Finite Group Databasebeta

← #164867

Subgroups — #164867

Order 512.

913 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36376C211yesmaximal, characteristic
2#21998C212yesmaximal, complemented
3#7730C212yesmaximal, complemented
4#7716C212yesmaximal, complemented
5#7687C212yesmaximal, complemented
6#7653C211yesmaximal, characteristic, complemented
7#7643C211yesmaximal, characteristic, complemented
8#7631C211yesmaximal, characteristic, complemented
9#7605C211yesmaximal, characteristic, complemented
10#7595C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#552C411yescharacteristic
13C2×#548C411yescharacteristic
14#2861C2211yescharacteristic
15#2857C2211yescharacteristic
16#2850C2211yescharacteristic
17#2565C411yescharacteristic
18#2556—22no—
19#2545C411yescharacteristic
20#2521—22no—
21#2501C411yescharacteristic
22#2499C411yescharacteristic
23#2434C411yescharacteristic
24#2432C411yescharacteristic
25#2425—22no—
26#2359—22no—
27C4×C4⋊Q8C2211yescharacteristic
28#2297C2211yescharacteristic
29C2×#391C2211yescharacteristic
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#1272C2212yes—
42#1272C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2211yescharacteristic
47#1272C2211yescharacteristic
48#1272C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51C22×#41C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53C22×M16—22no—
54#552—44no—
55#548—44no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#126—21no—
59C2×#125—21no—
60C2×#125C2×C411yescharacteristic
61C2×#124—21no—
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#123—21no—
65C2×#123—22no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×#121—21no—
71C2×#121C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4D412yes—
77C2×#117—21no—
78C2×#117C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C4×C8—22no—
88#399—22no—
89#398—22no—
90#397—22no—
91#396—22no—
92#395—22no—
93#394—22no—
94#391—22no—
95#391C2×C411yescharacteristic
96#391C2×C411yescharacteristic
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2311yescharacteristic
100#391C2311yescharacteristic
101#391C2311yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2311yescharacteristic
105#389—22no—
106#389—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—22no—
113#386—22no—
114#386—22no—
115#385—22no—
116#385—22no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#382C2×C411yescharacteristic
120#380—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D412yes—
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×Q8C22×C411yescharacteristic
152C22×D4—21no—
153C22×D4—21no—
154C22×D4C22⋊C411yescharacteristic
155C22×D4C22×C411yescharacteristic
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C2×M16—44no—
164C2×M16—82no—
165C2×M16—82no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—41no—
172C4⋊Q8—81no—
173#126—44no—
174#125—42no—
175#125—44no—
176#124—42no—
177#124—44no—
178#124—81no—
179#123—48no—
180#123—44no—
181C22⋊Q8—44no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
185#121—42no—
186#121—44no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×D4—24no—
190C4×D4—44no—
191C4×D4—44no—
192#117—44no—
193#117—42no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
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—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4C22⋊C411yescharacteristic
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C411yescharacteristic
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—81no—
232C2×C42—81no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—82no—
243C4⋊C8—82no—
244C22⋊C8—44no—
245C22⋊C8—82no—
246C22⋊C8—82no—
247C4×C8—82no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259C24—21no—
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C411yescharacteristic
262#41—84no—
263#41—84no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8#13112yes—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—42no—
291C2×D4—81no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D412yes—
332M16—88no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—44no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—82no—
363C22⋊C4—22no—
364C22⋊C4—24no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—44no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C22⋊C4#10014yes—
379C42—21no—
380C42—21no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—42no—
385C42—42no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C23—21no—
394C23—21no—
395C23—22no—
396C23—22no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23C2×#10012yes—
408C23C22×C22⋊C411yescharacteristic
409Q8—42no—
410Q8—42no—
411Q8—81no—
412Q8—81no—
413Q8—81no—
414Q8—81no—
415Q8—84no—
416D4—44no—
417D4—81no—
418D4—81no—
419D4—84no—
420D4—84no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4C2×#10012yes—
470C2×C4C2×#10012yes—
471C2×C4#42114yes—
472C2×C4C2×#13111yescharacteristic
473C2×C4#56811yescharacteristic
474C2×C4#57712yes—
475C2×C4#57812yes—
476C8—82no—
477C8—82no—
478C8—82no—
479C8—82no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22—41no—
486C22—41no—
487C22—42no—
488C22—42no—
489C22—41no—
490C22—81no—
491C22—81no—
492C22—82no—
493C22—82no—
494C22—81no—
495C22#288011yescharacteristic
496C22#288311yescharacteristic
497C22C22×#10011yescharacteristic
498C22#289812yes—
499C22C2×#42112yes—
500C4—21no—
501C4—21no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—24no—
507C4—41no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—81no—
518C2—42no—
519C2—81no—
520C2—161no—
521C2#1742111yescharacteristic
522C2#1752512yes—
523C2#1759812yes—
524C2#1766311yescharacteristic
525C2#1777911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#548C4
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. #7730C2semidirect
4. #7716C2semidirect
5. #7687C2semidirect
6. #7653C2semidirect
7. #7643C2semidirect
8. #7631C2semidirect
9. #7605C2semidirect
10. #7595C2semidirect
11. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#363762
#219982
#77302
#77162
#76872
#76532
#76432
#76312
#76052
#75952
#75632