Finite Group Databasebeta

← #361851

Subgroups — #361851

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#1826C214yesmaximal
3C4×#407C211yesmaximal, characteristic
4C4×#380C211yesmaximal, characteristic
5#11359C214yesmaximal, complemented
6#10851C214yesmaximal, complemented
7C22×#117C2212yes—
8C23×C42C2211yescharacteristic
9C2×#407C2212yes—
10C2×C4×M16C2212yes—
11#2285—21no—
12#2285—21no—
13#2285—21no—
14#2285—21no—
15#2285C412yescomplemented
16#2285C412yescomplemented
17#2285C2212yes—
18#2285C2212yes—
19C2×#384C411yescharacteristic, complemented
20C2×#384C411yescharacteristic, complemented
21C2×#382C412yescomplemented
22C2×#380—21no—
23C2×#380—21no—
24C2×#380—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1826—216no—
30#1749—21no—
31#1749—21no—
32C4×C22⋊C8—21no—
33C4×C22⋊C8—21no—
34C2×#339C22116yes—
35#1728C2212yes—
36C4⋊#95—21no—
37#1434—21no—
38C22⋊#95—21no—
39C4×#95C2212yes—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16C2314yes—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—28no—
47C2×#117—28no—
48C2×#117C2×C414yes—
49C2×#117C2314yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C42—28no—
60C22×C42—22no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2314yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#407—24no—
75#407D418yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C4×M16—28no—
79C2×#97—22no—
80C2×C4×C8—22no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384C2×C412yes—
86#384C2×C412yes—
87#384C2×C412yes—
88#384C2×C412yes—
89#382—22no—
90#382—22no—
91#382C2×C414yes—
92#382C2×C414yes—
93#380—28no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—28no—
111C4×C22⋊C4—28no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118#339—232no—
119#336—22no—
120#336—22no—
121C25C2×D411yescharacteristic
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D414yes—
150C23×C4C2×D411yescharacteristic
151C23×C4#4114yes—
152C23×C4C2411yescharacteristic
153C2×M16—24no—
154C2×M16—28no—
155C2×M16—28no—
156C2×M16C2×D418yes—
157C22×C8—24no—
158#117—216no—
159#117—28no—
160#117—28no—
161#117—48no—
162#117—44no—
163#117—44no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4C22×C414yes—
184C2×C4⋊C4C22×C414yes—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—28no—
206C2×C42—28no—
207C2×C42—28no—
208C2×C42—48no—
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4118yes—
226C22⋊C8—28no—
227C22⋊C8—28no—
228#97—44no—
229C4×C8—44no—
230#95—28no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—44no—
236#95—44no—
237#95—44no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24C2×C22⋊C411yescharacteristic
245C24C2×C22⋊C411yescharacteristic
246C24#12412yes—
247C24#12712yes—
248C24C22×D411yescharacteristic
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—216no—
261C22×C4—216no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—416no—
298C22×C4—416no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4#10414yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C4×D414yes—
323C22×C4C4×D414yes—
324C22×C4#12114yes—
325C22×C4#12114yes—
326C22×C4#12314yes—
327C22×C4#12314yes—
328C22×C4#12412yes—
329C22×C4#12712yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4114yes—
334C22×C4C2×#4112yes—
335C22×C42−1+414yes—
336M16—216no—
337M16—48no—
338C2×C8—28no—
339C2×C8—44no—
340C2×C8—44no—
341C4⋊C4—28no—
342C4⋊C4—28no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—416no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C22⋊C4—416no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—48no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—48no—
371C42—48no—
372C42—416no—
373C42#11714yes—
374C42#11714yes—
375C42#12314yes—
376C42#12314yes—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—21no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23#39012yes—
402C23#39012yes—
403C23C2×#10412yes—
404C23#42112yes—
405C23C22×C22⋊C411yescharacteristic
406C23C2×#12312yes—
407C23C2×#12411yescharacteristic
408C23C2×#12711yescharacteristic
409C23#53612yes—
410C2×C4—28no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—88no—
461C2×C4—88no—
462C2×C4—84no—
463C2×C4#38112yes—
464C2×C4#38112yes—
465C2×C4#38112yes—
466C2×C4#38112yes—
467C2×C4#39012yes—
468C2×C4#39012yes—
469C2×C4C2×#10412yes—
470C2×C4C2×#10412yes—
471C2×C4#42112yes—
472C2×C4#42112yes—
473C2×C4C2×#11712yes—
474C2×C4C2×C4×D412yes—
475C2×C4#51912yes—
476C2×C4C2×#12112yes—
477C2×C4C2×#12312yes—
478C2×C4#53612yes—
479C2×C4#54718yes—
480C2×C4#55218yes—
481C8—48no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—48no—
500C22—48no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22#198014yes—
506C22C2×#38111yescharacteristic
507C22C2×#39011yescharacteristic
508C22#231011yescharacteristic
509C22#250112yes—
510C22#250312yes—
511C22C22×#10411yescharacteristic
512C22C2×#42111yescharacteristic
513C22C2×#42111yescharacteristic
514C22C2×#42111yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—88no—
525C4—84no—
526C4—84no—
527C4#182614yescomplemented
528C4#234414yes—
529C4#248614yes—
530C2—22no—
531C2—24no—
532C2—22no—
533C2—42no—
534C2#1390411yescharacteristic
535C2C2×#182611yescharacteristic
536C2#2070511yescharacteristic
537C2C2×#198011yescharacteristic
538C2#2125211yescharacteristic
539C2#2184411yescharacteristic
540C2#2184511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11359C2semidirect
6. #10851C2semidirect
15. #2285C4semidirect
16. #2285C4semidirect
19. C2×#384C4semidirect
20. C2×#384C4semidirect
21. C2×#382C4semidirect
527. C4#1826semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#18262
C4×#4072
C4×#3802
#113592
#108512