Finite Group Databasebeta

← #194280

Subgroups — #194280

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4C2×#1987C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic
6#8976C214yesmaximal, complemented
7#8973C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#520—21no—
13#3227—24no—
14#3227C418yescomplemented
15C2×#498—21no—
16C2×#486—21no—
17C2×#481—22no—
18C2×#476C2214yes—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2367—22no—
23#2366—21no—
24#2365—21no—
25#2354C2212yes—
26#2352—22no—
27D4×C4⋊C4C2212yes—
28#2325—22no—
29#2311—21no—
30C4×#121—22no—
31#2285C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1987—22no—
34#1987C2214yes—
35#1981—22no—
36#1956C2212yes—
37#1956C2212yes—
38#1787—22no—
39C2×#340—21no—
40C2×#340—21no—
41#1480—24no—
42C2×D4⋊C8C2214yes—
43#1296C2212yes—
44#1296C2212yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8—21no—
48C23×C8—21no—
49#532—22no—
50#532—22no—
51#532C2×C414yes—
52#532D414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yescomplemented
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#127D411yescharacteristic
63C2×#124—21no—
64C2×#121—21no—
65C2×#121D412yes—
66#520—44no—
67#520—44no—
68C2×C4×Q8—22no—
69C2×C4×D4—24no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4C2312yes—
74C2×#117—21no—
75C2×#117—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#498—28no—
84#486—28no—
85#481—216no—
86#476—28no—
87#476C2×C4116yescomplemented
88#431—22no—
89#431—22no—
90#431—22no—
91#431C2×C414yescomplemented
92C2×#106—24no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99C2×#103—24no—
100C2×#103—24no—
101C2×#102—24no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2312yes—
106#391—22no—
107#390—22no—
108#390—22no—
109#389—22no—
110#388—24no—
111#388—24no—
112#387—24no—
113#386—24no—
114#386—24no—
115#385—24no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#382—21no—
119#380—21no—
120#380—22no—
121#380—22no—
122#380D412yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2312yes—
130C4×C22⋊C4—24no—
131C2×#95—21no—
132#340—24no—
133#340—24no—
134D4⋊C8—28no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8—21no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—41no—
145C22×Q8—42no—
146C22×Q8—82no—
147C22×Q8—82no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C2×D412yes—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—21no—
156C23×C4—41no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8—42no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8C22⋊C414yes—
177C4⋊Q8C22×C414yes—
178#127—22no—
179#127C22⋊C414yes—
180#124—24no—
181#124—24no—
182#121—24no—
183#121—24no—
184#121—24no—
185#121C22⋊C418yes—
186C4×Q8—48no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—48no—
192C4×D4—416no—
193C4×D4—84no—
194C4×D4C22×C414yes—
195C4×D4C2×D414yes—
196#117—44no—
197#117—42no—
198#117—42no—
199#117—44no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4SD1614yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—82no—
238C2×C42C22×C412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42C2411yescharacteristic
246#106—216no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8C22×C414yes—
253#103—84no—
254#103—84no—
255#102—216no—
256C22⋊C8—24no—
257C22⋊C8—42no—
258C4×C8—24no—
259C4×C8—22no—
260C4×C8—22no—
261#95—24no—
262#95—24no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267C24—22no—
268C24#12011yescharacteristic
269#41—48no—
270#41—48no—
271#41—48no—
272#41—48no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8—88no—
284C2×Q8—88no—
285C2×D4—28no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—81no—
296C2×D4—81no—
297C2×D4C2×C22⋊C414yes—
298C22×C4—22no—
299C22×C4—22no—
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—44no—
317C22×C4—42no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4C2×C22⋊C412yes—
327C22×C4C4×D412yes—
328C22×C4C4×D412yes—
329C22×C4#12011yescharacteristic
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12312yes—
335C22×C4C2×SD1612yes—
336C22×C4#13512yes—
337C22×C4AGL(1,ℤ/8)12yes—
338C22×C4#13712yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—84no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—48no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4C2×C22⋊C414yes—
375C4⋊C4C2×SD1614yes—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—28no—
379C22⋊C4—48no—
380C22⋊C4—44no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—48no—
384C42—44no—
385C42—44no—
386C42—82no—
387C42C2×C22⋊C412yes—
388C42C4×D412yes—
389C42C4×D412yes—
390C42#12112yes—
391C42#12112yes—
392C42#12312yes—
393C42C23×C411yescharacteristic
394C42C22×D411yescharacteristic
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—24no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23#38612yes—
409C23#45012yes—
410C23#45212yes—
411C23C2×#12011yescharacteristic
412Q8—48no—
413Q8—48no—
414Q8—84no—
415Q8—88no—
416D4—28no—
417D4—416no—
418D4—44no—
419D4—48no—
420D4—84no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—88no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—168no—
457C2×C4#38612yes—
458C2×C4#38612yes—
459C2×C4#38612yes—
460C2×C4C4×SD1614yes—
461C2×C4#44014yes—
462C2×C4#45012yes—
463C2×C4#45212yes—
464C2×C4#46114yes—
465C2×C4#46314yes—
466C2×C4C22×C22⋊C411yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×#12111yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4C22×SD1611yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
475C2×C4C2×#13711yescharacteristic
476C2×C4#57712yes—
477C2×C4#57812yes—
478C8—44no—
479C8—82no—
480C8—82no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22#187714yes—
495C22C2×#38611yescharacteristic
496C22C2×#45011yescharacteristic
497C22C2×#45211yescharacteristic
498C22#300911yescharacteristic
499C22#301111yescharacteristic
500C22#302111yescharacteristic
501C22#302111yescharacteristic
502C4—22no—
503C4—21no—
504C4—21no—
505C4—22no—
506C4—44no—
507C4—44no—
508C4—82no—
509C4—84no—
510C4—82no—
511C4—82no—
512C4#187714yes—
513C4#289812yes—
514C4C2×C4×SD1611yescharacteristic
515C4C2×#44011yescharacteristic
516C4C2×#46111yescharacteristic
517C4C2×#46311yescharacteristic
518C4#309712yes—
519C2—24no—
520C2—84no—
521C2#1889311yescharacteristic
522C2#1889911yescharacteristic
523C2#1891311yescharacteristic
524C2#1891911yescharacteristic
525C2#2091311yescharacteristic
526C2C2×#187711yescharacteristic
527C2#2094511yescharacteristic
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. #21804C2semidirect
4. C2×#1987C2semidirect
6. #8976C2semidirect
7. #8973C2semidirect
8. #8039C2semidirect
9. #7892C2semidirect
14. #3227C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
87. #476C2×C4semidirect
91. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#342062
#218042
C2×#19872
C2×#14802
#89762
#89732
#80392
#78922