Finite Group Databasebeta

← #160441

Subgroups — #160441

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3C4×#431C211yesmaximal, characteristic, complemented
4#7574C214yesmaximal, complemented
5#7573C212yesmaximal, complemented
6#7572C212yesmaximal
7#7560C214yesmaximal, complemented
8C22×C4×D4C411yescharacteristic
9C22×#117—21no—
10C2×#431—21no—
11C2×#431—21no—
12#2293—21no—
13#2293—21no—
14#2293—21no—
15#2293—21no—
16#2293C412yes—
17#2293C412yes—
18#2293C2212yes—
19#2293C2212yes—
20C2×#390C411yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C411yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388—21no—
25C2×#388—21no—
26C2×C4×C4⋊C4C411yescharacteristic
27C2×C4×C22⋊C4—21no—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831C2212yescomplemented
31#1797—21no—
32#1797C2212yes—
33C4×C4⋊C8—21no—
34C4×C4⋊C8—21no—
35C4×C4⋊C8C2212yescomplemented
36C4×C4⋊C8C2212yescomplemented
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39#1744C2212yescomplemented
40C42×C8—21no—
41#1271C2218yescomplemented
42#1270C2218yes—
43C23×D4C2×C411yescharacteristic
44C24×C4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—24no—
48C2×C4×D4C818yescomplemented
49C2×C4×D4C2×C412yes—
50C2×C4×D4C2×C412yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—44no—
56C2×#117—44no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75#431—28no—
76#431—44no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390C2×C412yes—
92#390C2×C412yes—
93#390C2×C412yes—
94#390C2312yes—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2312yes—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43C2311yescharacteristic
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336C2314yes—
133#336C2314yes—
134C25—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4C2×C814yes—
140C22×D4C22×C412yes—
141C23×C4—21no—
142C23×C4—28no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—24no—
157C22×C8—24no—
158C4×D4—28no—
159C4×D4—28no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4C2×C818yes—
163C4×D4C2×C818yes—
164#117—48no—
165#117—48no—
166#117—44no—
167#117—44no—
168#117—44no—
169#117—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C2×C814yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—44no—
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×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C2×C814yes—
224C2×C42M1614yes—
225C2×C42D814yes—
226C2×C42SD1614yes—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2411yescharacteristic
243C4⋊C8—28no—
244C4⋊C8—44no—
245C4⋊C8—44no—
246C4⋊C8—44no—
247C22⋊C8—28no—
248C22⋊C8—44no—
249C4×C8—28no—
250C4×C8—44no—
251C4×C8—44no—
252C4×C8—44no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259C24—22no—
260C24—24no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24C2×C22⋊C411yescharacteristic
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4C22×C814yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
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—42no—
298C22×C4—42no—
299C22×C4—432no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4C22⋊C814yes—
311C22×C4C22⋊C814yes—
312C22×C4#10214yes—
313C22×C4#10214yes—
314C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C22×C812yes—
327C22×C4C2×M1612yes—
328C22×C4C2×D812yes—
329C22×C4C2×SD1612yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C2×C8—28no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4C22×C814yes—
351C22⋊C4—28no—
352C22⋊C4—28no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—44no—
383C42—44no—
384C42C22⋊C814yes—
385C42C22⋊C814yes—
386C42#10214yes—
387C42#10214yes—
388C42C22×C812yes—
389C42C22×C812yes—
390C42C2×M1612yes—
391C42C2×M1612yes—
392C42C2×D812yes—
393C42C2×D812yes—
394C42C2×SD1612yes—
395C42C2×SD1612yes—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—24no—
401C23—48no—
402C23—48no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—48no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23—84no—
414C23C2×C22⋊C812yes—
415C23C2×#10212yes—
416C23C2×#10412yes—
417C23C22×C22⋊C411yescharacteristic
418D4—44no—
419D4—44no—
420D4—48no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—48no—
457C2×C4—416no—
458C2×C4—416no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4D4⋊C8116yes—
468C2×C4C2×C22⋊C812yes—
469C2×C4C2×C22⋊C812yes—
470C2×C4C2×C22⋊C812yes—
471C2×C4C2×C22⋊C812yes—
472C2×C4C2×C22⋊C812yes—
473C2×C4#41012yes—
474C2×C4#41312yes—
475C2×C4C2×#10212yes—
476C2×C4C2×#10212yes—
477C2×C4C2×#10212yes—
478C2×C4C2×#10212yes—
479C2×C4C2×#10212yes—
480C2×C4C2×#10412yes—
481C2×C4C2×#10412yes—
482C2×C4#42112yes—
483C2×C4C23×C811yescharacteristic
484C2×C4C22×M1611yescharacteristic
485C2×C4C22×D811yescharacteristic
486C2×C4C22×SD1611yescharacteristic
487C2×C4#57512yes—
488C8—48no—
489C8—84no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—816no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22C2×D4⋊C814yes—
505C22C22×C22⋊C811yescharacteristic
506C22C2×#41011yescharacteristic
507C22C2×#41311yescharacteristic
508C22C22×#10211yescharacteristic
509C22#289611yescharacteristic
510C22C22×#10411yescharacteristic
511C22C2×#42111yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—21no—
518C4—21no—
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—44no—
524C4—41no—
525C4—41no—
526C4—82no—
527C4—82no—
528C4#145814yes—
529C4C2×D4⋊C814yes—
530C4C2×D4⋊C814yes—
531C2—24no—
532C2—84no—
533C2C2×#145811yescharacteristic
534C2C22×D4⋊C811yescharacteristic
535C2#1743711yescharacteristic
536C2#1753711yescharacteristic
537C2#1757411yescharacteristic
538C2#1764211yescharacteristic
539C2#1770411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
3. C4×#431C2semidirect
4. #7574C2semidirect
5. #7573C2semidirect
7. #7560C2semidirect
30. #1831C22semidirect
35. C4×C4⋊C8C22semidirect
36. C4×C4⋊C8C22semidirect
39. #1744C22semidirect
41. #1271C22semidirect
48. C2×C4×D4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
#208082
C4×#4312
#75742
#75732
#75722
#75602