Finite Group Databasebeta

← #206058

Subgroups — #206058

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3247C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic, complemented
3#20543C211yesmaximal, characteristic, complemented
4#17544C211yesmaximal, characteristic, complemented
5#17537C211yesmaximal, characteristic, complemented
6#17384C211yesmaximal, characteristic, complemented
7#17380C211yesmaximal, characteristic, complemented
8#9815C214yesmaximal, complemented
9#9813C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#529C2211yescharacteristic
12#3247—22no—
13#3247—22no—
14#3247—22no—
15#3247—22no—
16C2×#477C411yescharacteristic, complemented
17C2×#478C412yescomplemented
18C2×#476C411yescharacteristic, complemented
19C2×#466C2212yes—
20C2×#465C2212yes—
21C2×#460C411yescharacteristic, complemented
22C2×#463C412yescomplemented
23C2×#459C411yescharacteristic, complemented
24#2920C2211yescharacteristic
25#2920C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1950—22no—
30#1950—22no—
31#1810C2214yes—
32C8×C4⋊C4—22no—
33C4×C4⋊C8—22no—
34C8×C22⋊C4C2214yes—
35#1509—22no—
36#1508C2212yes—
37#1507C2214yes—
38#1507C2212yes—
39#1505—22no—
40#1502—22no—
41#1500—22no—
42#1473—22no—
43#1472C2212yes—
44#1471C2214yes—
45#1471C2212yes—
46#1469—22no—
47#1460—22no—
48#1458—22no—
49C22×#41—21no—
50C22×SD16C2×C412yes—
51C22×D8C2×C412yes—
52C23×C8C2312yes—
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
63#529—21no—
64#529—21no—
65#529—21no—
66#529—21no—
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71C2×#126—21no—
72C2×#125C2×C411yescharacteristic
73C2×#124—21no—
74C2×#124C2×C411yescharacteristic
75C2×#123—21no—
76C2×C22⋊Q8C2311yescharacteristic
77C2×#121—21no—
78C2×#121C2311yescharacteristic
79C2×#121C2312yes—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4C2×C412yes—
84C2×C4×D4C2×C411yescharacteristic
85C2×#117—21no—
86C2×#117—21no—
87C22×C42C2311yescharacteristic
88#478—24no—
89#478—24no—
90#477—22no—
91#477—22no—
92#476—22no—
93#476—22no—
94#466—24no—
95#466—24no—
96#465—24no—
97#465—24no—
98#463—24no—
99#463—24no—
100#460—22no—
101#460—22no—
102#459—22no—
103#459—22no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431D412yes—
115#431D412yes—
116C2×#108—24no—
117C2×#107C2×C412yes—
118C2×#106C2×C412yes—
119#423—21no—
120#423—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8C2×C412yes—
123C2×C4⋊C8C2×C411yescharacteristic
124C2×C4⋊C8C2×C411yescharacteristic
125C2×C4⋊C8D412yes—
126C2×C4⋊C8D412yes—
127C2×C4⋊C8D412yes—
128C2×C4⋊C8D412yes—
129C2×#103C2×C412yes—
130C2×#102C2×C412yes—
131C2×#102C2×C412yes—
132C2×#102C2×C412yes—
133#407C2312yes—
134#407C2312yes—
135C2×C22⋊C8C2312yes—
136C4×M16—21no—
137C4×M16—21no—
138C2×C4×C8—21no—
139C2×C4×C8—24no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4D412yes—
142C4×C22⋊C4—21no—
143C4×C22⋊C4C2312yes—
144#376D412yes—
145C43—21no—
146#341—24no—
147#336—24no—
148#328—24no—
149#327—24no—
150#327—28no—
151#324—24no—
152#323—24no—
153#323—28no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—41no—
157C2×#41—41no—
158C2×#41—84no—
159C22×Q8—21no—
160C22×Q8C22×C411yescharacteristic
161C22×D4—21no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C22×D4C22×C412yes—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4#4111yescharacteristic
170C23×C4#4111yescharacteristic
171C23×C4C2411yescharacteristic
172C2×SD16—24no—
173C2×SD16—82no—
174C2×D8—24no—
175C2×D8—82no—
176C2×M16—22no—
177C2×M16—22no—
178C22×C8—22no—
179C22×C8—82no—
180C22×C8C22×C412yes—
181C22×C8C22×C412yes—
182C22×C8C2×D414yes—
183C22×C8C2×D414yes—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188#127—21no—
189#127—21no—
190#127—21no—
191#127—21no—
192#126—41no—
193#126—41no—
194#126—41no—
195#126—41no—
196#125—21no—
197#125—21no—
198#125—21no—
199#125—21no—
200#124—21no—
201#124—21no—
202#124—21no—
203#124—21no—
204#124—41no—
205#124—41no—
206#124—41no—
207#124—41no—
208#123—41no—
209#123—41no—
210#123—41no—
211#123—41no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216#121—21no—
217#121—21no—
218#121—21no—
219#121—22no—
220#121—22no—
221#121—22no—
222#121—21no—
223#121—22no—
224#121—41no—
225#121—41no—
226#121—41no—
227#121—41no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248#117—28no—
249#117—41no—
250#117—41no—
251#117—41no—
252#117—41no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42#4111yescharacteristic
286#108—84no—
287#107—24no—
288#106—24no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—41no—
292C4⋊C8—41no—
293C4⋊C8—41no—
294C4⋊C8—41no—
295C4⋊C8C22⋊C412yes—
296C4⋊C8C22⋊C412yes—
297C4⋊C8C22⋊C412yes—
298C4⋊C8C22⋊C412yes—
299C4⋊C8C22⋊C414yes—
300C4⋊C8C22⋊C414yes—
301#103—24no—
302#102—24no—
303#102—24no—
304#102—24no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8C2×D414yes—
311#97—41no—
312#97—41no—
313C4×C8—22no—
314C4×C8—22no—
315C4×C8—41no—
316C4×C8—41no—
317C4×C8—84no—
318#95—22no—
319#95C2×D414yes—
320C24—21no—
321C24—22no—
322C24C2×#4111yescharacteristic
323#41—84no—
324#41—84no—
325#41—84no—
326#41—84no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—82no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—44no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—84no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—84no—
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12411yescharacteristic
378C22×C4#12411yescharacteristic
379C22×C4#12411yescharacteristic
380C22×C4#12411yescharacteristic
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387SD16—84no—
388D8—84no—
389M16—42no—
390M16—42no—
391M16—42no—
392M16—42no—
393C2×C8—24no—
394C2×C8—44no—
395C2×C8—44no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8C2×C22⋊C414yes—
403C2×C8C2×C22⋊C414yes—
404C4⋊C4—28no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C22⋊C4—28no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C42—28no—
437C42—42no—
438C42—42no—
439C42—41no—
440C42—41no—
441C42—82no—
442C42C2×C22⋊C411yescharacteristic
443C42C2×C22⋊C411yescharacteristic
444C42C2×C22⋊C411yescharacteristic
445C42C2×C22⋊C411yescharacteristic
446C42C4×D411yescharacteristic
447C42C4×D411yescharacteristic
448C42C4×D411yescharacteristic
449C42C4×D411yescharacteristic
450C42#12111yescharacteristic
451C42#12111yescharacteristic
452C42#12111yescharacteristic
453C42#12111yescharacteristic
454C42#12711yescharacteristic
455C42#12711yescharacteristic
456C42#12711yescharacteristic
457C42#12711yescharacteristic
458C23—22no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—82no—
466C23—81no—
467C23—82no—
468C23—81no—
469C23C2×C4×D411yescharacteristic
470C23C2×#12111yescharacteristic
471C23C2×#12411yescharacteristic
472Q8—41no—
473Q8—41no—
474Q8—81no—
475Q8—82no—
476Q8—82no—
477D4—41no—
478D4—42no—
479D4—41no—
480D4—42no—
481D4—81no—
482D4—82no—
483D4—82no—
484D4—82no—
485D4—82no—
486D4—84no—
487D4—82no—
488D4—84no—
489D4—82no—
490D4—82no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4#39011yescharacteristic
527C2×C4#39011yescharacteristic
528C2×C4#39011yescharacteristic
529C2×C4#39011yescharacteristic
530C2×C4#39011yescharacteristic
531C2×C4#39011yescharacteristic
532C2×C4#39011yescharacteristic
533C2×C4#39011yescharacteristic
534C2×C4#44314yes—
535C2×C4C22×C22⋊C411yescharacteristic
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×#12711yescharacteristic
539C8—24no—
540C8—24no—
541C8—44no—
542C8—82no—
543C8—82no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—44no—
552C22—82no—
553C22—81no—
554C22—81no—
555C22—82no—
556C22—81no—
557C22—82no—
558C22—162no—
559C22—164no—
560C22C2×#39011yescharacteristic
561C22C2×#44312yes—
562C22#306812yes—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—48no—
573C4—82no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4#197412yes—
578C4#197412yes—
579C4#288311yescharacteristic
580C4#288411yescharacteristic
581C2—22no—
582C2—22no—
583C2—41no—
584C2—162no—
585C2—161no—
586C2#2121411yescharacteristic
587C2C2×#197411yescharacteristic
588C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3247C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21969C2semidirect
3. #20543C2semidirect
4. #17544C2semidirect
5. #17537C2semidirect
6. #17384C2semidirect
7. #17380C2semidirect
8. #9815C2semidirect
9. #9813C2semidirect
16. C2×#477C4semidirect
17. C2×#478C4semidirect
18. C2×#476C4semidirect
21. C2×#460C4semidirect
22. C2×#463C4semidirect
23. C2×#459C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32472
#219692
#205432
#175442
#175372
#173842
#173802
#98152
#98132