Finite Group Databasebeta

← #196550

Subgroups — #196550

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic
4C2×#1534C212yesmaximal, complemented
5#9125C214yesmaximal, complemented
6#9122C214yesmaximal, complemented
7#8030C212yesmaximal, complemented
8C22×#125C2211yescharacteristic
9C23×C42C2211yescharacteristic
10#2920—28no—
11C2×#432C2212yes—
12C2×#432C2212yes—
13C2×#432C2212yes—
14C2×#431—21no—
15C2×#431—21no—
16C2×#423—21no—
17C2×#407—21no—
18C2×C4×M16C2212yes—
19#2851—21no—
20#2851—21no—
21#2851—21no—
22#2851—21no—
23#2851C412yes—
24#2851C412yes—
25#2851C2212yes—
26#2851C2212yes—
27C2×#389C411yescharacteristic
28C2×#389C411yescharacteristic
29C2×#387—21no—
30C2×#387—21no—
31C2×#385—21no—
32C2×#385—21no—
33C2×#382C2211yescharacteristic
34C2×#380C411yescharacteristic
35C2×#380C411yescharacteristic
36#1817—21no—
37#1797—21no—
38#1797—21no—
39#1779—21no—
40#1749C2212yes—
41#1749C2212yes—
42#1534—28no—
43C2×#330C2218yescomplemented
44#1295C2218yescomplemented
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C22×M16—22no—
49C2×#125—21no—
50C2×#125—21no—
51C2×#125—24no—
52C2×#125—24no—
53C2×#125C2×C412yes—
54C2×#125C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42—24no—
66C22×C42—24no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#432—24no—
80#432—24no—
81#432—24no—
82#432—28no—
83#432D418yes—
84#431—28no—
85#431—28no—
86#423—28no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90#407—28no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C4×M16—28no—
97C2×#97C2314yes—
98C2×#97C2314yes—
99C2×C4×C8—22no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#382—21no—
125#382—21no—
126#382C2×C412yes—
127#382C2312yes—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380C2×C412yes—
133#380C2×C412yes—
134#380C2×C412yes—
135#380C2×C412yes—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142#336—22no—
143#336—22no—
144#330—216no—
145C25C22⋊C411yescharacteristic
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C2×M16—28no—
179C2×M16—44no—
180C2×M16—44no—
181C22×C8—24no—
182C22×C8—24no—
183#125—28no—
184#125—24no—
185#125—24no—
186#125—42no—
187#125—44no—
188#125—42no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D414yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4114yes—
253C2×C42C2411yescharacteristic
254C4⋊C8—28no—
255C4⋊C8—28no—
256C4⋊C8—44no—
257C22⋊C8—28no—
258C22⋊C8—28no—
259C22⋊C8—44no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262#97—24no—
263#97—24no—
264#97—24no—
265#97C2×D418yes—
266C4×C8—28no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—41no—
285C24—41no—
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288C24C2×C22⋊C411yescharacteristic
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
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—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—416no—
346C22×C4—48no—
347C22×C4—416no—
348C22×C4—48no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4#10114yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C4×D414yes—
372C22×C4#12012yes—
373C22×C4#12114yes—
374C22×C4#12312yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378M16—416no—
379C2×C8—28no—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—44no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C42—22no—
407C42—22no—
408C42—21no—
409C42—21no—
410C42—22no—
411C42—22no—
412C42—21no—
413C42—21no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42C4×D414yes—
421C42#12012yes—
422C42#12114yes—
423C42#12312yes—
424C42C22×D412yes—
425C42C2×#4112yes—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—28no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23#38612yes—
450C23C2×#10112yes—
451C23#41312yes—
452C23#42112yes—
453C23#42112yes—
454C23C22×C22⋊C411yescharacteristic
455C2×C4—24no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4#38612yes—
519C2×C4#38612yes—
520C2×C4#38612yes—
521C2×C4C2×#10112yes—
522C2×C4C2×#10112yes—
523C2×C4#42112yes—
524C2×C4#42112yes—
525C2×C4C2×C4×D412yes—
526C2×C4C2×#12011yescharacteristic
527C2×C4C2×#12112yes—
528C2×C4C2×#12311yescharacteristic
529C8—48no—
530C8—84no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—416no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22#190114yes—
553C22C2×#38611yescharacteristic
554C22C22×#10111yescharacteristic
555C22C2×#41311yescharacteristic
556C22C2×#42111yescharacteristic
557C22C2×#42111yescharacteristic
558C22C2×#42111yescharacteristic
559C22C2×#42111yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—41no—
572C4—41no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C2—24no—
578C2—42no—
579C2—42no—
580C2#1765911yescharacteristic
581C2#1766411yescharacteristic
582C2#1766711yescharacteristic
583C2#1766911yescharacteristic
584C2#2090211yescharacteristic
585C2#2097911yescharacteristic
586C2C2×#190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1534C2semidirect
5. #9125C2semidirect
6. #9122C2semidirect
7. #8030C2semidirect
43. C2×#330C22semidirect
44. #1295C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#28512
#207152
C2×#15342
#91252
#91222
#80302