Finite Group Databasebeta

← #204479

Subgroups — #204479

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2#17806C211yesmaximal, characteristic, complemented
3#17464C211yesmaximal, characteristic, complemented
4C2×#1478C211yesmaximal, characteristic
5#17385C211yesmaximal, characteristic, complemented
6#17364C211yesmaximal, characteristic
7#17325C211yesmaximal, characteristic, complemented
8#9672C214yesmaximal, complemented
9#9667C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3101—22no—
13#3101—22no—
14#3101—21no—
15#3101—21no—
16#3101—22no—
17C2×#483C412yes—
18C2×#476C412yes—
19C2×#461C412yes—
20C2×#450C412yes—
21C2×C4×SD16C2214yes—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#1574—22no—
28#1573—22no—
29#1564C2214yescomplemented
30#1485—22no—
31#1481—22no—
32#1479C2212yescomplemented
33#1479C2212yescomplemented
34#1478—22no—
35#1478C2212yes—
36#1478C2212yes—
37C2×D4⋊C8—22no—
38#1472—22no—
39#1470—22no—
40#1466—22no—
41#1462—22no—
42#1461C2214yescomplemented
43#1460C2214yes—
44#1453—22no—
45#1452—21no—
46#1452C2212yescomplemented
47#1450—21no—
48#1450C2212yescomplemented
49C23×D4C2×C411yescharacteristic
50C24×C4—21no—
51C22×SD16C2×C414yes—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8C2×C412yes—
65C2×C4×Q8—21no—
66C2×C4×Q8C2312yes—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2312yes—
79C2×#117—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C22⋊C4—21no—
82C22×C42C2311yescharacteristic
83#483—22no—
84#483—22no—
85#483—24no—
86#476—24no—
87#476—22no—
88#476—22no—
89#461—24no—
90#461—22no—
91#461—22no—
92#450—22no—
93#450—22no—
94#450—24no—
95C4×SD16—24no—
96C4×SD16—24no—
97C4×SD16—24no—
98C4×SD16—24no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432D412yes—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431D412yes—
111#431D412yes—
112#431D412yes—
113#431D411yescharacteristic
114#431D411yescharacteristic
115#431D412yes—
116#431C2312yes—
117#431C2312yes—
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120#431C2312yes—
121C2×#106C2×C414yes—
122#423—21no—
123#423—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8C2×C412yes—
126C2×C4⋊C8D412yes—
127C2×#103C2×C414yes—
128C2×#102C2×C414yes—
129#407—21no—
130#407D412yes—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8C2×C412yes—
134C4×M16—21no—
135C4×M16—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8C2312yes—
139#335—24no—
140#334—24no—
141Q8⋊C8—24no—
142Q8⋊C8—24no—
143D4⋊C8—24no—
144D4⋊C8—24no—
145#324—28no—
146#323—28no—
147#322—22no—
148#322—22no—
149#322—22no—
150C82—22no—
151C25—21no—
152C22×Q8—21no—
153C22×Q8C22×C412yes—
154C22×D4—24no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—42no—
161C22×D4C22×C412yes—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—84no—
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C2×SD16—28no—
174C2×SD16—24no—
175C2×SD16—24no—
176C2×SD16—44no—
177C2×M16—22no—
178C2×M16—41no—
179C22×C8—22no—
180C22×C8—41no—
181C22×C8C22×C414yes—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190#125—41no—
191#125—41no—
192#125—41no—
193#125—41no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—41no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C42—22no—
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42C2411yescharacteristic
262#106—24no—
263#106—24no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8C22⋊C414yes—
272C4⋊C8C22⋊C412yes—
273C4⋊C8C22⋊C412yes—
274C4⋊C8C2×D412yes—
275C4⋊C8C2×D412yes—
276#103—24no—
277#103—24no—
278#102—24no—
279#102—24no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—41no—
287C22⋊C8—41no—
288C22⋊C8C22⋊C412yes—
289C22⋊C8C22⋊C412yes—
290C22⋊C8C22⋊C414yes—
291C22⋊C8C2×D414yes—
292#97—22no—
293#97—41no—
294C4×C8—22no—
295C4×C8—21no—
296C4×C8—22no—
297C4×C8—21no—
298C4×C8—22no—
299C4×C8—22no—
300C4×C8—41no—
301C4×C8C2×D412yes—
302C4×C8C2×D412yes—
303C4×C8C2×D414yes—
304C24—22no—
305C24—44no—
306C24—41no—
307C24—41no—
308C24—82no—
309C24—82no—
310C24C4×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—82no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4#13114yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—81no—
351C22×C4—84no—
352C22×C4—82no—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12411yescharacteristic
365C22×C4#12411yescharacteristic
366C22×C4#12711yescharacteristic
367C22×C4#12711yescharacteristic
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371SD16—48no—
372SD16—48no—
373M16—82no—
374M16—82no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—44no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—82no—
384C2×C8C2×C22⋊C414yes—
385C2×C8C2×C22⋊C414yes—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C42—42no—
409C42—42no—
410C42C2×C22⋊C411yescharacteristic
411C42C2×C22⋊C411yescharacteristic
412C42C4×D411yescharacteristic
413C42C4×D411yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12111yescharacteristic
417C42#12111yescharacteristic
418C42#12411yescharacteristic
419C42#12411yescharacteristic
420C42#12711yescharacteristic
421C42#12711yescharacteristic
422C42C22×D411yescharacteristic
423C42C22×D411yescharacteristic
424C42C22×D411yescharacteristic
425C42C2×#4111yescharacteristic
426C23—22no—
427C23—22no—
428C23—24no—
429C23—41no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—816no—
440C23—82no—
441C23#39011yescharacteristic
442C23#39011yescharacteristic
443C23C2×C4×D411yescharacteristic
444Q8—44no—
445Q8—42no—
446Q8—42no—
447Q8—82no—
448Q8—82no—
449D4—24no—
450D4—24no—
451D4—44no—
452D4—44no—
453D4—42no—
454D4—42no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#39011yescharacteristic
486C2×C4#39011yescharacteristic
487C2×C4#39011yescharacteristic
488C2×C4#39011yescharacteristic
489C2×C4#39011yescharacteristic
490C2×C4#39011yescharacteristic
491C2×C4#47114yes—
492C2×C4C22×C22⋊C411yescharacteristic
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C2×#12111yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12411yescharacteristic
497C2×C4C2×#12711yescharacteristic
498C2×C4C2×#13112yes—
499C2×C4#56812yes—
500C8—24no—
501C8—24no—
502C8—44no—
503C8—42no—
504C8—42no—
505C8—82no—
506C8—82no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—44no—
515C22—44no—
516C22—84no—
517C22—84no—
518C22—88no—
519C22—88no—
520C22—82no—
521C22—82no—
522C22#197712yes—
523C22C2×#39011yescharacteristic
524C22#293312yes—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4#197712yes—
537C4#288012yes—
538C4#292911yescharacteristic
539C4#293011yescharacteristic
540C4C2×#47112yes—
541C2—22no—
542C2—22no—
543C2—41no—
544C2—84no—
545C2#2119111yescharacteristic
546C2#2124411yescharacteristic
547C2C2×#197711yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17806C2semidirect
3. #17464C2semidirect
5. #17385C2semidirect
7. #17325C2semidirect
8. #9672C2semidirect
9. #9667C2semidirect
29. #1564C22semidirect
32. #1479C22semidirect
33. #1479C22semidirect
42. #1461C22semidirect
46. #1452C22semidirect
48. #1450C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
#178062
#174642
C2×#14782
#173852
#173642
#173252
#96722
#96672