Finite Group Databasebeta

← #481986

Subgroups — #481986

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2632C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3#17365C211yesmaximal, characteristic, complemented
4#11833C214yesmaximal, complemented
5#11815C212yesmaximal, complemented
6#11802C212yesmaximal
7#11738C214yesmaximal, complemented
8C22×#124C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11#2920—21no—
12#2632—21no—
13#2632—21no—
14#2632—21no—
15#2632—21no—
16#2632C412yes—
17#2632C412yes—
18#2632C2212yes—
19#2632C2212yes—
20C2×#399—21no—
21C2×#395C411yescharacteristic
22C2×#395C2211yescharacteristic
23C2×#394C411yescharacteristic
24C2×#394C2211yescharacteristic
25C2×#393—21no—
26C2×#388—21no—
27C2×#388—21no—
28C2×#383C411yescharacteristic
29#1821C2212yes—
30#1801—21no—
31#1801C2212yescomplemented
32#1784—21no—
33C4×#101—21no—
34C4×#101C2212yescomplemented
35#1467—21no—
36#1467C2212yescomplemented
37#1463—21no—
38#1463C2212yes—
39#1461—21no—
40#1461C2212yescomplemented
41#1354C2218yescomplemented
42#1345C2218yes—
43C23×Q8C2×C411yescharacteristic
44C23×Q8D411yescharacteristic
45C23×D4C2×C411yescharacteristic
46C2×#124—21no—
47C2×#124—21no—
48C2×#124—41no—
49C2×#124—41no—
50C2×#124—41no—
51C2×#124—41no—
52C2×#124C2×C412yes—
53C2×#124C2×C412yes—
54C2×C4×Q8—41no—
55C2×C4×Q8—41no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80#432—22no—
81#432—41no—
82#432—41no—
83#432—41no—
84#431—22no—
85#431—41no—
86#431—41no—
87#431—41no—
88#423—22no—
89#423—41no—
90C2×#101—22no—
91C2×#101—22no—
92#407—22no—
93#407—22no—
94C4×M16—22no—
95C4×M16—41no—
96#399—22no—
97#399—22no—
98#399—22no—
99#399—22no—
100#395—21no—
101#395—21no—
102#395—21no—
103#395—21no—
104#395C2×C412yes—
105#395C2×C412yes—
106#395C2×C412yes—
107#395C2312yes—
108#394—21no—
109#394—21no—
110#394—21no—
111#394—21no—
112#394C2×C412yes—
113#394C2×C412yes—
114#394C2×C412yes—
115#394C2312yes—
116#393—22no—
117#393—22no—
118#393—22no—
119#393—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#383—21no—
129#383—21no—
130#383C2×C412yes—
131#383C2×C412yes—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95C2×C411yescharacteristic
137C2×#95C2×C411yescharacteristic
138#343—22no—
139#343C2314yes—
140#324—22no—
141#324—22no—
142#324—22no—
143#324C2314yes—
144C25—21no—
145C22×Q8—21no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—82no—
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22×C412yes—
156C22×Q8C2×D411yescharacteristic
157C22×Q8C2×D411yescharacteristic
158C22×D4—21no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—82no—
162C22×D4C22×C412yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C2×M16—24no—
174C2×M16—42no—
175#124—42no—
176#124—42no—
177#124—42no—
178#124—42no—
179#124—42no—
180#124—42no—
181#124—42no—
182#124—42no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2411yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C4⋊C8—42no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272C4⋊C8—42no—
273#101—44no—
274#101—44no—
275C22⋊C8—24no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279#97—42no—
280#97—42no—
281C4×C8—42no—
282C4×C8—42no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95C22×C412yes—
294#95C22×C412yes—
295C24—22no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—41no—
303C24—41no—
304C24C2×C22⋊C411yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—82no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—82no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
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—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4#9911yescharacteristic
367C22×C4#9911yescharacteristic
368C22×C4#9911yescharacteristic
369C22×C4#9911yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384M16—44no—
385M16—44no—
386M16—84no—
387C2×C8—44no—
388C2×C8—44no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—82no—
455C23—84no—
456C23—84no—
457C23—84no—
458C23—84no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23C2×#9911yescharacteristic
465C23C2×#9911yescharacteristic
466C23C22×C22⋊C411yescharacteristic
467Q8—44no—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471Q8—84no—
472Q8—84no—
473D4—84no—
474D4—84no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
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—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4C2×#9911yescharacteristic
535C2×C4C2×#9911yescharacteristic
536C2×C4C2×#9911yescharacteristic
537C2×C4C2×#9911yescharacteristic
538C2×C4#42112yes—
539C2×C4#57512yes—
540C8—84no—
541C8—84no—
542C22—24no—
543C22—24no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—164no—
558C22—164no—
559C22#212212yes—
560C22#213312yes—
561C22C22×#9911yescharacteristic
562C22#289611yescharacteristic
563C22C2×#42111yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—82no—
581C4—82no—
582C4—82no—
583C2—21no—
584C2—21no—
585C2—22no—
586C2—22no—
587C2—162no—
588C2#1755911yescharacteristic
589C2C2×#212211yescharacteristic
590C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #17365C2semidirect
4. #11833C2semidirect
5. #11815C2semidirect
7. #11738C2semidirect
31. #1801C22semidirect
34. C4×#101C22semidirect
36. #1467C22semidirect
40. #1461C22semidirect
41. #1354C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26322
#207272
#173652
#118332
#118152
#118022
#117382