Finite Group Databasebeta

← #204470

Subgroups — #204470

Order 512.

1136 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#17806C211yesmaximal, characteristic, complemented
3#17478C211yesmaximal, characteristic, complemented
4#17448C211yesmaximal, characteristic, complemented
5#17384C211yesmaximal, characteristic, complemented
6C2×#1460C211yesmaximal, characteristic, complemented
7#17326C211yesmaximal, characteristic, complemented
8#9670C214yesmaximal, complemented
9#9668C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3100—22no—
13#3100—22no—
14#3100—21no—
15#3100—21no—
16#3100—22no—
17C2×#480C412yes—
18C2×#474C412yes—
19C2×#459C412yes—
20C2×#447C412yes—
21C2×C4×D8C2214yes—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#1574—22no—
28#1573—22no—
29#1564C2214yescomplemented
30#1496—22no—
31#1492—22no—
32#1490—22no—
33#1486—22no—
34#1478C2214yes—
35#1478C2214yescomplemented
36#1471—22no—
37#1469—22no—
38#1460—22no—
39#1460C2212yes—
40#1460C2212yes—
41#1460C2212yescomplemented
42#1460C2212yescomplemented
43C2×#323—22no—
44#1454—22no—
45#1452C2212yescomplemented
46#1451—21no—
47#1450C2212yescomplemented
48C8×M16—21no—
49C22×#41—21no—
50C23×D4C2×C411yescharacteristic
51C24×C4—21no—
52C22×D8C2×C414yes—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#127C2×C411yescharacteristic
63C2×#124—21no—
64C2×#121—21no—
65C2×#121C2×C412yes—
66C2×C4×D4—21no—
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×C4×D4C2312yes—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C22⋊C4—21no—
82C22×C42C2311yescharacteristic
83#480—22no—
84#480—22no—
85#480—24no—
86#474—24no—
87#474—22no—
88#474—22no—
89#459—24no—
90#459—22no—
91#459—22no—
92#447—22no—
93#447—22no—
94#447—24no—
95C4×D8—24no—
96C4×D8—24no—
97C4×D8—24no—
98C4×D8—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#431C2312yes—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#107C2×C414yes—
122#423—21no—
123#423D412yes—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2×C412yes—
127C2×#102C2×C414yes—
128C2×#102C2×C414yes—
129#407—21no—
130#407—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8C2×C412yes—
133C2×C22⋊C8D412yes—
134C4×M16—21no—
135C4×M16—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8C2312yes—
139#335—24no—
140#334—24no—
141D4⋊C8—28no—
142D4⋊C8—28no—
143#323—24no—
144#323—24no—
145#323—24no—
146#323—24no—
147#322—22no—
148#322—22no—
149#322—22no—
150C82—22no—
151C25—21no—
152C2×#41—41no—
153C2×#41—41no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—84no—
157C22×Q8—21no—
158C22×D4—24no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—42no—
168C22×D4C22×C412yes—
169C22×D4C22×C412yes—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4—84no—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×D8—28no—
183C2×D8—24no—
184C2×D8—24no—
185C2×D8—44no—
186C2×M16—22no—
187C2×M16—41no—
188C22×C8—22no—
189C22×C8—41no—
190C22×C8C22×C414yes—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195#127—21no—
196#127—21no—
197#127—21no—
198#127—21no—
199#124—41no—
200#124—41no—
201#124—41no—
202#124—41no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—22no—
207#121—41no—
208#121—41no—
209#121—41no—
210#121—41no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42C2411yescharacteristic
263#107—24no—
264#107—24no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8C22⋊C414yes—
274C4⋊C8C22⋊C412yes—
275C4⋊C8C22⋊C412yes—
276C4⋊C8C2×D414yes—
277#102—24no—
278#102—24no—
279#102—24no—
280#102—24no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—41no—
287C22⋊C8—41no—
288C22⋊C8C22⋊C412yes—
289C22⋊C8C22⋊C412yes—
290C22⋊C8C22⋊C414yes—
291C22⋊C8C2×D412yes—
292C22⋊C8C2×D412yes—
293#97—22no—
294#97—41no—
295C4×C8—22no—
296C4×C8—21no—
297C4×C8—22no—
298C4×C8—21no—
299C4×C8—22no—
300C4×C8—22no—
301C4×C8—41no—
302C4×C8C2×D412yes—
303C4×C8C2×D412yes—
304C4×C8C2×D414yes—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—42no—
309C24—41no—
310C24—41no—
311C24—82no—
312C24—82no—
313C24C4×D411yescharacteristic
314#41—84no—
315#41—84no—
316#41—84no—
317#41—84no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—82no—
345C2×D4—82no—
346C2×D4—84no—
347C2×D4#13114yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—81no—
368C22×C4—84no—
369C22×C4—82no—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12411yescharacteristic
382C22×C4#12411yescharacteristic
383C22×C4#12711yescharacteristic
384C22×C4#12711yescharacteristic
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4111yescharacteristic
388D8—48no—
389D8—48no—
390M16—44no—
391M16—82no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—82no—
400C2×C8—82no—
401C2×C8C2×C22⋊C414yes—
402C2×C8C2×C22⋊C414yes—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C42C2×C22⋊C411yescharacteristic
420C42C2×C22⋊C411yescharacteristic
421C42C4×D411yescharacteristic
422C42C4×D411yescharacteristic
423C42#12111yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12411yescharacteristic
428C42#12411yescharacteristic
429C42#12711yescharacteristic
430C42#12711yescharacteristic
431C42C22×D411yescharacteristic
432C42C22×D411yescharacteristic
433C42C22×D411yescharacteristic
434C42C2×#4111yescharacteristic
435C23—21no—
436C23—21no—
437C23—22no—
438C23—24no—
439C23—41no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—88no—
455C23—88no—
456C23—82no—
457C23#39011yescharacteristic
458C23#39011yescharacteristic
459C23C2×C4×D411yescharacteristic
460Q8—82no—
461Q8—82no—
462D4—24no—
463D4—24no—
464D4—44no—
465D4—44no—
466D4—42no—
467D4—44no—
468D4—42no—
469D4—44no—
470D4—82no—
471D4—82no—
472D4—82no—
473D4—84no—
474D4—84no—
475D4—82no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4#39011yescharacteristic
506C2×C4#39011yescharacteristic
507C2×C4#39011yescharacteristic
508C2×C4#39011yescharacteristic
509C2×C4#39011yescharacteristic
510C2×C4#39011yescharacteristic
511C2×C4#47214yes—
512C2×C4C22×C22⋊C411yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12111yescharacteristic
516C2×C4C2×#12411yescharacteristic
517C2×C4C2×#12711yescharacteristic
518C2×C4C2×#13112yes—
519C2×C4#56812yes—
520C8—24no—
521C8—24no—
522C8—42no—
523C8—42no—
524C8—44no—
525C8—82no—
526C8—82no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—84no—
538C22—88no—
539C22—84no—
540C22—88no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—162no—
547C22—162no—
548C22#197812yes—
549C22C2×#39011yescharacteristic
550C22#292911yescharacteristic
551C22#293011yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—82no—
561C4—82no—
562C4#197812yes—
563C4#288012yes—
564C4#293312yes—
565C4C2×#47212yes—
566C2—21no—
567C2—21no—
568C2—22no—
569C2—41no—
570C2—84no—
571C2—162no—
572C2#2119111yescharacteristic
573C2#2124411yescharacteristic
574C2C2×#197811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC2×#3100C2
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. #17478C2semidirect
4. #17448C2semidirect
5. #17384C2semidirect
6. C2×#1460C2semidirect
7. #17326C2semidirect
8. #9670C2semidirect
9. #9668C2semidirect
29. #1564C22semidirect
35. #1478C22semidirect
41. #1460C22semidirect
42. #1460C22semidirect
45. #1452C22semidirect
47. #1450C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#178062
#174782
#174482
#173842
C2×#14602
#173262
#96702
#96682