Finite Group Databasebeta

← #164385

Subgroups — #164385

Order 512.

1121 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic, complemented
3#20567C211yesmaximal, characteristic, complemented
4#7710C212yesmaximal, complemented
5#7679C211yesmaximal, characteristic, complemented
6#7679C211yesmaximal, characteristic, complemented
7#7670C212yesmaximal, complemented
8#7635C211yesmaximal, characteristic, complemented
9#7606C212yesmaximal, complemented
10#7572C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13C2×D42C411yescharacteristic
14C2×#436—21no—
15#2858C2211yescharacteristic
16#2850C2212yes—
17#2563C411yescharacteristic
18#2548C411yescharacteristic
19#2504C412yes—
20#2490—21no—
21#2490—21no—
22#2472—22no—
23#2472—22no—
24#2435C412yes—
25D4×C22⋊C4—21no—
26D4×C22⋊C4—21no—
27C4×#124C2211yescharacteristic
28C2×#390C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1817C2212yescomplemented
34#1797C2212yescomplemented
35C8×C4⋊C4—21no—
36#1767C2211yescharacteristic, complemented
37#1767C2211yescharacteristic, complemented
38#1753—22no—
39C4×C22⋊C8C2212yescomplemented
40#1744C2211yescharacteristic, complemented
41#1744C2211yescharacteristic, complemented
42#1728C2212yescomplemented
43#1272C2211yescharacteristic, complemented
44#1272C2212yescomplemented
45#1272C2211yescharacteristic, complemented
46#1270C2211yescharacteristic
47#1270C2211yescharacteristic
48#1270C2212yes—
49#1270C2212yescomplemented
50#1270C2212yescomplemented
51#1270C2212yescomplemented
52#1270C2212yescomplemented
53C23×D4—21no—
54C23×D4C2×C412yes—
55C22×M16—22no—
56#554—44no—
57D42—44no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#127C2×C411yescharacteristic
60C2×#125—21no—
61C2×#124—21no—
62C2×#124—21no—
63C2×#124C2×C411yescharacteristic
64C2×#124C2×C412yes—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8C2×C412yes—
67C2×#121—21no—
68C2×#121C2×C412yes—
69C2×#120—22no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C22×C4⋊C4C2×C412yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42C2312yes—
80#436—28no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×#97—21no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—22no—
92#397—22no—
93#397—22no—
94#394—22no—
95#394—22no—
96#394—22no—
97#393—22no—
98#391C2311yescharacteristic
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#390—22no—
104#390—22no—
105#390C2×C412yes—
106#390C2×C412yes—
107#390C2×C412yes—
108#390C2312yes—
109#390C2311yescharacteristic
110#388—22no—
111#388—22no—
112#388—21no—
113#388—21no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#382C2×C412yes—
119#382C2×C411yescharacteristic
120#380—21no—
121#380—21no—
122#379—22no—
123#379—22no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4C2×C412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143C25—22no—
144C22×Q8—21no—
145C22×Q8C22×C411yescharacteristic
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—42no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—82no—
157C22×D4C22⋊C411yescharacteristic
158C22×D4C22×C412yes—
159C22×D4C22×C411yescharacteristic
160C23×C4—22no—
161C23×C4—21no—
162C23×C4C22×C412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C2×M16—28no—
166C2×M16—82no—
167C2×M16—82no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C4⋊Q8—42no—
175#127—42no—
176#125—44no—
177#124—22no—
178#124—41no—
179#124—81no—
180#124—81no—
181#124—81no—
182#124—81no—
183#124—82no—
184C22⋊Q8—48no—
185C22⋊Q8—44no—
186#121—44no—
187#121—44no—
188#120—48no—
189C4×Q8—42no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—44no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C42—21no—
231C2×C42—82no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—24no—
243C4⋊C8—42no—
244C4⋊C8—82no—
245C22⋊C8—28no—
246C22⋊C8—82no—
247C22⋊C8—82no—
248#97—81no—
249#97—81no—
250C4×C8—24no—
251C4×C8—42no—
252C4×C8—82no—
253#95—21no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—21no—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262C24—21no—
263C24—22no—
264C24—44no—
265C24—44no—
266C24—44no—
267C24—44no—
268C24—82no—
269C24—82no—
270C24—82no—
271C24C2×C22⋊C412yes—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—44no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—82no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—88no—
301C2×D4—88no—
302C2×D4—816no—
303C2×D4#10414yes—
304C2×D4#13112yes—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337M16—88no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—24no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4#10014yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—44no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—42no—
400C42—42no—
401C42—41no—
402C42—42no—
403C42—42no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—88no—
425C23—88no—
426C23—88no—
427C23—88no—
428C23—164no—
429C23—164no—
430C23—164no—
431C23C2×#10414yes—
432C23C22×C22⋊C411yescharacteristic
433Q8—81no—
434Q8—81no—
435Q8—162no—
436D4—28no—
437D4—42no—
438D4—42no—
439D4—88no—
440D4—82no—
441D4—82no—
442D4—88no—
443D4—84no—
444D4—162no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—164no—
488C2×C4C2×#10014yes—
489C2×C4C2×#10012yes—
490C2×C4C2×#10412yes—
491C2×C4#42112yes—
492C2×C4C2×#13111yescharacteristic
493C2×C4#56811yescharacteristic
494C2×C4#57611yescharacteristic
495C2×C4#57611yescharacteristic
496C2×C4#57711yescharacteristic
497C2×C4#57711yescharacteristic
498C8—42no—
499C8—44no—
500C8—42no—
501C8—81no—
502C8—82no—
503C8—81no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—1616no—
526C22#288011yescharacteristic
527C22#288311yescharacteristic
528C22C22×#10011yescharacteristic
529C22#289711yescharacteristic
530C22#289711yescharacteristic
531C22C22×#10411yescharacteristic
532C22C2×#42111yescharacteristic
533C4—21no—
534C4—21no—
535C4—24no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—42no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4—82no—
548C4—81no—
549C4#152714yes—
550C2—44no—
551C2—82no—
552C2—162no—
553C2#1742111yescharacteristic
554C2#1751711yescharacteristic
555C2#1752211yescharacteristic
556C2#1754711yescharacteristic
557C2#1759911yescharacteristic
558C2C2×#152711yescharacteristic
559C2#1778211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×D42C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22012C2semidirect
3. #20567C2semidirect
4. #7710C2semidirect
5. #7679C2semidirect
6. #7679C2semidirect
7. #7670C2semidirect
8. #7635C2semidirect
9. #7606C2semidirect
10. #7572C2semidirect
11. #7567C2semidirect
33. #1817C22semidirect
34. #1797C22semidirect
36. #1767C22semidirect
37. #1767C22semidirect
39. C4×C22⋊C8C22semidirect
40. #1744C22semidirect
41. #1744C22semidirect
42. #1728C22semidirect
43. #1272C22semidirect
44. #1272C22semidirect
45. #1272C22semidirect
49. #1270C22semidirect
50. #1270C22semidirect
51. #1270C22semidirect
52. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#362892
#220122
#205672
#77102
#76792
#76702
#76352
#76062
#75722
#75672