Finite Group Databasebeta

← #164541

Subgroups — #164541

Order 512.

1015 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36492C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic, complemented
3#20630C211yesmaximal, characteristic, complemented
4#7731C212yesmaximal, complemented
5#7711C212yesmaximal, complemented
6#7688C211yesmaximal, characteristic, complemented
7#7665C211yesmaximal, characteristic, complemented
8#7616C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7599C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#436—21no—
15C2×#424—21no—
16C2×#408—21no—
17#2857C2211yescharacteristic
18#2851C2212yes—
19#2563C411yescharacteristic
20#2562C412yes—
21#2537C411yescharacteristic
22#2535C412yes—
23#2477—22no—
24#2463—22no—
25#2421—21no—
26#2407—21no—
27#2361—21no—
28#2355C411yescharacteristic
29#2352—21no—
30C4×#121C2211yescharacteristic
31C4×#121C2212yes—
32#1984—21no—
33#1870—22no—
34#1817C2212yes—
35#1798—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40#1728C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C22×M16—22no—
56C23×C8—21no—
57#554—44no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—22no—
63C2×#121—21no—
64C2×#121C2×C411yescharacteristic
65C2×#121C2×C412yes—
66C4×#41—44no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—22no—
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C412yes—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D412yes—
77C2×#117D411yescharacteristic
78C2×#117D412yes—
79C22×C4⋊C4—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#436—44no—
83#424—44no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87#408—44no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C4×C8—21no—
94#400—22no—
95#397—22no—
96#394—22no—
97#394—22no—
98#393—22no—
99#392—22no—
100#391—21no—
101#391—21no—
102#390—21no—
103#390C2311yescharacteristic
104#389—21no—
105#389C2311yescharacteristic
106#389C2312yes—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—21no—
111#387—22no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—21no—
116#386—22no—
117#386—22no—
118#385—22no—
119#385—22no—
120#384C2×C411yescharacteristic
121#382—21no—
122#382C2×C412yes—
123#380—21no—
124#380—21no—
125#380C2×C411yescharacteristic
126#380C2×C412yes—
127#380C2×C412yes—
128#380C2×C412yes—
129#380C2×C412yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336—22no—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2312yes—
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C2×#41—42no—
149C22×Q8—21no—
150C22×Q8C22⋊C411yescharacteristic
151C22×D4—22no—
152C22×D4—21no—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22⋊C412yes—
155C22×D4C22×C411yescharacteristic
156C23×C4—21no—
157C23×C4—22no—
158C23×C4C22×C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C2×M16—44no—
165C2×M16—42no—
166C2×M16—42no—
167C2×M16—44no—
168C2×M16—42no—
169C2×M16—82no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—42no—
179C4⋊Q8—42no—
180#125—44no—
181C22⋊Q8—48no—
182C22⋊Q8—44no—
183C22⋊Q8—81no—
184C22⋊Q8—81no—
185#121—22no—
186#121—24no—
187#121—41no—
188#121—42no—
189#121—44no—
190C4×Q8—24no—
191C4×Q8—42no—
192C4×D4—24no—
193C4×D4—44no—
194C4×D4—42no—
195C4×D4—44no—
196C4×D4—42no—
197C4×D4—44no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—82no—
201C4×D4—82no—
202#117—24no—
203#117—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C412yes—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—42no—
238C2×C42—44no—
239C2×C42C22×C412yes—
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2411yescharacteristic
246C4⋊C8—44no—
247C4⋊C8—44no—
248C4⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—44no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253C22⋊C8—81no—
254C4×C8—42no—
255#95—22no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—21no—
266C24—21no—
267C24C2×C22⋊C412yes—
268C24C2×C22⋊C411yescharacteristic
269#41—48no—
270#41—48no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—42no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—42no—
290C2×D4—82no—
291C2×D4—81no—
292C2×D4#13112yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4AGL(1,ℤ/8)12yes—
333C22×C4#13712yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337M16—88no—
338M16—84no—
339C2×C8—22no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—84no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—81no—
377C4⋊C4#9914yes—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C42—22no—
395C42—24no—
396C42—24no—
397C42—22no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—42no—
402C42—82no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23#41712yes—
424C23C22×C22⋊C411yescharacteristic
425Q8—44no—
426Q8—162no—
427D4—42no—
428D4—42no—
429D4—48no—
430D4—44no—
431D4—84no—
432D4—88no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—84no—
481C2×C4—164no—
482C2×C4C2×#9914yes—
483C2×C4C2×#9912yes—
484C2×C4#41712yes—
485C2×C4#42112yes—
486C2×C4C2×#13111yescharacteristic
487C2×C4#56811yescharacteristic
488C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
489C2×C4C2×#13711yescharacteristic
490C2×C4#57611yescharacteristic
491C2×C4#57711yescharacteristic
492C8—42no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—82no—
497C8—82no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22#288011yescharacteristic
520C22C22×#9911yescharacteristic
521C22#288911yescharacteristic
522C22C2×#41711yescharacteristic
523C22#289711yescharacteristic
524C22C2×#42111yescharacteristic
525C22#290111yescharacteristic
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—24no—
533C4—41no—
534C4—42no—
535C4—42no—
536C4—41no—
537C4—81no—
538C4—81no—
539C4—82no—
540C4—81no—
541C4—82no—
542C4#151314yes—
543C2—42no—
544C2—82no—
545C2—82no—
546C2#1741411yescharacteristic
547C2#1751211yescharacteristic
548C2#1752211yescharacteristic
549C2C2×#151311yescharacteristic
550C2#1760311yescharacteristic
551C2#1768211yescharacteristic
552C2#1776511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36492C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22473C2semidirect
3. #20630C2semidirect
4. #7731C2semidirect
5. #7711C2semidirect
6. #7688C2semidirect
7. #7665C2semidirect
8. #7616C2semidirect
9. #7616C2semidirect
10. #7599C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#364922
#224732
#206302
#77312
#77112
#76882
#76652
#76162
#75992
#75652