Finite Group Databasebeta

← #481634

Subgroups — #481634

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#11789C214yesmaximal, complemented
5#11788C212yesmaximal, complemented
6#11787C212yesmaximal, complemented
7#11715C214yesmaximal, complemented
8C2×#533C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10#2920—21no—
11C2×#410—21no—
12#2568C411yescharacteristic, complemented
13#2567C411yescharacteristic, complemented
14#2566—21no—
15#2565—21no—
16#2564C412yescomplemented
17#2564C2212yes—
18#2563C2211yescharacteristic
19#2542C2211yescharacteristic
20#2417—21no—
21#2417—21no—
22#2396C412yescomplemented
23#2396C2212yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26#2297—21no—
27#2285—21no—
28C4×#117—21no—
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34#1740C2212yes—
35#1467—21no—
36#1466C2212yes—
37#1465—21no—
38#1465C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1352C2218yes—
42#1351C2218yes—
43C22×#41D411yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C4×#41—41no—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C412yes—
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×C42—21no—
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#432—22no—
80#432—41no—
81#431—22no—
82#431—41no—
83#423—41no—
84#410—28no—
85C2×#99—22no—
86C2×#99—22no—
87#407—22no—
88C4×M16—41no—
89#400C2×C412yes—
90#400C2×C412yes—
91#399C2×C412yes—
92#399C2312yes—
93#398—22no—
94#398—22no—
95#396—22no—
96#396—22no—
97#395C2×C412yes—
98#395C2×C412yes—
99#393C2×C412yes—
100#393C2312yes—
101#391—21no—
102#391—21no—
103#391—21no—
104#391—21no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#387—22no—
110#387—22no—
111#386—22no—
112#386—22no—
113#385—22no—
114#385—22no—
115#384—21no—
116#384—21no—
117#382—21no—
118#382—21no—
119#380—21no—
120#380—21no—
121#380—22no—
122#380—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—41no—
128C4×C4⋊C4—41no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C412yes—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136#376—41no—
137C2×#95C2×C412yes—
138C43—41no—
139#342—22no—
140#342C2314yes—
141#324—22no—
142#324—22no—
143#323—22no—
144#323C2314yes—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—41no—
148C2×#41—41no—
149C22×Q8—21no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22×C412yes—
152C22×D4C22⋊C411yescharacteristic
153C22×D4C2×D411yescharacteristic
154C22×D4C2×D411yescharacteristic
155C23×C4—22no—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×M16—42no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170#125—42no—
171#125—42no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200#117—28no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—42no—
208#117—42no—
209#117—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C22×C411yescharacteristic
240C2×C22⋊C4C2×D411yescharacteristic
241C2×C22⋊C4C2×D411yescharacteristic
242C2×C22⋊C4C2411yescharacteristic
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42—44no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C4⋊C8—42no—
274C4⋊C8—42no—
275#99—28no—
276#99—28no—
277C22⋊C8—24no—
278C22⋊C8—42no—
279#97—42no—
280C4×C8—42no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—42no—
291#95—42no—
292#95C22×C412yes—
293#95C22×C412yes—
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24C22×D411yescharacteristic
297#41—48no—
298#41—48no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—82no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—82no—
352C22×C4—81no—
353C22×C4#10414yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4#13714yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367M16—84no—
368C2×C8—44no—
369C4⋊C4—24no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C22⋊C4—28no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—48no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—42no—
428C42—84no—
429C42—84no—
430C42#9911yescharacteristic
431C42#9911yescharacteristic
432C42#9911yescharacteristic
433C42#9911yescharacteristic
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23C2×#10412yes—
448C23C22×C22⋊C411yescharacteristic
449C23C2×#13712yes—
450Q8—44no—
451Q8—82no—
452Q8—84no—
453Q8—82no—
454D4—44no—
455D4—44no—
456D4—44no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
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—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—88no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4C2×#9911yescharacteristic
513C2×C4C2×#9911yescharacteristic
514C2×C4C2×#9911yescharacteristic
515C2×C4C2×#9911yescharacteristic
516C2×C4C2×#9911yescharacteristic
517C2×C4C2×#9911yescharacteristic
518C2×C4#41912yes—
519C2×C4#41912yes—
520C2×C4C2×#10412yes—
521C2×C4C2×#10412yes—
522C8—84no—
523C22—21no—
524C22—21no—
525C22—22no—
526C22—22no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—81no—
539C22—81no—
540C22C22×#9911yescharacteristic
541C22C2×#41911yescharacteristic
542C22C22×#10411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—44no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—82no—
557C4—88no—
558C4—82no—
559C4—82no—
560C4#150812yes—
561C4#150812yes—
562C4#212612yes—
563C2—21no—
564C2—21no—
565C2—42no—
566C2—42no—
567C2—81no—
568C2C2×#150811yescharacteristic
569C2#3334711yescharacteristic
570C2C2×#212611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #11789C2semidirect
5. #11788C2semidirect
6. #11787C2semidirect
7. #11715C2semidirect
8. C2×#533C4semidirect
12. #2568C4semidirect
13. #2567C4semidirect
16. #2564C4semidirect
22. #2396C4semidirect
29. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#349472
C4×#4102
#173612
#117892
#117882
#117872
#117152