Finite Group Databasebeta

← #480393

Subgroups — #480393

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#532C211yesmaximal, characteristic
2#17366C212yesmaximal, complemented
3#11531C212yesmaximal, complemented
4#11529C212yesmaximal, complemented
5#11513C212yesmaximal, complemented
6#11507C212yesmaximal, complemented
7#11478C212yesmaximal, complemented
8#11476C212yesmaximal, complemented
9C2×#532C411yescharacteristic
10C2×C4×#41C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2920—22no—
13C4×C4⋊Q8C411yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×#127C411yescharacteristic
16C4×#127C2211yescharacteristic
17C4×#124C411yescharacteristic
18C4×#124C411yescharacteristic
19C4×#124C2211yescharacteristic
20C4×#124C2211yescharacteristic
21C4×#121—22no—
22C4×#121—22no—
23#2297C411yescharacteristic
24#2297C2211yescharacteristic
25#2293—22no—
26#2285C2211yescharacteristic
27C42×D4—22no—
28C2×C43C411yescharacteristic
29#1465—22no—
30#1465C2214yes—
31#1464—22no—
32#1464C2214yes—
33#1461C2214yes—
34#1460—22no—
35#1325C2214yes—
36#1324C2214yes—
37#1319C2214yes—
38#1318C2214yes—
39C22×#41C2×C411yescharacteristic
40C22×#41C2×C411yescharacteristic
41#532—24no—
42#532—24no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C411yescharacteristic
47C2×#121—22no—
48C4×#41—41no—
49C4×#41—41no—
50C4×#41—41no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D412yes—
69C2×C4×D4D412yes—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117C2×C411yescharacteristic
73C2×#117D412yes—
74C2×#117C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D412yes—
82#432—24no—
83#432—42no—
84#431—24no—
85#431—42no—
86#423—42no—
87#407—24no—
88C4×M16—42no—
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#391C2×C411yescharacteristic
92#391C2311yescharacteristic
93#390—22no—
94#390—22no—
95#390C2×C411yescharacteristic
96#390C2×C411yescharacteristic
97#390C2×C411yescharacteristic
98#390C2311yescharacteristic
99#389—22no—
100#389—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#386—22no—
106#386—22no—
107#384C2×C411yescharacteristic
108#384C2311yescharacteristic
109#382C2×C411yescharacteristic
110#382C2311yescharacteristic
111#380—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C43—21no—
122C43—21no—
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125C43C2×C411yescharacteristic
126C43C2×C411yescharacteristic
127#324—24no—
128#324C2318yes—
129#323—24no—
130#323—24no—
131C2×#41—22no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—41no—
136C2×#41—41no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—41no—
140C2×#41—41no—
141C22×Q8C22×C411yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×D4—22no—
144C22×D4—21no—
145C22×D4C22⋊C412yes—
146C22×D4C22×C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D412yes—
159C2×M16—44no—
160C4⋊Q8—22no—
161C4⋊Q8M1614yescomplemented
162#127—22no—
163#127—22no—
164#124—22no—
165#124—22no—
166#124—22no—
167#124M1614yescomplemented
168#121—216no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×D4—216no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—48no—
187C4×D4—48no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196#117—42no—
197#117—42no—
198#117—41no—
199#117—41no—
200#117—42no—
201#117—42no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2411yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C2×D412yes—
226C2×C22⋊C4C2×D412yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
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—44no—
251C2×C42—44no—
252C2×C42—42no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C412yes—
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C4⋊C8—44no—
269C4⋊C8—44no—
270C22⋊C8—28no—
271C22⋊C8—44no—
272#97—44no—
273C4×C8—44no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280C24—21no—
281C24C2×C22⋊C412yes—
282C24C2×C22⋊C411yescharacteristic
283#41—44no—
284#41—44no—
285#41—44no—
286#41—44no—
287#41—44no—
288#41—44no—
289#41—44no—
290#41—44no—
291C2×Q8—21no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8C2×M1612yes—
297C2×Q8C2×M1612yes—
298C2×Q8C2×M1612yes—
299C2×D4—28no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—24no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4C2×M1612yes—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D412yes—
369M16—88no—
370C2×C8—48no—
371C4⋊C4—28no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C22⋊C4—28no—
383C22⋊C4—28no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—42no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—21no—
398C42—21no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—21no—
403C42—21no—
404C42—22no—
405C42—22no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—22no—
410C42—21no—
411C42—41no—
412C42—41no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42#9911yescharacteristic
427C42#9911yescharacteristic
428C42#9911yescharacteristic
429C42#9911yescharacteristic
430C42C2×M1612yes—
431C42C2×M1612yes—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23#41312yes—
447C23C22×C22⋊C411yescharacteristic
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452D4—48no—
453D4—42no—
454D4—416no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—48no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—48no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4#40714yes—
517C2×C4C2×#9911yescharacteristic
518C2×C4C2×#9911yescharacteristic
519C2×C4C2×#9911yescharacteristic
520C2×C4C2×#9911yescharacteristic
521C2×C4C2×#9911yescharacteristic
522C2×C4C2×#9911yescharacteristic
523C2×C4C22×M1611yescharacteristic
524C2×C4C22×M1611yescharacteristic
525C8—88no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—44no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—82no—
540C22—82no—
541C22—81no—
542C22—81no—
543C22C2×#40711yescharacteristic
544C22C22×#9911yescharacteristic
545C22C2×#41311yescharacteristic
546C4—24no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—24no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—81no—
564C4#212612yes—
565C4#213212yes—
566C2—21no—
567C2—21no—
568C2—42no—
569C2—82no—
570C2—82no—
571C2#1736211yescharacteristic
572C2C2×#212611yescharacteristic
573C2C2×#213211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17366C2semidirect
3. #11531C2semidirect
4. #11529C2semidirect
5. #11513C2semidirect
6. #11507C2semidirect
7. #11478C2semidirect
8. #11476C2semidirect
161. C4⋊Q8M16semidirect
167. #124M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5322
#173662
#115312
#115292
#115132
#115072
#114782
#114762