Finite Group Databasebeta

← #191119

Subgroups — #191119

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic, complemented
3#13950C211yesmaximal, characteristic, complemented
4#8717C211yesmaximal, characteristic, complemented
5#8717C211yesmaximal, characteristic, complemented
6#8575C212yesmaximal, complemented
7#8573C212yesmaximal, complemented
8#8332C211yesmaximal, characteristic
9#8332C211yesmaximal, characteristic
10#8220C212yesmaximal, complemented
11C4×#327C212yesmaximal, complemented
12C2×#532C2211yescharacteristic
13C2×#520—21no—
14#2367—22no—
15#2366C412yescomplemented
16#2365C412yescomplemented
17#2354C412yescomplemented
18#2352—22no—
19D4×C4⋊C4C412yescomplemented
20#2325—21no—
21#2325—21no—
22#2311—21no—
23C4×#121C2212yes—
24C4×#121C2212yes—
25#2285C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1916—22no—
28#1916—22no—
29#1817C2211yescharacteristic
30#1817C2211yescharacteristic
31#1797C2212yes—
32C4×C22⋊C8C2212yes—
33C2×#327C2212yes—
34C2×#327C2212yes—
35C2×#327C2212yes—
36C2×#327C2212yes—
37#1445—21no—
38#1443—21no—
39#1436—21no—
40#1434—21no—
41#1434C2211yescharacteristic
42#1434C2211yescharacteristic
43#1434C2211yescharacteristic
44#1434C2211yescharacteristic
45#1434C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C22×#41—21no—
52#532—24no—
53#532—24no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#121—21no—
58C2×#121C2312yes—
59#520—44no—
60#520—44no—
61C2×C4×Q8—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C412yes—
69C2×#117—21no—
70C2×#117—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—22no—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×C4×C8—22no—
82#391—22no—
83#390—22no—
84#390C2×C412yes—
85#390C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#387—22no—
93#387—22no—
94#386—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#385—22no—
99#385—22no—
100#384C2×C411yescharacteristic
101#384C2×C411yescharacteristic
102#382—21no—
103#380—21no—
104#380—22no—
105#380—22no—
106#380C2312yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130C2×#95C2311yescharacteristic
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133C2×#95C2311yescharacteristic
134#336—22no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—22no—
140#327—24no—
141#327—24no—
142#327—24no—
143#327—24no—
144C2×#41—24no—
145C2×#41—24no—
146C2×#41—42no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C22×Q8—21no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4C22×C412yes—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C4⋊Q8—24no—
172#127—24no—
173#124—28no—
174#121—24no—
175#121—24no—
176#121—22no—
177#121—22no—
178#121#4114yes—
179#121#4114yes—
180C4×Q8—48no—
181C4×D4—28no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—48no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4—44no—
189C4×D4—82no—
190C4×D4—82no—
191#117—48no—
192#117—42no—
193#117—42no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
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×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—81no—
230C2×C42—81no—
231C2×C42—81no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C4⋊C8—82no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C4×C8—82no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—81no—
265#95—81no—
266#95—81no—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271C24—22no—
272C24C22×D411yescharacteristic
273#41—48no—
274#41—48no—
275#41—48no—
276#41—48no—
277C2×Q8—24no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4#11714yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—28no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C412yes—
368C22×C4#13512yes—
369C22×C4#13512yes—
370C22×C4AGL(1,ℤ/8)12yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4112yes—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—81no—
395C4⋊C4—81no—
396C4⋊C4#11714yes—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—44no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—44no—
417C42—44no—
418C42—44no—
419C42—84no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23C22×C22⋊C411yescharacteristic
432C23#52511yescharacteristic
433C23#52511yescharacteristic
434C23C2×#13511yescharacteristic
435C23C2×#13511yescharacteristic
436C23C2×AGL(1,ℤ/8)11yescharacteristic
437C23C2×AGL(1,ℤ/8)11yescharacteristic
438Q8—48no—
439D4—44no—
440D4—44no—
441D4—48no—
442D4—44no—
443D4—44no—
444D4—44no—
445D4—44no—
446D4—84no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—416no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—164no—
527C2×C4—164no—
528C2×C4#41612yes—
529C2×C4#41612yes—
530C2×C4#41814yes—
531C2×C4C2×#11712yes—
532C2×C4#52512yes—
533C8—82no—
534C8—82no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22#228011yescharacteristic
555C22#288311yescharacteristic
556C22#288411yescharacteristic
557C22C2×#41611yescharacteristic
558C22C2×#41811yescharacteristic
559C22#290111yescharacteristic
560C22#290111yescharacteristic
561C22#310511yescharacteristic
562C22#310511yescharacteristic
563C22#310511yescharacteristic
564C22#310511yescharacteristic
565C4—24no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—44no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—161no—
581C4—161no—
582C2—21no—
583C2—21no—
584C2—21no—
585C2—21no—
586C2—84no—
587C2#1761611yescharacteristic
588C2#1761711yescharacteristic
589C2#1762011yescharacteristic
590C2#1763111yescharacteristic
591C2#2072511yescharacteristic
592C2#2073311yescharacteristic
593C2#2074911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21912C2semidirect
3. #13950C2semidirect
4. #8717C2semidirect
5. #8717C2semidirect
6. #8575C2semidirect
7. #8573C2semidirect
10. #8220C2semidirect
11. C4×#327C2semidirect
15. #2366C4semidirect
16. #2365C4semidirect
17. #2354C4semidirect
19. D4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219122
#139502
#87172
#85752
#85732
#83322
#82202
C4×#3272