4#45212yes—
479C2×C4#47414yes—
480C2×C4#47814yes—
481C2×C4C22×C4⋊C411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×C22⋊Q811yescharacteristic
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C22×D811yescharacteristic
488C2×C4C2×#13511yescharacteristic
489C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
490C2×C4C2×#13711yescharacteristic
491C2×C4#57611yescharacteristic
492C2×C4#57711yescharacteristic
493C2×C4#57711yescharacteristic
494C2×C4#57811yescharacteristic
495C8—24no—
496C8—42no—
497C8—44no—
498C8—82no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22#186214yes—
513C22C2×#38511yescharacteristic
514C22C2×#44711yescharacteristic
515C22C2×#45211yescharacteristic
516C22#301211yescharacteristic
517C22#301311yescharacteristic
518C22#301811yescharacteristic
519C22#301811yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—44no—
525C4—44no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—161no—
530C4—161no—
531C4#186214yes—
532C4#291211yescharacteristic
533C4#291311yescharacteristic
534C4C2×C4×D811yescharacteristic
535C4C2×#44011yescharacteristic
536C4C2×#47411yescharacteristic
537C4C2×#47811yescharacteristic
538C4#309611yescharacteristic
539C4#309711yescharacteristic
540C2—24no—
541C2—84no—
542C2#1866211yescharacteristic
543C2#1866911yescharacteristic
544C2#1867411yescharacteristic
545C2#1868111yescharacteristic
546C2#2084911yescharacteristic
547C2#2086311yescharacteristic
548C2C2×#186211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21170C2semidirect
4. #20817C2semidirect
6. #8897C2semidirect
7. #8894C2semidirect
8. #8885C2semidirect
9. #8882C2semidirect
10. #8045C2semidirect
11. C2×#1279C2semidirect
45. #1492C4semidirect
87. #423C2×C4semidirect
98. C2×C4×C8C2×C4semidirect
140. D4⋊C8C2×C4semidirect
141. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#218842
#211702
#208172
C2×#14922
#88972
#88942
#88852
#88822
#80452
C2×#12792