Finite Group Databasebeta

← #164636

Subgroups — #164636

Order 512.

1124 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36259C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic, complemented
3#20684C211yesmaximal, characteristic, complemented
4#7718C212yesmaximal, complemented
5#7694C212yesmaximal, complemented
6#7685C211yesmaximal, characteristic, complemented
7#7668C211yesmaximal, characteristic, complemented
8#7616C211yesmaximal, characteristic, complemented
9#7607C212yesmaximal, complemented
10#7593C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic, complemented
12C2×#547—21no—
13C2×#547C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#436—21no—
16C2×#408—21no—
17#2857C2211yescharacteristic
18#2854C2212yes—
19#2546C411yescharacteristic
20#2538C411yescharacteristic
21#2486C412yes—
22#2429—21no—
23#2425—22no—
24#2416C412yes—
25#2415—21no—
26#2413—22no—
27D4×C4⋊C4C411yescharacteristic
28#2339—21no—
29C4×#121C2211yescharacteristic
30C2×#389C2212yes—
31#1984—21no—
32#1870—22no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38C4×C22⋊C8C2211yescharacteristic
39#1729—21no—
40#1728C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41—21no—
54C22×#41D411yescharacteristic
55C22×M16—21no—
56C22×M16—22no—
57C23×C8—21no—
58#547—44no—
59#547—44no—
60#547—44no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×#123—22no—
64C2×#123—22no—
65C2×#123—22no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×#121—22no—
70C2×#121C2×C411yescharacteristic
71C2×#121C2×C412yes—
72C4×#41—44no—
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C412yes—
80C2×C4×D4D411yescharacteristic
81C2×C4×D4D412yes—
82C2×#117D412yes—
83C2×#117D411yescharacteristic
84C22×C4⋊C4—22no—
85C22×C4⋊C4C2×C412yes—
86C22×C4⋊C4C2×C412yes—
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89#436—44no—
90C2×C4⋊C8—21no—
91#408—44no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C4×C8—21no—
98#398—22no—
99#398—22no—
100#397—22no—
101#396—22no—
102#396—22no—
103#394—22no—
104#391—21no—
105#391—21no—
106#390C2×C412yes—
107#390C2311yescharacteristic
108#389—21no—
109#389—21no—
110#389—22no—
111#389—22no—
112#389C2×C412yes—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2312yes—
116#389C2311yescharacteristic
117#388—22no—
118#388—22no—
119#388—21no—
120#386—22no—
121#386—21no—
122#386—22no—
123#385—22no—
124#384C2×C411yescharacteristic
125#381—22no—
126#380—21no—
127#380—21no—
128#380C2×C412yes—
129#380C2×C412yes—
130#380C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C22⋊C4—21no—
133C2×#95—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—21no—
138#336—21no—
139#336—22no—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2312yes—
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146C2×#41—24no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—42no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—44no—
153C22×Q8—21no—
154C22×Q8C22⋊C411yescharacteristic
155C22×D4—22no—
156C22×D4—21no—
157C22×D4C22⋊C412yes—
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—44no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—44no—
173C2×M16—42no—
174C2×M16—82no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—42no—
184C4⋊Q8—42no—
185C4⋊Q8—44no—
186#123—48no—
187#123—44no—
188#123—44no—
189#123—48no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194#121—22no—
195#121—41no—
196#121—44no—
197#121—48no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—42no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—42no—
207C4×D4—42no—
208#117—44no—
209#117—42no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C22⋊C412yes—
245C2×C22⋊C4C22⋊C412yes—
246C2×C22⋊C4C22⋊C412yes—
247C2×C42—21no—
248C2×C42—42no—
249C2×C42—44no—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2411yescharacteristic
257C4⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—44no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262C22⋊C8—81no—
263C4×C8—42no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—21no—
272#95—22no—
273C24—21no—
274C24C2×C22⋊C412yes—
275C24C2×C22⋊C411yescharacteristic
276#41—48no—
277#41—48no—
278#41—816no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—42no—
299C2×D4—81no—
300C2×D4#9914yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—82no—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4AGL(1,ℤ/8)12yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348M16—88no—
349M16—84no—
350C2×C8—22no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—84no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—81no—
391C4⋊C4#13112yes—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—82no—
407C42—22no—
408C42—22no—
409C42—41no—
410C42—42no—
411C42—41no—
412C42—42no—
413C42—42no—
414C42—42no—
415C23—22no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—21no—
420C23—22no—
421C23—21no—
422C23—22no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—21no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23C2×#9914yes—
435C23#41812yes—
436C23C22×C22⋊C411yescharacteristic
437Q8—44no—
438Q8—84no—
439D4—28no—
440D4—48no—
441D4—81no—
442D4—81no—
443D4—84no—
444D4—88no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—88no—
494C2×C4—88no—
495C2×C4—84no—
496C2×C4C2×#9912yes—
497C2×C4#41812yes—
498C2×C4#42112yes—
499C2×C4C2×#13111yescharacteristic
500C2×C4#56811yescharacteristic
501C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4#57711yescharacteristic
504C2×C4#57811yescharacteristic
505C8—42no—
506C8—81no—
507C8—81no—
508C8—81no—
509C8—82no—
510C8—82no—
511C22—24no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—24no—
516C22—24no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—41no—
524C22—41no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22#288011yescharacteristic
533C22C22×#9911yescharacteristic
534C22#288911yescharacteristic
535C22C2×#41811yescharacteristic
536C22#289811yescharacteristic
537C22C2×#42111yescharacteristic
538C22#290111yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—41no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—41no—
551C4—81no—
552C4—82no—
553C4—81no—
554C4—82no—
555C4#151014yes—
556C2—44no—
557C2—42no—
558C2—161no—
559C2#1741611yescharacteristic
560C2#1751111yescharacteristic
561C2#1752511yescharacteristic
562C2C2×#151011yescharacteristic
563C2#1760811yescharacteristic
564C2#1768211yescharacteristic
565C2#1775911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36259C2
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. #20684C2semidirect
4. #7718C2semidirect
5. #7694C2semidirect
6. #7685C2semidirect
7. #7668C2semidirect
8. #7616C2semidirect
9. #7607C2semidirect
10. #7593C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#362592
#224732
#206842
#77182
#76942
#76852
#76682
#76162
#76072
#75932
#75652