Finite Group Databasebeta

← #190875

Subgroups — #190875

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#117C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#8710C212yesmaximal, complemented
5#8569C212yesmaximal, complemented
6#8565C212yesmaximal, complemented
7#8310C212yesmaximal
8#8183C211yesmaximal, characteristic, complemented
9#8177C211yesmaximal, characteristic, complemented
10#8174C211yesmaximal, characteristic, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14#2354C412yes—
15#2352—22no—
16D4×C4⋊C4C412yes—
17#2344C412yes—
18#2342—22no—
19D4×C22⋊C4C412yes—
20#2293C2211yescharacteristic
21#2293C2211yescharacteristic
22#2290—21no—
23#2290—21no—
24#2285C2211yescharacteristic
25#2280—21no—
26C42×D4C2211yescharacteristic
27C42×D4C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1923—22no—
30#1916—22no—
31#1840—21no—
32#1831C2212yescomplemented
33#1818—21no—
34#1797C2212yes—
35#1767C2211yescharacteristic, complemented
36C4×C4⋊C8C2211yescharacteristic, complemented
37C4×C4⋊C8C2212yescomplemented
38#1750—21no—
39C4×C22⋊C8C2212yes—
40#1746—21no—
41#1744C2211yescharacteristic, complemented
42C42×C8C2211yescharacteristic, complemented
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47#1279C2212yescomplemented
48#1279C2212yescomplemented
49#1279C2212yescomplemented
50#1279C2212yescomplemented
51C23×D4C2×C411yescharacteristic
52C24×C4—21no—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2311yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—44no—
69C2×#117—44no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×#97—21no—
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87C2×C4×C8C2312yes—
88C2×C4×C8C2312yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#386—22no—
100#386—22no—
101#385—22no—
102#385—22no—
103#384C2×C412yes—
104#382—21no—
105#381—22no—
106#380—21no—
107#380—22no—
108#380—22no—
109#380C2×C412yes—
110#379—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4C2×C412yes—
125C4×C22⋊C4D412yes—
126#376D411yescharacteristic
127#376D411yescharacteristic
128C43C2311yescharacteristic
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138D4⋊C8—24no—
139D4⋊C8—24no—
140D4⋊C8—24no—
141D4⋊C8—24no—
142C25—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—24no—
146C22×D4—21no—
147C22×D4—42no—
148C22×D4C22×C411yescharacteristic
149C22×D4C22×C411yescharacteristic
150C23×C4—28no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4#4114yes—
181C4×D4#4114yes—
182#117—24no—
183#117—22no—
184#117—22no—
185#117—42no—
186#117—42no—
187#117—48no—
188#117—48no—
189#117—42no—
190#117—42no—
191#117—42no—
192#117—42no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C22⋊C4—28no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C412yes—
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2411yescharacteristic
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—44no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251#97—42no—
252C4×C8—22no—
253C4×C8—22no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—44no—
257C4×C8—42no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269C24—21no—
270C24—24no—
271C24—21no—
272C24—44no—
273C24—42no—
274C24—42no—
275C24—82no—
276C24C2×C22⋊C411yescharacteristic
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4#11712yes—
288C2×D4#11712yes—
289C2×D4#13114yes—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—416no—
311C22×C4—416no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4#13514yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C2×C8—24no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—24no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—82no—
364C4⋊C4#11712yes—
365C4⋊C4#11712yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—21no—
380C42—21no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—21no—
390C42—21no—
391C42—42no—
392C42—42no—
393C42—44no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—44no—
399C42AGL(1,ℤ/8)12yes—
400C42AGL(1,ℤ/8)12yes—
401C42C2×#4112yes—
402C42C2×#4112yes—
403C23—24no—
404C23—22no—
405C23—21no—
406C23—22no—
407C23—22no—
408C23—21no—
409C23—22no—
410C23—48no—
411C23—48no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—42no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—88no—
420C23—84no—
421C23—88no—
422C23#41612yes—
423C23C22×C22⋊C411yescharacteristic
424D4—22no—
425D4—24no—
426D4—24no—
427D4—48no—
428D4—41no—
429D4—41no—
430D4#43314yes—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—48no—
454C2×C4—48no—
455C2×C4—48no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—88no—
475C2×C4—88no—
476C2×C4#41612yes—
477C2×C4#41814yes—
478C2×C4C2×#11711yescharacteristic
479C2×C4C2×#11711yescharacteristic
480C2×C4#52512yes—
481C2×C4#52512yes—
482C2×C4C2×#13112yes—
483C2×C4#56812yes—
484C2×C4C2×#13512yes—
485C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
487C8—44no—
488C8—82no—
489C8—82no—
490C8—82no—
491C22—24no—
492C22—24no—
493C22—22no—
494C22—41no—
495C22—42no—
496C22—44no—
497C22—44no—
498C22—42no—
499C22—41no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—816no—
505C22—82no—
506C22—82no—
507C22#228011yescharacteristic
508C22#288012yes—
509C22C2×#41611yescharacteristic
510C22C2×#41811yescharacteristic
511C22#290112yes—
512C4—22no—
513C4—24no—
514C4—22no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—82no—
524C4C2×#43311yescharacteristic
525C4#292111yescharacteristic
526C4#292412yes—
527C4#310512yes—
528C4#310512yes—
529C2—42no—
530C2—44no—
531C2—82no—
532C2#1750812yes—
533C2#1751112yes—
534C2#2071611yescharacteristic
535C2#2073311yescharacteristic
536C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21430C2semidirect
3. #20786C2semidirect
4. #8710C2semidirect
5. #8569C2semidirect
6. #8565C2semidirect
8. #8183C2semidirect
9. #8177C2semidirect
10. #8174C2semidirect
11. C4×D4⋊C8C2semidirect
32. #1831C22semidirect
35. #1767C22semidirect
36. C4×C4⋊C8C22semidirect
37. C4×C4⋊C8C22semidirect
41. #1744C22semidirect
42. C42×C8C22semidirect
47. #1279C22semidirect
48. #1279C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect

Maximal subgroups

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

subgroupindex
D4×#1172
#214302
#207862
#87102
#85692
#85652
#83102
#81832
#81772
#81742
C4×D4⋊C82