Finite Group Databasebeta

← #164213

Subgroups — #164213

Order 512.

1175 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36444C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic, complemented
3#20378C211yesmaximal, characteristic, complemented
4#7705C212yesmaximal, complemented
5#7688C212yesmaximal, complemented
6#7685C211yesmaximal, characteristic, complemented
7#7668C211yesmaximal, characteristic, complemented
8#7633C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7582C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic, complemented
12C2×D4×Q8C411yescharacteristic
13C2×#548—21no—
14C2×C4×#41C2211yescharacteristic
15C2×#436—21no—
16C2×#408—21no—
17C2×#405—21no—
18#2857C2211yescharacteristic
19#2857C2212yes—
20#2547C412yes—
21#2547C411yescharacteristic
22#2537C411yescharacteristic
23#2537C412yes—
24#2427—21no—
25#2427—22no—
26#2415—21no—
27#2415—22no—
28Q8×C4⋊C4C411yescharacteristic
29Q8×C22⋊C4—21no—
30C4×#121C2211yescharacteristic
31C4×#121C2212yes—
32#1984—21no—
33#1870—22no—
34#1817C2212yes—
35#1797C2211yescharacteristic
36#1750—22no—
37#1749C2212yes—
38C4×C22⋊C8C2211yescharacteristic
39C4×C22⋊C8C2212yes—
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
54C23×Q8—21no—
55C22×M16—21no—
56C22×M16—22no—
57C23×C8—21no—
58D4×Q8—44no—
59#548—44no—
60#548—44no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C412yes—
63C2×#124—21no—
64C2×#124—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#121—22no—
71C2×#121C2×C411yescharacteristic
72C2×#121C2×C412yes—
73C4×#41—44no—
74C2×C4×Q8—21no—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×D4—22no—
78C2×C4×D4—21no—
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C412yes—
81C2×C4×D4C2×C412yes—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D412yes—
85C2×#117D411yescharacteristic
86C2×#117D412yes—
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89#436—44no—
90C2×C4⋊C8—21no—
91#408—28no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97#405—44no—
98C2×#97—22no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101#397—22no—
102#397—22no—
103#397—22no—
104#394—22no—
105#394—22no—
106#394—22no—
107#391—21no—
108#391—22no—
109#391—21no—
110#391—22no—
111#390C2311yescharacteristic
112#389—22no—
113#389—21no—
114#389—22no—
115#389—21no—
116#389C2312yes—
117#389C2311yescharacteristic
118#388—22no—
119#388—21no—
120#386—22no—
121#386—21no—
122#384C2×C411yescharacteristic
123#384C2×C412yes—
124#380—22no—
125#380—21no—
126#380—21no—
127#380—22no—
128#380C2×C412yes—
129#380C2×C412yes—
130#380C2×C412yes—
131#380C2×C411yescharacteristic
132#380C2×C412yes—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C412yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—22no—
137#336—22no—
138#336—22no—
139#336—21no—
140#336—21no—
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146#336C2312yes—
147C2×#41—24no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—42no—
151C22×Q8—22no—
152C22×Q8—21no—
153C22×Q8—44no—
154C22×Q8—44no—
155C22×Q8—44no—
156C22×Q8C22⋊C411yescharacteristic
157C22×D4—22no—
158C22×D4—21no—
159C22×D4C22⋊C411yescharacteristic
160C22×D4C22⋊C412yes—
161C22×D4C22×C411yescharacteristic
162C23×C4—22no—
163C23×C4—21no—
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—24no—
171C2×M16—44no—
172C2×M16—42no—
173C2×M16—42no—
174C2×M16—44no—
175C2×M16—82no—
176C22×C8—24no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—44no—
187#124—48no—
188#124—44no—
189C22⋊Q8—48no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C22⋊Q8—44no—
195#121—22no—
196#121—24no—
197#121—41no—
198#121—42no—
199#121—44no—
200#121—48no—
201C4×Q8—24no—
202C4×Q8—42no—
203C4×Q8—44no—
204C4×D4—24no—
205C4×D4—44no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—42no—
213C4×D4—42no—
214#117—24no—
215#117—44no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4C22⋊C412yes—
245C2×C22⋊C4C22⋊C412yes—
246C2×C22⋊C4C22⋊C412yes—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—44no—
250C2×C42—42no—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42C2411yescharacteristic
258C4⋊C8—42no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261C22⋊C8—42no—
262C22⋊C8—44no—
263C22⋊C8—81no—
264#97—44no—
265C4×C8—44no—
266C4×C8—42no—
267#95—21no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—21no—
275#95—22no—
276C24—21no—
277C24C2×C22⋊C412yes—
278C24C2×C22⋊C411yescharacteristic
279#41—48no—
280#41—48no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—44no—
288C2×Q8—42no—
289C2×Q8—816no—
290C2×Q8#10414yes—
291C2×D4—24no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—81no—
305C2×D4—82no—
306C2×D4#13112yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4AGL(1,ℤ/8)12yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351M16—48no—
352M16—88no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—22no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—48no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—41no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—82no—
391C4⋊C4—81no—
392C4⋊C4#9914yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—44no—
409C42—24no—
410C42—22no—
411C42—24no—
412C42—22no—
413C42—41no—
414C42—41no—
415C42—42no—
416C42—44no—
417C42—42no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23#41812yes—
439C23C22×C22⋊C411yescharacteristic
440Q8—28no—
441Q8—88no—
442Q8—84no—
443D4—48no—
444D4—42no—
445D4—42no—
446D4—44no—
447D4—84no—
448D4—88no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—88no—
496C2×C4—84no—
497C2×C4C2×#9914yes—
498C2×C4C2×#9912yes—
499C2×C4#41812yes—
500C2×C4C2×#10414yes—
501C2×C4C2×#10412yes—
502C2×C4C2×#13111yescharacteristic
503C2×C4#56811yescharacteristic
504C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
505C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
506C2×C4#57711yescharacteristic
507C2×C4#57811yescharacteristic
508C8—42no—
509C8—42no—
510C8—44no—
511C8—42no—
512C8—81no—
513C8—82no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22#288011yescharacteristic
536C22C22×#9911yescharacteristic
537C22#288911yescharacteristic
538C22C2×#41811yescharacteristic
539C22#289811yescharacteristic
540C22C22×#10411yescharacteristic
541C22#290111yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—24no—
547C4—21no—
548C4—21no—
549C4—24no—
550C4—41no—
551C4—42no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—82no—
556C4—82no—
557C4—81no—
558C4#150714yes—
559C2—42no—
560C2—82no—
561C2—82no—
562C2#1741411yescharacteristic
563C2#1751111yescharacteristic
564C2#1751811yescharacteristic
565C2C2×#150711yescharacteristic
566C2#1760811yescharacteristic
567C2#1765411yescharacteristic
568C2#1775911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36444C2
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. #20378C2semidirect
4. #7705C2semidirect
5. #7688C2semidirect
6. #7685C2semidirect
7. #7668C2semidirect
8. #7633C2semidirect
9. #7616C2semidirect
10. #7582C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#364442
#224732
#203782
#77052
#76882
#76852
#76682
#76332
#76162
#75822
#75652