Finite Group Databasebeta

← #164869

Subgroups — #164869

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36412C211yesmaximal, characteristic
2#21998C212yesmaximal, complemented
3#7724C212yesmaximal, complemented
4#7719C212yesmaximal, complemented
5#7684C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7645C211yesmaximal, characteristic, complemented
8#7631C211yesmaximal, characteristic, complemented
9#7601C211yesmaximal, characteristic, complemented
10#7597C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#555C411yescharacteristic
13C2×#550C411yescharacteristic
14#2858C2211yescharacteristic
15#2856C2211yescharacteristic
16#2856C2211yescharacteristic
17#2567C411yescharacteristic
18#2555—22no—
19#2555—22no—
20#2541C411yescharacteristic
21#2500C411yescharacteristic
22#2500C411yescharacteristic
23#2431C411yescharacteristic
24#2431C411yescharacteristic
25#2428—22no—
26#2363—22no—
27C4×#124C2211yescharacteristic
28#2297C2211yescharacteristic
29#2297C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1923—22no—
32#1916—22no—
33#1870—22no—
34#1851—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1817C2212yes—
38#1797C2212yes—
39C4×C22⋊C8C2212yes—
40#1728C2212yes—
41#1272C2211yescharacteristic
42#1272C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1270C2212yes—
46#1270C2212yes—
47#1270C2212yes—
48#1270C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic
53C22×M16—22no—
54#555—44no—
55#550—44no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127—21no—
58C2×#127C2×C411yescharacteristic
59C2×#127C2×C411yescharacteristic
60C2×#127C2×C411yescharacteristic
61C2×#124—21no—
62C2×#124—21no—
63C2×#124—22no—
64C2×#124C2×C411yescharacteristic
65C2×#121—21no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D412yes—
80C2×#117—21no—
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C4×C8—22no—
91#395—22no—
92#394—22no—
93#394—22no—
94#392—22no—
95#392—22no—
96#392—22no—
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2311yescharacteristic
100#390—22no—
101#390—22no—
102#390—22no—
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2311yescharacteristic
108#390C2311yescharacteristic
109#390C2311yescharacteristic
110#389—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#382C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D412yes—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C411yescharacteristic
127C4×C22⋊C4C2×C411yescharacteristic
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2312yes—
139#336C2312yes—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—41no—
143C2×#41—41no—
144C2×#41—42no—
145C2×#41—42no—
146C2×#41—81no—
147C2×#41—81no—
148C22×Q8—21no—
149C22×Q8C22×C411yescharacteristic
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C22×D4C22×C411yescharacteristic
163C22×D4C22×C411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C2×M16—44no—
172C2×M16—82no—
173C2×M16—82no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C4⋊Q8—42no—
179#127—42no—
180#127—44no—
181#127—81no—
182#127—81no—
183#124—22no—
184#124—41no—
185#124—44no—
186#124—44no—
187#124—82no—
188#124—82no—
189#121—44no—
190#121—44no—
191#121—48no—
192#121—44no—
193#121—42no—
194#121—42no—
195#121—42no—
196#121—42no—
197#121—82no—
198#121—82no—
199#121—82no—
200#121—82no—
201C4×Q8—42no—
202C4×D4—24no—
203C4×D4—42no—
204C4×D4—44no—
205C4×D4—44no—
206#117—44no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—81no—
242C2×C42—81no—
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2411yescharacteristic
252C4⋊C8—82no—
253C4⋊C8—82no—
254C22⋊C8—44no—
255C22⋊C8—82no—
256C22⋊C8—82no—
257C4×C8—82no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265C24—21no—
266C24—21no—
267C24—21no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270#41—84no—
271#41—84no—
272#41—84no—
273#41—84no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—82no—
279C2×Q8#13112yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—22no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—44no—
297C2×D4—42no—
298C2×D4—44no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4—82no—
312C2×D4—81no—
313C2×D4—81no—
314C2×D4—82no—
315C2×D4—82no—
316C2×D4—82no—
317C2×D4—82no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C412yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D412yes—
356M16—88no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—24no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4#10014yes—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—24no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—42no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C42—21no—
402C42—21no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—42no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—42no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23C22×C22⋊C411yescharacteristic
441Q8—42no—
442Q8—42no—
443Q8—162no—
444D4—44no—
445D4—81no—
446D4—81no—
447D4—81no—
448D4—81no—
449D4—81no—
450D4—81no—
451D4—84no—
452D4—84no—
453D4—84no—
454D4—164no—
455D4—162no—
456D4—164no—
457D4—162no—
458D4—162no—
459D4—162no—
460D4—162no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4C2×#10012yes—
503C2×C4C2×#10012yes—
504C2×C4C2×#10012yes—
505C2×C4#42114yes—
506C2×C4C2×#13111yescharacteristic
507C2×C4#56811yescharacteristic
508C2×C4#57612yes—
509C2×C4#57712yes—
510C8—82no—
511C8—82no—
512C8—82no—
513C8—82no—
514C22—41no—
515C22—41no—
516C22—42no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22#288011yescharacteristic
540C22#288311yescharacteristic
541C22C22×#10011yescharacteristic
542C22#289712yes—
543C22C2×#42112yes—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—24no—
551C4—22no—
552C4—42no—
553C4—42no—
554C4—41no—
555C4—42no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C2—81no—
561C2—81no—
562C2—161no—
563C2—161no—
564C2—161no—
565C2#1742111yescharacteristic
566C2#1752612yes—
567C2#1759712yes—
568C2#1766311yescharacteristic
569C2#1778211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36412C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21998C2semidirect
3. #7724C2semidirect
4. #7719C2semidirect
5. #7684C2semidirect
6. #7649C2semidirect
7. #7645C2semidirect
8. #7631C2semidirect
9. #7601C2semidirect
10. #7597C2semidirect
11. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#364122
#219982
#77242
#77192
#76842
#76492
#76452
#76312
#76012
#75972
#75632