Finite Group Databasebeta

← #194037

Subgroups — #194037

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3036C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#21884C211yesmaximal, characteristic, complemented
4#21401C211yesmaximal, characteristic, complemented
5C22×D4⋊C8C211yesmaximal, characteristic, complemented
6#8985C214yesmaximal, complemented
7#8977C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C22×AGL(1,ℤ/8)—21no—
11C2×#532C2211yescharacteristic
12C2×#518—21no—
13C22×C4×D4C2211yescharacteristic
14#3036—24no—
15#3036C418yescomplemented
16C2×#463—22no—
17C2×#461C2212yes—
18C2×#459C2212yes—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#417—21no—
22C22×C4×C8C2211yescharacteristic
23#2349—22no—
24#2349—22no—
25D4×C4⋊C4—22no—
26D4×C4⋊C4C2214yes—
27#2316—21no—
28#2315—22no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31C2×#380—22no—
32#1990C2212yescomplemented
33#1988C2212yescomplemented
34#1982C2212yescomplemented
35#1957C2212yescomplemented
36#1893—22no—
37#1893—22no—
38#1802—22no—
39C4×#102—22no—
40C2×D4⋊C8—24no—
41C2×D4⋊C8C2214yes—
42#1296C2214yescomplemented
43#1279C2214yescomplemented
44C22×#41—21no—
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C2×AGL(1,ℤ/8)—28no—
48C2×AGL(1,ℤ/8)—44no—
49C2×AGL(1,ℤ/8)—44no—
50C22×SD16—22no—
51C22×D8—22no—
52C22×M16—21no—
53C23×C8—21no—
54C23×C8—21no—
55#532—22no—
56#532—22no—
57#532—22no—
58#532C2×C414yescomplemented
59C2×C4⋊Q8C2311yescharacteristic
60C2×#127C2311yescharacteristic
61C2×#124—21no—
62C2×#121—21no—
63C2×#121—21no—
64#518—44no—
65#518—44no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C414yes—
75C2×C4×D4D414yes—
76C2×C4×D4D412yes—
77C2×C4×D4C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4D411yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—24no—
84C22×C42C2311yescharacteristic
85#463—216no—
86#461—24no—
87#461C2×C418yescomplemented
88#459—24no—
89#459C2×C418yescomplemented
90#423—22no—
91#423C2×C414yescomplemented
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98#417—28no—
99C2×#103—22no—
100C2×#102—22no—
101C2×#102—24no—
102C2×#102—24no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8C2312yes—
107#389—24no—
108#389—24no—
109#385—28no—
110#385—28no—
111#384D412yes—
112#384C2312yes—
113#382—21no—
114#382—21no—
115#380—24no—
116#380—24no—
117#380—21no—
118#380—21no—
119#380—24no—
120#380D412yes—
121#380D412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4C2312yes—
126C2×#95—24no—
127#340—24no—
128#340—24no—
129D4⋊C8—28no—
130C25—21no—
131C2×#41—24no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—42no—
137C22×Q8—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—41no—
146C22×D4C22⋊C414yes—
147C22×D4D814yes—
148C22×D4SD1614yes—
149C22×D4C2×D412yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—28no—
156C23×C4—41no—
157C23×C4—82no—
158C23×C4—82no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162AGL(1,ℤ/8)—432no—
163C2×SD16—28no—
164C2×SD16—44no—
165C2×SD16—44no—
166C2×D8—28no—
167C2×D8—44no—
168C2×D8—44no—
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—44no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—82no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C22×C412yes—
183#127—21no—
184#127—21no—
185#127—21no—
186#127C22×C412yes—
187#124—24no—
188#124—42no—
189#121—24no—
190#121—24no—
191#121—42no—
192#121—42no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—48no—
203C4×D4—416no—
204C4×D4—48no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4C22⋊C418yes—
208C4×D4C22×C414yes—
209C4×D4C2×D414yes—
210#117—42no—
211#117—42no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—84no—
224C2×C4⋊C4—84no—
225C2×C4⋊C4C22⋊C414yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—416no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42C22×C412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42C2411yescharacteristic
249C4⋊C8—24no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—42no—
254C4⋊C8C22×C414yes—
255#103—28no—
256#102—28no—
257#102—48no—
258#102—84no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—42no—
262#95—28no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267C24—22no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—41no—
272C24—82no—
273C24—82no—
274C24#12011yescharacteristic
275#41—48no—
276#41—48no—
277#41—84no—
278#41—84no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—28no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—28no—
296C2×D4—28no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4#102116yescomplemented
311C2×D4C2×C22⋊C414yes—
312C2×D4C2×D814yes—
313C2×D4C2×SD1614yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—41no—
327C22×C4—416no—
328C22×C4—44no—
329C22×C4—88no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—82no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4C2×C22⋊C412yes—
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12312yes—
347C22×C4C2×D812yes—
348C22×C4C2×SD1612yes—
349C22×C4AGL(1,ℤ/8)12yes—
350C22×C4#13712yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354SD16—416no—
355D8—416no—
356M16—48no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—48no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—44no—
366C2×C8—84no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—44no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—82no—
382C4⋊C4—88no—
383C4⋊C4C2×C22⋊C414yes—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—28no—
387C22⋊C4—416no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C42—42no—
393C42—42no—
394C42—44no—
395C42C2×C22⋊C412yes—
396C42C4×D412yes—
397C42C4×D412yes—
398C42#12112yes—
399C42#12112yes—
400C42#12312yes—
401C42C23×C411yescharacteristic
402C42C22×D411yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—216no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—84no—
415C23—84no—
416C23—88no—
417C23—88no—
418C23—81no—
419C23—81no—
420C23—82no—
421C23—82no—
422C23#38612yes—
423C23#44712yes—
424C23#44812yes—
425C23C2×#12011yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—81no—
429Q8—81no—
430D4—24no—
431D4—24no—
432D4—24no—
433D4—216no—
434D4—44no—
435D4—42no—
436D4—42no—
437D4—44no—
438D4—82no—
439D4—88no—
440D4—81no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4C2×#10218yes—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—48no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—88no—
466C2×C4—84no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—88no—
470C2×C4—1616no—
471C2×C4#38612yes—
472C2×C4#38612yes—
473C2×C4#38612yes—
474C2×C4C2×#10214yes—
475C2×C4#41714yes—
476C2×C4#44712yes—
477C2×C4#44812yes—
478C2×C4#48014yes—
479C2×C4#48114yes—
480C2×C4C22×C22⋊C411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12311yescharacteristic
486C2×C4C22×D811yescharacteristic
487C2×C4C22×SD1611yescharacteristic
488C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
489C2×C4C2×#13711yescharacteristic
490C2×C4#57612yes—
491C2×C4#57712yes—
492C8—44no—
493C8—44no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—24no—
498C22—24no—
499C22—28no—
500C22—44no—
501C22—44no—
502C22—48no—
503C22—48no—
504C22—816no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—162no—
509C22—162no—
510C22#187414yes—
511C22C2×#38611yescharacteristic
512C22C2×#44711yescharacteristic
513C22C2×#44811yescharacteristic
514C22#301212yes—
515C22#301812yes—
516C4—22no—
517C4—21no—
518C4—21no—
519C4—22no—
520C4—44no—
521C4—82no—
522C4—84no—
523C4—84no—
524C4#187414yes—
525C4C22×#10211yescharacteristic
526C4C2×#41711yescharacteristic
527C4#294612yes—
528C4#304111yescharacteristic
529C4#304211yescharacteristic
530C4C2×#48011yescharacteristic
531C4C2×#48111yescharacteristic
532C2—24no—
533C2—48no—
534C2—162no—
535C2#1866212yes—
536C2#1866412yes—
537C2C2×#187411yescharacteristic
538C2#2092911yescharacteristic
539C2#2094211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3036C2
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
2. #34181C2semidirect
3. #21884C2semidirect
4. #21401C2semidirect
5. C22×D4⋊C8C2semidirect
6. #8985C2semidirect
7. #8977C2semidirect
8. #8045C2semidirect
9. #7897C2semidirect
15. #3036C4semidirect
32. #1990C22semidirect
33. #1988C22semidirect
34. #1982C22semidirect
35. #1957C22semidirect
42. #1296C22semidirect
43. #1279C22semidirect
58. #532C2×C4semidirect
87. #461C2×C4semidirect
89. #459C2×C4semidirect
91. #423C2×C4semidirect
310. C2×D4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#30362
#341812
#218842
#214012
C22×D4⋊C82
#89852
#89772
#80452
#78972