Finite Group Databasebeta

← #493968

Subgroups — #493968

Order 512.

1215 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2#20724C212yesmaximal, complemented
3#12856C212yesmaximal, complemented
4#12799C212yesmaximal, complemented
5#12798C212yesmaximal, complemented
6#12696C212yesmaximal
7#12627C212yesmaximal, complemented
8C4×#353C212yesmaximal, complemented
9C2×#535C2211yescharacteristic
10C2×#520C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2363—22no—
13#2361—22no—
14#2358—22no—
15#2358—22no—
16#2328C411yescharacteristic, complemented
17#2328C2211yescharacteristic
18#2326C411yescharacteristic, complemented
19#2326C2211yescharacteristic
20#2317C411yescharacteristic, complemented
21#2312—22no—
22C4×#127C411yescharacteristic, complemented
23C4×#127C2211yescharacteristic
24C4×#124C411yescharacteristic, complemented
25C4×#124C2211yescharacteristic
26#2297C411yescharacteristic, complemented
27C2×#390C2211yescharacteristic
28C4×#117C411yescharacteristic, complemented
29C2×#354C2214yes—
30C2×#353C2214yes—
31#1819—22no—
32#1799—22no—
33#1799C2214yes—
34#1781C2214yes—
35C4×#99—22no—
36C4×#99C2214yes—
37#1393C2214yes—
38#1393C2214yes—
39C22×#41C2×C411yescharacteristic
40C23×D4C2311yescharacteristic
41#535—24no—
42#535—24no—
43C2×#127C2×C411yescharacteristic
44C2×#127C2311yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2311yescharacteristic
47C2×#121—22no—
48C2×#120—22no—
49#520—44no—
50#518—44no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×C4×D4C2311yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×#117—21no—
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C22⋊C4D412yes—
74C22×C42C2×C411yescharacteristic
75C2×#99—24no—
76C2×#99C2318yes—
77#391C2×C411yescharacteristic
78#391C2×C411yescharacteristic
79#390—22no—
80#390—22no—
81#390—22no—
82#390C2×C411yescharacteristic
83#390C2×C411yescharacteristic
84#390C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#390C2311yescharacteristic
87#390C2311yescharacteristic
88#389—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—48no—
94#386—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#384C2×C411yescharacteristic
103#384C2×C411yescharacteristic
104#382C2×C411yescharacteristic
105#382C2×C411yescharacteristic
106#380—24no—
107#379—24no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—24no—
116#376C2×C411yescharacteristic
117#376C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120#354—28no—
121#353—28no—
122#342—24no—
123#342—24no—
124C25C2×D412yes—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—42no—
128C2×#41—42no—
129C22×Q8C22×C411yescharacteristic
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4C22×C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C22×D4C2×D412yes—
140C22×D4C2411yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151#127—22no—
152#127—22no—
153#127—22no—
154#127#4114yes—
155#124—22no—
156#124—22no—
157#124—22no—
158#124#4114yes—
159#121—216no—
160#120—216no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—44no—
170C4×D4—44no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—84no—
174C4×D4—84no—
175#117—24no—
176#117—22no—
177#117—22no—
178#117—42no—
179#117—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—84no—
212C2×C22⋊C4C22⋊C411yescharacteristic
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C411yescharacteristic
217C2×C22⋊C4C2×D412yes—
218C2×C22⋊C4C2×D412yes—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—42no—
227C2×C42—44no—
228C2×C42C22⋊C412yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235#99—28no—
236#99—28no—
237#99—48no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—44no—
247#95—44no—
248C24—21no—
249C24—24no—
250C24—24no—
251C24—42no—
252C24—82no—
253C24—82no—
254C24C2×C22⋊C412yes—
255C24C2×C22⋊C411yescharacteristic
256C24C22×D412yes—
257#41—48no—
258#41—48no—
259C2×Q8—21no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8#11712yes—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—28no—
266C2×D4—28no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—24no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4#11712yes—
287C2×D4C2×#4112yes—
288C2×D4C2×#4112yes—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—48no—
313C22×C4—81no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C23×C411yescharacteristic
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—164no—
342C22⋊C4—24no—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—24no—
346C22⋊C4—22no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—44no—
355C22⋊C4—42no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—164no—
362C22⋊C4—164no—
363C22⋊C4#9914yes—
364C42—21no—
365C42—21no—
366C42—22no—
367C42—22no—
368C42—42no—
369C42—41no—
370C42—41no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—81no—
377C42—81no—
378C42—164no—
379C42#11712yes—
380C42#11712yes—
381C23—22no—
382C23—21no—
383C23—21no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—41no—
390C23—41no—
391C23—48no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—84no—
401C23—84no—
402C23—88no—
403C23—164no—
404C23C2×#9912yes—
405C23C22×C22⋊C411yescharacteristic
406C23#52514yes—
407Q8—42no—
408Q8—42no—
409D4—42no—
410D4—42no—
411D4—48no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—48no—
417D4—88no—
418D4—88no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—48no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—88no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4—162no—
462C2×C4—162no—
463C2×C4—162no—
464C2×C4—162no—
465C2×C4—162no—
466C2×C4C2×#9912yes—
467C2×C4C2×#9912yes—
468C2×C4#41012yes—
469C2×C4C2×#11711yescharacteristic
470C2×C4C2×#11711yescharacteristic
471C22—21no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—24no—
477C22—44no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—81no—
492C22—81no—
493C22—168no—
494C22#228011yescharacteristic
495C22C22×#9911yescharacteristic
496C22C2×#41011yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—21no—
501C4—21no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—41no—
509C4—42no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—168no—
514C2—21no—
515C2—21no—
516C2—44no—
517C2—42no—
518C2—82no—
519C2—82no—
520C2#2072211yescharacteristic
521C2#3335911yescharacteristic
522C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20724C2semidirect
3. #12856C2semidirect
4. #12799C2semidirect
5. #12798C2semidirect
7. #12627C2semidirect
8. C4×#353C2semidirect
10. C2×#520C4semidirect
16. #2328C4semidirect
18. #2326C4semidirect
20. #2317C4semidirect
22. C4×#127C4semidirect
24. C4×#124C4semidirect
26. #2297C4semidirect
28. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
#342752
#207242
#128562
#127992
#127982
#126962
#126272
C4×#3532