Finite Group Databasebeta

← #196053

Subgroups — #196053

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2974C211yesmaximal, characteristic
2C2×#2580C211yesmaximal, characteristic
3#21788C211yesmaximal, characteristic
4C2×#1467C211yesmaximal, characteristic, complemented
5C2×#1461C211yesmaximal, characteristic, complemented
6#9085C214yesmaximal, complemented
7#9081C214yesmaximal, complemented
8#7914C211yesmaximal, characteristic, complemented
9#7914C211yesmaximal, characteristic, complemented
10C22×C4×Q8C2211yescharacteristic
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#2580—21no—
24#2580—22no—
25#2580—21no—
26#2580—21no—
27#2580—21no—
28#2580—22no—
29C2×#399C412yes—
30C2×#393C412yes—
31C2×#390C412yes—
32C2×#389C412yes—
33C2×#385C2214yes—
34C2×#382C2211yescharacteristic
35#1955—22no—
36#1954—21no—
37#1954—21no—
38#1779C2214yes—
39#1467—24no—
40#1461—24no—
41C2×#324C2212yes—
42C2×#324C2214yes—
43C2×#324C2212yes—
44#1280C2212yes—
45#1280C2214yes—
46#1280C2212yes—
47C23×Q8C2×C411yescharacteristic
48C23×D4C2×C411yescharacteristic
49C24×C4C2312yes—
50C22×M16—22no—
51C23×C8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—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×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C414yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4C2×C414yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C414yes—
78C22×C42C2×C412yes—
79C22×C42C2311yescharacteristic
80#436—28no—
81#435—216no—
82C8×D4—28no—
83#432—22no—
84#432—22no—
85#432—22no—
86#432D414yes—
87#431—22no—
88#431—22no—
89#431—22no—
90#431D414yes—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95#407—24no—
96#407—24no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8C2312yes—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2311yescharacteristic
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2311yescharacteristic
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112#399—22no—
113#399—22no—
114#399—22no—
115#399—22no—
116#393—22no—
117#393—22no—
118#393—22no—
119#393—22no—
120#390—22no—
121#390—24no—
122#390—22no—
123#389—24no—
124#389—22no—
125#389—22no—
126#385—24no—
127#385—24no—
128#385—24no—
129#385—24no—
130#382—21no—
131#382—21no—
132#382—21no—
133#382—21no—
134C2×#95C2×C412yes—
135C2×#95C2×C414yes—
136C2×#95C2312yes—
137#324—24no—
138#324—24no—
139#324—28no—
140C25C22×C412yes—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—24no—
146C22×Q8—42no—
147C22×D4—24no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—42no—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22⋊C412yes—
153C22×D4C22⋊C411yescharacteristic
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—84no—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C414yes—
164C23×C4C22×C412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2411yescharacteristic
170C2×M16—28no—
171C2×M16—84no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—28no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—84no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×D4—22no—
188C4×D4—24no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—24no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—24no—
198C4×D4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
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—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—84no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—84no—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4112yes—
244C2×C42#4112yes—
245C4⋊C8—24no—
246C4⋊C8—24no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C22⋊C8—28no—
250C22⋊C8—28no—
251C22⋊C8—21no—
252C22⋊C8—22no—
253C22⋊C8—21no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—84no—
259C22⋊C8—84no—
260C22⋊C8C2×D412yes—
261C22⋊C8C2×D414yes—
262C22⋊C8C2×D412yes—
263#97—24no—
264#97—42no—
265C4×C8—24no—
266C4×C8—42no—
267#95—22no—
268#95—22no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273C24—24no—
274C24—22no—
275C24—22no—
276C24—24no—
277C24—24no—
278C24—44no—
279C24—42no—
280C24C2×C22⋊C412yes—
281C24C2×C22⋊C412yes—
282C24C23×C411yescharacteristic
283C2×Q8—28no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—48no—
287C2×Q8—44no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—42no—
300C2×D4#10114yes—
301C22×C4—22no—
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—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4#13114yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358M16—88no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—44no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—44no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—88no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C42—44no—
393C42—44no—
394C42#12012yes—
395C42#12012yes—
396C42#12112yes—
397C42#12112yes—
398C42#12112yes—
399C42#12112yes—
400C42#12312yes—
401C42#12312yes—
402C23—22no—
403C23—21no—
404C23—21no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—44no—
415C23—44no—
416C23—48no—
417C23—48no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—82no—
422C23—82no—
423C23—84no—
424C23—84no—
425C23C2×#10114yes—
426C23C22×C22⋊C411yescharacteristic
427C23C2×C4×D411yescharacteristic
428C23C2×C4×D411yescharacteristic
429C23C2×#13112yes—
430C23#56812yes—
431Q8—48no—
432Q8—48no—
433D4—28no—
434D4—48no—
435D4—44no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#38612yes—
483C2×C4#38612yes—
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4C2×#10112yes—
487C2×C4#41312yes—
488C2×C4C2×#12011yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C8—42no—
493C8—44no—
494C8—42no—
495C8—82no—
496C8—82no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—28no—
506C22—24no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—84no—
516C22—84no—
517C22—88no—
518C22C2×#38611yescharacteristic
519C22#288012yes—
520C22#288312yes—
521C22C22×#10111yescharacteristic
522C22C2×#41311yescharacteristic
523C22#292911yescharacteristic
524C22#293011yescharacteristic
525C22#293312yes—
526C4—22no—
527C4—22no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—22no—
533C4—44no—
534C4—42no—
535C4—42no—
536C4—84no—
537C4—82no—
538C4#190114yes—
539C2—21no—
540C2—21no—
541C2—22no—
542C2—44no—
543C2—82no—
544C2#1742612yes—
545C2#1742912yes—
546C2#2089911yescharacteristic
547C2#2098511yescharacteristic
548C2C2×#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×#1467C2semidirect
5. C2×#1461C2semidirect
6. #9085C2semidirect
7. #9081C2semidirect
8. #7914C2semidirect
9. #7914C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29742
C2×#25802
#217882
C2×#14672
C2×#14612
#90852
#90812
#79142