Finite Group Databasebeta

← #164913

Subgroups — #164913

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36396C211yesmaximal, characteristic
2#22785C212yesmaximal, complemented
3#7670C212yesmaximal, complemented
4#7664C212yesmaximal, complemented
5#7656C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7645C211yesmaximal, characteristic, complemented
8#7643C211yesmaximal, characteristic, complemented
9#7567C211yesmaximal, characteristic, complemented
10#7563C211yesmaximal, characteristic, complemented
11C2×#1272C211yesmaximal, characteristic, complemented
12C2×#553C411yescharacteristic
13C2×#548C411yescharacteristic
14#2861C2211yescharacteristic
15#2856C2211yescharacteristic
16#2850C2211yescharacteristic
17#2565C411yescharacteristic
18#2560—22no—
19#2557—22no—
20#2544C411yescharacteristic
21#2543—22no—
22#2504C411yescharacteristic
23#2499C411yescharacteristic
24#2436C411yescharacteristic
25#2431C411yescharacteristic
26#2364—22no—
27C4×#124C2211yescharacteristic
28#2297C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1923—22no—
32#1916—22no—
33#1869—22no—
34#1839C2212yes—
35#1817C2212yes—
36#1797C2212yes—
37C8×C22⋊C4—22no—
38C4×C22⋊C8C2212yes—
39#1744C2212yes—
40C2×#336C2212yes—
41#1272C2211yescharacteristic
42#1272C2211yescharacteristic
43#1272C2212yes—
44#1272C2211yescharacteristic
45#1272C2211yescharacteristic
46#1272C2212yes—
47#1270C2212yes—
48#1270C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51C22×#41C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53C23×C8—22no—
54#553—44no—
55#548—44no—
56C2×#127C2×C411yescharacteristic
57C2×#126—21no—
58C2×#125C2×C411yescharacteristic
59C2×#124—21no—
60C2×#124C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#123—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—22no—
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4D412yes—
78C2×#117C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88#398—22no—
89#397—22no—
90#397—22no—
91#394—22no—
92#393—22no—
93#392—22no—
94#391—22no—
95#391C2×C411yescharacteristic
96#391C2×C411yescharacteristic
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2311yescharacteristic
100#391C2311yescharacteristic
101#390—22no—
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2311yescharacteristic
105#390C2311yescharacteristic
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#384C2×C411yescharacteristic
119#382C2×C411yescharacteristic
120#382C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D412yes—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4C2×C411yescharacteristic
126C4×C22⋊C4C2×C411yescharacteristic
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129C2×#95—22no—
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—41no—
142C2×#41—42no—
143C2×#41—81no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—41no—
147C22×Q8—42no—
148C22×Q8—81no—
149C22×Q8C22×C411yescharacteristic
150C22×Q8C22×C411yescharacteristic
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4C22⋊C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—44no—
172C22×C8—44no—
173C22×C8—82no—
174#127—81no—
175#126—44no—
176#125—42no—
177#124—22no—
178#124—41no—
179#124—42no—
180#124—44no—
181#124—81no—
182#123—44no—
183C22⋊Q8—44no—
184C22⋊Q8—42no—
185#121—42no—
186#121—42no—
187#121—44no—
188#121—44no—
189#121—44no—
190#121—44no—
191#121—42no—
192C4×Q8—42no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—42no—
197C4×D4—44no—
198C4×D4—44no—
199#117—42no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4C22⋊C411yescharacteristic
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—81no—
234C2×C42—81no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2411yescharacteristic
244C4⋊C8—82no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—82no—
248C4×C8—44no—
249C4×C8—82no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261C24—21no—
262C24—21no—
263C24C2×C22⋊C411yescharacteristic
264C24C2×C22⋊C411yescharacteristic
265#41—84no—
266#41—84no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—81no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8#13112yes—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—42no—
297C2×D4—44no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—81no—
301C2×D4—82no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—81no—
329C22×C4—82no—
330C22×C4—81no—
331C22×C4—82no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4#13514yes—
340C22×C4#13514yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D412yes—
343C2×C8—24no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—88no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—44no—
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—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4#10014yes—
388C42—21no—
389C42—21no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—42no—
400C42—42no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23C2×#10012yes—
420C23#41614yes—
421C23C22×C22⋊C411yescharacteristic
422Q8—42no—
423Q8—42no—
424Q8—81no—
425Q8—81no—
426Q8—84no—
427D4—44no—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—84no—
433D4—84no—
434D4—84no—
435D4—164no—
436D4—162no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
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—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—162no—
482C2×C4—162no—
483C2×C4C2×#10012yes—
484C2×C4C2×#10012yes—
485C2×C4#41614yes—
486C2×C4C2×#13111yescharacteristic
487C2×C4#56811yescharacteristic
488C2×C4C2×#13512yes—
489C2×C4C2×#13512yes—
490C8—44no—
491C8—82no—
492C8—82no—
493C8—82no—
494C22—22no—
495C22—22no—
496C22—24no—
497C22—24no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—42no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22#154314yes—
514C22#288011yescharacteristic
515C22#288311yescharacteristic
516C22C22×#10011yescharacteristic
517C22C2×#41612yes—
518C22#290112yes—
519C4—21no—
520C4—21no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—24no—
526C4—41no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—81no—
536C2—24no—
537C2—81no—
538C2—161no—
539C2—161no—
540C2#1742111yescharacteristic
541C2#1751012yes—
542C2#1761512yes—
543C2#1768511yescharacteristic
544C2C2×#154311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36396C2
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. #22785C2semidirect
3. #7670C2semidirect
4. #7664C2semidirect
5. #7656C2semidirect
6. #7649C2semidirect
7. #7645C2semidirect
8. #7643C2semidirect
9. #7567C2semidirect
10. #7563C2semidirect
11. C2×#1272C2semidirect

Maximal subgroups

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

subgroupindex
#363962
#227852
#76702
#76642
#76562
#76492
#76452
#76432
#75672
#75632
C2×#12722