Finite Group Databasebeta

← #196801

Subgroups — #196801

Order 512.

1831 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3C2×#1817C211yesmaximal, characteristic, complemented
4C2×#1501C212yesmaximal
5#9149C212yesmaximal, complemented
6#9148C212yesmaximal, complemented
7#9145C212yesmaximal, complemented
8#9144C212yesmaximal, complemented
9#8012C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C22×#117—21no—
12C23×C42C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14#2855C412yescomplemented
15#2855C2211yescharacteristic
16#2855C2211yescharacteristic
17#2854C412yescomplemented
18#2854C2211yescharacteristic
19#2854C2211yescharacteristic
20#2853—22no—
21#2849—21no—
22#2849—21no—
23C4×#123—22no—
24C4×#123—21no—
25C4×#123—21no—
26#2297C412yescomplemented
27#2296C412yescomplemented
28#2280C2211yescharacteristic
29C4×#117—28no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4—21no—
33C2×C4×C22⋊C4C2212yes—
34C2×#376C2212yes—
35C2×C43—21no—
36#1817—22no—
37#1817C2212yes—
38#1817C2212yes—
39C2×#336—22no—
40#1501—28no—
41C2×#328C2214yes—
42C2×#327C2214yes—
43#1294C2214yes—
44#1294C2214yes—
45C22×#41C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50#525—24no—
51#525—24no—
52C2×#123—22no—
53C2×#123—21no—
54C2×#123—21no—
55C2×#123—21no—
56C2×C22⋊Q8C2×C411yescharacteristic
57C2×C22⋊Q8C2311yescharacteristic
58C2×#121C2×C411yescharacteristic
59C2×#121C2311yescharacteristic
60C2×#120—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×#117—216no—
64C2×#117—28no—
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C412yes—
67C22×C4⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—24no—
70C22×C42—22no—
71C22×C42—24no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42C2×C412yes—
75C22×C42C2×C412yes—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D412yes—
79C22×C42D412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8C2314yes—
86#391C2×C412yes—
87#391C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#386—21no—
106#386—21no—
107#385—22no—
108#385—22no—
109#385—22no—
110#385—21no—
111#385—21no—
112#381—22no—
113#381—21no—
114#381—21no—
115#381—21no—
116#380C2×C412yes—
117#380C2311yescharacteristic
118#380C2311yescharacteristic
119#379—21no—
120C4×C4⋊C4—28no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—28no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4D418yes—
131#376—28no—
132C2×#95—22no—
133C2×#95C2314yes—
134C43—28no—
135#336—24no—
136#336—24no—
137#328—28no—
138#327—28no—
139C25C2×D411yescharacteristic
140C2×#41—24no—
141C2×#41—44no—
142C22×Q8C22×C411yescharacteristic
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C411yescharacteristic
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—28no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4C22⋊C412yes—
158C23×C4C22⋊C412yes—
159C23×C4C22⋊C412yes—
160C23×C4C22⋊C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D414yes—
171C23×C4#4114yes—
172C23×C4C2411yescharacteristic
173C22×C8—24no—
174C22×C8—48no—
175C22×C8—44no—
176#123—44no—
177#123—48no—
178#123—44no—
179#123—44no—
180C22⋊Q8—24no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183#121—24no—
184#121—22no—
185#121—22no—
186#120—42no—
187#120—42no—
188C4×D4—42no—
189C4×D4—44no—
190C4×D4—42no—
191#117—264no—
192#117—24no—
193#117—44no—
194C2×C4⋊C4—216no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C22⋊C4—216no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C42—28no—
222C2×C42—216no—
223C2×C42—28no—
224C2×C42—28no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—28no—
228C2×C42—28no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—28no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—28no—
239C2×C42—44no—
240C2×C42—48no—
241C2×C42—44no—
242C2×C42C22⋊C412yes—
243C2×C42C22⋊C412yes—
244C2×C42C22⋊C412yes—
245C2×C42C22⋊C412yes—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C22⋊C8—24no—
251C22⋊C8—24no—
252C22⋊C8—44no—
253#95—28no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—24no—
264#95—21no—
265#95—21no—
266#95C2×D418yes—
267C24—22no—
268C24—22no—
269C24—24no—
270C24—21no—
271C24—42no—
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C411yescharacteristic
274C24#12012yes—
275C24#12312yes—
276C24C22×D411yescharacteristic
277#41—88no—
278C2×Q8—22no—
279C2×Q8—42no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—44no—
290C22×C4—216no—
291C22×C4—216no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—216no—
310C22×C4—21no—
311C22×C4—28no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4#10418yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C4×D414yes—
346C22×C4C4×D414yes—
347C22×C4#12012yes—
348C22×C4#12114yes—
349C22×C4#12114yes—
350C22×C4#12312yes—
351C22×C4#13514yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C2×#4112yes—
356C2×C8—48no—
357C2×C8—44no—
358C2×C8—88no—
359C4⋊C4—232no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C22⋊C4—232no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C42—232no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—416no—
395C42—416no—
396C42—42no—
397C42—44no—
398C23—22no—
399C23—22no—
400C23—21no—
401C23—21no—
402C23—22no—
403C23—24no—
404C23—22no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—42no—
412C23—42no—
413C23—48no—
414C23—48no—
415C23—42no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23#38612yes—
421C23#38612yes—
422C23C2×#10414yes—
423C23C22×C22⋊C411yescharacteristic
424C23C2×#12011yescharacteristic
425C23C2×#12311yescharacteristic
426C23C2×#13512yes—
427C23#57512yes—
428Q8—82no—
429D4—82no—
430D4—84no—
431D4—82no—
432D4—84no—
433D4—82no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—216no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—216no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—216no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—48no—
464C2×C4—48no—
465C2×C4—48no—
466C2×C4—416no—
467C2×C4—416no—
468C2×C4—42no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—84no—
485C2×C4#38612yes—
486C2×C4#38612yes—
487C2×C4#41014yes—
488C2×C4#41614yes—
489C2×C4C2×#10414yes—
490C2×C4C2×#10414yes—
491C2×C4C2×C4×D412yes—
492C2×C4C2×#12112yes—
493C8—84no—
494C22—24no—
495C22—24no—
496C22—24no—
497C22—24no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—48no—
508C22—48no—
509C22—42no—
510C22—42no—
511C22—82no—
512C22—82no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#189614yes—
517C22C2×#38611yescharacteristic
518C22C2×#41012yes—
519C22C2×#41611yescharacteristic
520C22#289611yescharacteristic
521C22C22×#10412yes—
522C22#300212yes—
523C22#309112yes—
524C4—24no—
525C4—24no—
526C4—24no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—416no—
531C4—42no—
532C4—82no—
533C4—82no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4#189914yes—
538C2—22no—
539C2—22no—
540C2—24no—
541C2—42no—
542C2—161no—
543C2#1753312yes—
544C2#1754312yes—
545C2#2090811yescharacteristic
546C2C2×#189611yescharacteristic
547C2C2×#189911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#117C2semidirect
3. C2×#1817C2semidirect
5. #9149C2semidirect
6. #9148C2semidirect
7. #9145C2semidirect
8. #9144C2semidirect
9. #8012C2semidirect
14. #2855C4semidirect
17. #2854C4semidirect
26. #2297C4semidirect
27. #2296C4semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×C4×#1172
C2×#18172
C2×#15012
#91492
#91482
#91452
#91442
#80122