Finite Group Databasebeta

← #482052

Subgroups — #482052

Order 512.

1070 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#35316C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11878C214yesmaximal, complemented
5#11874C214yesmaximal, complemented
6#11694C214yesmaximal, complemented
7C2×#536—21no—
8C22×C4⋊Q8C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10C2×#431—21no—
11C2×#410—21no—
12#2857—21no—
13#2857—21no—
14#2621C412yescomplemented
15#2621C2212yes—
16#2547—21no—
17#2547C2212yes—
18#2546—21no—
19#2546C412yescomplemented
20#2486—21no—
21#2486—21no—
22#2486C412yescomplemented
23#2486C2212yes—
24#2310—21no—
25#2285—21no—
26#2285C411yescharacteristic, complemented
27#1799—21no—
28#1799—21no—
29#1799C2212yes—
30#1799C2212yes—
31#1740C2212yes—
32#1461—21no—
33#1460C2212yes—
34#1459—21no—
35#1459C2212yes—
36#1458—21no—
37#1458C2212yes—
38#1344C22116yes—
39C22×#41D411yescharacteristic
40C23×Q8—21no—
41#536—44no—
42#536—44no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C412yes—
52C2×#124—22no—
53C2×#123—22no—
54C2×#123—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C4×#41—41no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—22no—
71C2×#117C2×C411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2×C412yes—
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#431—22no—
80#431—22no—
81#431—41no—
82#431—41no—
83C2×C4⋊C8—41no—
84#410—44no—
85C2×#99—22no—
86C2×#99—22no—
87C2×C22⋊C8—22no—
88C2×C4×C8—41no—
89#398—22no—
90#398—22no—
91#398C2×C414yes—
92#398C2×C414yes—
93#397—22no—
94#397—22no—
95#397C2×C414yes—
96#397C2314yes—
97#391—22no—
98#391—22no—
99#391—22no—
100#391—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#384—21no—
106#384—41no—
107#382—41no—
108#382C2×C412yes—
109#381—22no—
110#381—22no—
111#380—22no—
112#380—21no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—41no—
121C4×C4⋊C4—41no—
122C4×C4⋊C4C2×C412yes—
123#342—22no—
124#342C2314yes—
125#324—22no—
126#324—22no—
127#323—22no—
128#323C2314yes—
129C2×#41—41no—
130C2×#41—41no—
131C2×#41—42no—
132C2×#41D814yes—
133C2×#41SD1614yes—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—44no—
139C22×Q8—44no—
140C22×Q8C22⋊C411yescharacteristic
141C22×D4C22⋊C411yescharacteristic
142C22×D4C2×D411yescharacteristic
143C22×D4C2×D411yescharacteristic
144C23×C4—22no—
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C22×C8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—48no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166#124—44no—
167#124—44no—
168#123—48no—
169#123—44no—
170#123—44no—
171C22⋊Q8—48no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190#117—41no—
191#117—41no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—44no—
197#117—44no—
198#117—44no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4C22⋊C411yescharacteristic
226C2×C22⋊C4C22⋊C411yescharacteristic
227C2×C22⋊C4C22×C411yescharacteristic
228C2×C22⋊C4C2×D411yescharacteristic
229C2×C22⋊C4C2×D411yescharacteristic
230C2×C22⋊C4C2411yescharacteristic
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C4⋊C8—42no—
256C4⋊C8—42no—
257#99—44no—
258#99—44no—
259C22⋊C8—24no—
260C22⋊C8—42no—
261C4×C8—42no—
262C4×C8—42no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—42no—
269#95—42no—
270#95C22×C414yes—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C24C22×D411yescharacteristic
274#41—44no—
275#41—44no—
276#41—44no—
277#41—44no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—88no—
288C2×Q8—88no—
289C2×Q8#10214yes—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4#10214yes—
297C2×D4C2×D812yes—
298C2×D4C2×D812yes—
299C2×D4C2×SD1612yes—
300C2×D4C2×SD1612yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4#10214yes—
335C22×C4#10214yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×D812yes—
347C22×C4C2×SD1612yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C2×C8—44no—
351C2×C8—84no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—44no—
403C42—84no—
404C42#9911yescharacteristic
405C42#9911yescharacteristic
406C42#9911yescharacteristic
407C42#9911yescharacteristic
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23C2×#10212yes—
421C23C2×#10212yes—
422C23C22×C22⋊C411yescharacteristic
423C23C22×D811yescharacteristic
424C23C22×SD1611yescharacteristic
425Q8—44no—
426Q8—88no—
427Q8—84no—
428Q8—84no—
429D4—44no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—42no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
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—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—88no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4C2×#9911yescharacteristic
485C2×C4C2×#9911yescharacteristic
486C2×C4C2×#9911yescharacteristic
487C2×C4C2×#9911yescharacteristic
488C2×C4C2×#9911yescharacteristic
489C2×C4C2×#9911yescharacteristic
490C2×C4C2×#10212yes—
491C2×C4C2×#10212yes—
492C2×C4C2×#10212yes—
493C2×C4C2×#10212yes—
494C2×C4#42112yes—
495C8—84no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—81no—
514C22—81no—
515C22C22×#9911yescharacteristic
516C22C22×#10211yescharacteristic
517C22C2×#42111yescharacteristic
518C4—21no—
519C4—21no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—84no—
530C4—84no—
531C4—164no—
532C4#151912yes—
533C4#151912yes—
534C4#212312yes—
535C4#212912yes—
536C2—21no—
537C2—21no—
538C2—41no—
539C2—42no—
540C2—41no—
541C2—42no—
542C2C2×#151911yescharacteristic
543C2C2×#212311yescharacteristic
544C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11878C2semidirect
5. #11874C2semidirect
6. #11694C2semidirect
8. C22×C4⋊Q8C4semidirect
14. #2621C4semidirect
19. #2546C4semidirect
22. #2486C4semidirect
26. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#353162
#206322
#173492
#118782
#118742
#116942