Finite Group Databasebeta

← #164026

Subgroups — #164026

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35254C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic, complemented
3#22530C211yesmaximal, characteristic, complemented
4#7714C212yesmaximal, complemented
5#7698C212yesmaximal, complemented
6#7674C212yesmaximal, complemented
7#7658C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7572C211yesmaximal, characteristic, complemented
10#7572C211yesmaximal, characteristic, complemented
11#7564C211yesmaximal, characteristic, complemented
12C2×#535C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#446—21no—
15C2×C8×Q8—21no—
16#2854C2212yes—
17#2854C2212yes—
18#2624C412yes—
19#2624C412yes—
20#2448—21no—
21#2448—21no—
22#2445—21no—
23#2445—21no—
24#2418—22no—
25#2418—22no—
26#2316C411yescharacteristic
27#2311C411yescharacteristic
28#2309C411yescharacteristic
29#2296C2211yescharacteristic
30C2×#390C2211yescharacteristic
31#1955—22no—
32#1954—22no—
33#1954—21no—
34#1954—21no—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1797C2212yescomplemented
38#1797C2212yescomplemented
39#1749C2212yescomplemented
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2212yescomplemented
43#1271C2211yescharacteristic, complemented
44#1271C2212yescomplemented
45#1271C2211yescharacteristic, complemented
46#1271C2212yescomplemented
47#1271C2211yescharacteristic, complemented
48#1271C2211yescharacteristic, complemented
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41C2×C411yescharacteristic
54C23×D4C2×C411yescharacteristic
55#535—44no—
56C2×#127—21no—
57C2×#124—21no—
58C2×#123—22no—
59C2×#123—22no—
60C2×#121C2×C412yes—
61C2×#121C2×C412yes—
62C2×#120—22no—
63C4×#41—44no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4D412yes—
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C412yes—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79#446—44no—
80C8×Q8—44no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×#97—22no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97#399—22no—
98#399—22no—
99#397—22no—
100#397—22no—
101#394—22no—
102#394—22no—
103#394—22no—
104#394—22no—
105#391—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#390C2×C412yes—
112#390C2×C412yes—
113#390C2×C412yes—
114#390C2312yes—
115#389—21no—
116#389—21no—
117#389C2×C412yes—
118#389C2×C412yes—
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#389C2311yescharacteristic
122#389C2311yescharacteristic
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#385—21no—
131#385—21no—
132#380—22no—
133#380C2×C412yes—
134#380C2×C412yes—
135#379—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C25—21no—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C22⋊C411yescharacteristic
156C22×D4C22⋊C412yes—
157C22×D4C22×C412yes—
158C23×C4—21no—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171#127—44no—
172#124—44no—
173#123—82no—
174#123—82no—
175#123—82no—
176#123—82no—
177#121—44no—
178#121—44no—
179#120—48no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×D4—44no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—44no—
186#117—24no—
187#117—42no—
188#117—44no—
189#117—42no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—81no—
222C2×C22⋊C4—81no—
223C2×C22⋊C4—81no—
224C2×C22⋊C4—81no—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—44no—
232C2×C42C22×C412yes—
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2411yescharacteristic
239C4⋊C8—42no—
240C4⋊C8—44no—
241C4⋊C8—42no—
242C4⋊C8—44no—
243C22⋊C8—42no—
244C22⋊C8—44no—
245C22⋊C8—44no—
246C22⋊C8—42no—
247C22⋊C8—81no—
248C22⋊C8—81no—
249C22⋊C8—82no—
250C22⋊C8—82no—
251#97—44no—
252C4×C8—42no—
253C4×C8—44no—
254C4×C8—42no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—21no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—81no—
268C24—81no—
269C24—81no—
270C24—81no—
271C24C2×C22⋊C412yes—
272C24C2×C22⋊C411yescharacteristic
273#41—416no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8#9914yes—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—88no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—82no—
295C2×D4—82no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4#13114yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C42—24no—
391C42—22no—
392C42—22no—
393C42—24no—
394C42—41no—
395C42—41no—
396C42—42no—
397C42—42no—
398C42—42no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—84no—
420C23—84no—
421C23—84no—
422C23—162no—
423C23—162no—
424C23—162no—
425C23—162no—
426C23#40812yes—
427C23C22×C22⋊C411yescharacteristic
428Q8—28no—
429D4—48no—
430D4—44no—
431D4—84no—
432D4—84no—
433D4—164no—
434D4—164no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—28no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—48no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—162no—
481C2×C4#40812yes—
482C2×C4C2×#9914yes—
483C2×C4C2×#9912yes—
484C2×C4#41312yes—
485C2×C4C2×#13112yes—
486C2×C4#57611yescharacteristic
487C2×C4#57611yescharacteristic
488C2×C4#57711yescharacteristic
489C2×C4#57711yescharacteristic
490C8—42no—
491C8—42no—
492C8—81no—
493C8—82no—
494C8—81no—
495C8—82no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—168no—
518C22C2×#40811yescharacteristic
519C22C22×#9911yescharacteristic
520C22C2×#41311yescharacteristic
521C22#289711yescharacteristic
522C22#289711yescharacteristic
523C22#290111yescharacteristic
524C22#290111yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—22no—
530C4—24no—
531C4—41no—
532C4—41no—
533C4—42no—
534C4—42no—
535C4—81no—
536C4—81no—
537C4—82no—
538C4—81no—
539C4—81no—
540C4#146814yes—
541C2—42no—
542C2—82no—
543C2—162no—
544C2C2×#146811yescharacteristic
545C2#1746811yescharacteristic
546C2#1748311yescharacteristic
547C2#1761111yescharacteristic
548C2#1761111yescharacteristic
549C2#1769111yescharacteristic
550C2#1777611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22532C2semidirect
3. #22530C2semidirect
4. #7714C2semidirect
5. #7698C2semidirect
6. #7674C2semidirect
7. #7658C2semidirect
8. #7598C2semidirect
9. #7572C2semidirect
10. #7572C2semidirect
11. #7564C2semidirect
37. #1797C22semidirect
38. #1797C22semidirect
39. #1749C22semidirect
42. C4×C22⋊C8C22semidirect
43. #1271C22semidirect
44. #1271C22semidirect
45. #1271C22semidirect
46. #1271C22semidirect
47. #1271C22semidirect
48. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#225322
#225302
#77142
#76982
#76742
#76582
#75982
#75722
#75642