Finite Group Databasebeta

← #196067

Subgroups — #196067

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2974C211yesmaximal, characteristic
2C2×#2566C211yesmaximal, characteristic
3#21788C211yesmaximal, characteristic
4C2×#1470C211yesmaximal, characteristic, complemented
5C2×#1470C211yesmaximal, characteristic, complemented
6#9085C214yesmaximal, complemented
7#9081C214yesmaximal, complemented
8#7912C211yesmaximal, characteristic, complemented
9#7908C211yesmaximal, characteristic, complemented
10C22×#125C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#2974—24no—
13#2974—24no—
14C2×#436—21no—
15C2×#435—22no—
16C2×C8×D4—21no—
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#407C2212yes—
22C22×C22⋊C8C2212yes—
23#2566—21no—
24#2566—22no—
25#2566—21no—
26#2566—21no—
27#2566—21no—
28#2566—22no—
29C2×#396C414yes—
30C2×#390C411yescharacteristic
31C2×#390C411yescharacteristic
32C2×#385C2214yes—
33C2×#384C411yescharacteristic
34C2×#382C411yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1955—22no—
37#1954—21no—
38#1954—21no—
39#1779C2214yes—
40#1470—24no—
41#1470—24no—
42C2×#324C2212yes—
43C2×#324C2214yes—
44C2×#324C2212yes—
45#1280C2212yes—
46#1280C2214yes—
47#1280C2212yes—
48C23×D4C2×C411yescharacteristic
49C24×C4C2312yes—
50C22×M16—22no—
51C23×C8—22no—
52C2×#125—21no—
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D412yes—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D412yes—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C414yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4C2×C414yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C414yes—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2311yescharacteristic
83#436—28no—
84#435—216no—
85C8×D4—28no—
86#432—22no—
87#432—22no—
88#432—22no—
89#432D414yes—
90#431—22no—
91#431—22no—
92#431—22no—
93#431D414yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98#407—24no—
99#407—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2311yescharacteristic
109C2×C22⋊C8C2312yes—
110C2×C22⋊C8C2311yescharacteristic
111C2×#97—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#396—24no—
116#396—24no—
117#396—24no—
118#396—24no—
119#390—21no—
120#390—22no—
121#390—21no—
122#390—21no—
123#390—22no—
124#390—21no—
125#385—24no—
126#385—24no—
127#385—24no—
128#385—24no—
129#384—22no—
130#384—21no—
131#384—21no—
132#382—22no—
133#382—21no—
134#382—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C2×#95C2×C414yes—
140C2×#95C2312yes—
141#324—28no—
142#324—24no—
143#324—24no—
144C25C22×C412yes—
145C22×D4—24no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—42no—
149C22×D4C22⋊C411yescharacteristic
150C22×D4C22⋊C412yes—
151C22×D4C22⋊C411yescharacteristic
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—84no—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2411yescharacteristic
169C2×M16—28no—
170C2×M16—84no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—28no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—84no—
178#125—42no—
179#125—42no—
180#125—42no—
181#125—42no—
182#125—42no—
183#125—42no—
184#125—42no—
185#125—42no—
186C4×D4—22no—
187C4×D4—24no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—24no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4—24no—
197C4×D4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—84no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—84no—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C412yes—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4112yes—
254C2×C42#4112yes—
255C4⋊C8—24no—
256C4⋊C8—24no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261C22⋊C8—21no—
262C22⋊C8—22no—
263C22⋊C8—21no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—84no—
269C22⋊C8—84no—
270C22⋊C8C2×D412yes—
271C22⋊C8C2×D414yes—
272C22⋊C8C2×D412yes—
273#97—24no—
274#97—42no—
275C4×C8—24no—
276C4×C8—42no—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—22no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—44no—
287C24—42no—
288C24C2×C22⋊C412yes—
289C24C2×C22⋊C412yes—
290C24C23×C411yescharacteristic
291C2×D4—24no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—42no—
301C2×D4#10114yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
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—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C4×D412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4C4×D412yes—
354C22×C4#13114yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361M16—88no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—42no—
366C2×C8—44no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—44no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—88no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42#12012yes—
404C42#12012yes—
405C42#12112yes—
406C42#12112yes—
407C42#12112yes—
408C42#12112yes—
409C42#12312yes—
410C42#12312yes—
411C23—22no—
412C23—21no—
413C23—21no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—44no—
424C23—44no—
425C23—48no—
426C23—48no—
427C23—42no—
428C23—42no—
429C23—44no—
430C23—82no—
431C23—82no—
432C23—84no—
433C23—84no—
434C23C2×#10114yes—
435C23C22×C22⋊C411yescharacteristic
436C23C2×C4×D411yescharacteristic
437C23C2×C4×D411yescharacteristic
438C23C2×#13112yes—
439C23#56812yes—
440D4—28no—
441D4—48no—
442D4—44no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#38612yes—
492C2×C4#38612yes—
493C2×C4#38612yes—
494C2×C4#38612yes—
495C2×C4C2×#10112yes—
496C2×C4#41312yes—
497C2×C4C2×#12011yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C8—42no—
502C8—44no—
503C8—42no—
504C8—82no—
505C8—82no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—28no—
515C22—24no—
516C22—44no—
517C22—42no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—84no—
525C22—84no—
526C22—88no—
527C22C2×#38611yescharacteristic
528C22#288012yes—
529C22#288312yes—
530C22C22×#10111yescharacteristic
531C22C2×#41311yescharacteristic
532C22#293111yescharacteristic
533C22#293211yescharacteristic
534C22#293512yes—
535C4—22no—
536C4—22no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—22no—
542C4—44no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4#190114yes—
549C2—21no—
550C2—21no—
551C2—22no—
552C2—44no—
553C2—82no—
554C2#1742412yes—
555C2#1742612yes—
556C2#2090011yescharacteristic
557C2#2098511yescharacteristic
558C2C2×#190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1470C2semidirect
5. C2×#1470C2semidirect
6. #9085C2semidirect
7. #9081C2semidirect
8. #7912C2semidirect
9. #7908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29742
C2×#25662
#217882
C2×#14702
#90852
#90812
#79122
#79082