Finite Group Databasebeta

← #481522

Subgroups — #481522

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34627C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11731C214yesmaximal, complemented
5#11728C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11694C212yesmaximal, complemented
8C2×#533C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10C2×#431—21no—
11C2×#410—21no—
12#2638C412yescomplemented
13#2638C2212yes—
14#2634—21no—
15#2633—21no—
16#2628C411yescharacteristic, complemented
17#2628C2211yescharacteristic
18#2625—21no—
19#2621C411yescharacteristic, complemented
20#2621C2211yescharacteristic
21#2616—21no—
22#2399C412yescomplemented
23#2398C2212yes—
24#2395—21no—
25#2395—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28#2283C411yescharacteristic, complemented
29C4×#117—21no—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34#1740C2212yes—
35#1461—21no—
36#1460C2212yes—
37#1459—21no—
38#1459C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1351C2218yes—
42#1344C2218yes—
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×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77#431—22no—
78#431—22no—
79#431—41no—
80#431—41no—
81C2×C4⋊C8—41no—
82#410—28no—
83C2×#99—22no—
84C2×#99—22no—
85C2×C22⋊C8—22no—
86C2×C4×C8—41no—
87#400—21no—
88#400—22no—
89#400—21no—
90#399—21no—
91#399—21no—
92#398—21no—
93#398—22no—
94#398—21no—
95#398C2×C412yes—
96#398C2×C412yes—
97#397—21no—
98#397—22no—
99#397—21no—
100#397—22no—
101#397C2×C412yes—
102#397C2312yes—
103#395—22no—
104#395C2×C412yes—
105#395C2×C412yes—
106#394—22no—
107#394—22no—
108#394—22no—
109#393C2×C412yes—
110#393C2312yes—
111#391—21no—
112#391—21no—
113#389—21no—
114#389—21no—
115#387—22no—
116#387—22no—
117#385—22no—
118#385—22no—
119#383—21no—
120#382C2×C411yescharacteristic
121#382C2×C411yescharacteristic
122#381C2×C412yes—
123#380—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—41no—
130C4×C4⋊C4—41no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136#376—41no—
137#376C2×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×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C22×C8—42no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169#125—42no—
170#125—42no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×D4—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—
199#117—28no—
200#117—41no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—42no—
207#117—42no—
208#117—44no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
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—42no—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22×C411yescharacteristic
238C2×C22⋊C4C2×D411yescharacteristic
239C2×C22⋊C4C2×D411yescharacteristic
240C2×C22⋊C4C2411yescharacteristic
241C2×C42—21no—
242C2×C42—21no—
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—22no—
254C2×C42—22no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C4⋊C8—42no—
269C4⋊C8—42no—
270#99—28no—
271#99—28no—
272C22⋊C8—24no—
273C22⋊C8—42no—
274C4×C8—42no—
275C4×C8—42no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—42no—
289#95—42no—
290#95C22×C412yes—
291#95C22×C412yes—
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C24C22×D411yescharacteristic
295#41—48no—
296#41—48no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—41no—
300C2×Q8—42no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—82no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4#10414yes—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4#13514yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C2×C8—44no—
363C2×C8—84no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—48no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C22⋊C4—28no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
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—44no—
418C42—48no—
419C42—42no—
420C42—42no—
421C42—84no—
422C42—84no—
423C42#9911yescharacteristic
424C42#9911yescharacteristic
425C42#9911yescharacteristic
426C42#9911yescharacteristic
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C2×#10412yes—
441C23C22×C22⋊C411yescharacteristic
442C23C2×#13512yes—
443Q8—44no—
444Q8—84no—
445Q8—82no—
446Q8—82no—
447D4—44no—
448D4—44no—
449D4—44no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4C2×#9911yescharacteristic
505C2×C4C2×#9911yescharacteristic
506C2×C4C2×#9911yescharacteristic
507C2×C4C2×#9911yescharacteristic
508C2×C4C2×#9911yescharacteristic
509C2×C4C2×#9911yescharacteristic
510C2×C4#41612yes—
511C2×C4#41612yes—
512C2×C4C2×#10412yes—
513C2×C4C2×#10412yes—
514C8—84no—
515C22—21no—
516C22—21no—
517C22—22no—
518C22—22no—
519C22—42no—
520C22—42no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22C22×#9911yescharacteristic
533C22C2×#41611yescharacteristic
534C22C22×#10411yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—22no—
539C4—22no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—82no—
548C4—82no—
549C4—88no—
550C4—82no—
551C4—82no—
552C4#150112yes—
553C4#150112yes—
554C4#212912yes—
555C2—21no—
556C2—21no—
557C2—42no—
558C2—42no—
559C2—81no—
560C2C2×#150111yescharacteristic
561C2#3334711yescharacteristic
562C2C2×#212911yescharacteristic
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. #11731C2semidirect
5. #11728C2semidirect
6. #11717C2semidirect
7. #11694C2semidirect
8. C2×#533C4semidirect
12. #2638C4semidirect
16. #2628C4semidirect
19. #2621C4semidirect
22. #2399C4semidirect
28. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#346272
C4×#4102
#173492
#117312
#117282
#117172
#116942