Finite Group Databasebeta

← #190882

Subgroups — #190882

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#117C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#8710C212yesmaximal, complemented
5#8567C212yesmaximal, complemented
6#8563C212yesmaximal, complemented
7#8310C212yesmaximal
8#8181C211yesmaximal, characteristic, complemented
9#8179C211yesmaximal, characteristic, complemented
10#8172C211yesmaximal, characteristic, complemented
11#8170C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C2×#433—21no—
15#2354C412yes—
16#2352—22no—
17D4×C4⋊C4C412yes—
18#2344C412yes—
19#2342—22no—
20D4×C22⋊C4C412yes—
21#2293C2211yescharacteristic
22#2293C2211yescharacteristic
23#2290—21no—
24#2290—21no—
25#2285C2211yescharacteristic
26#2280—21no—
27C42×D4C2211yescharacteristic
28C42×D4C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1924—22no—
31#1917—22no—
32#1842C2212yescomplemented
33#1832—21no—
34#1798—21no—
35#1797C2212yes—
36#1768—21no—
37#1767C2211yescharacteristic, complemented
38C4×C4⋊C8C2211yescharacteristic, complemented
39C4×C4⋊C8C2212yescomplemented
40C4×C22⋊C8C2212yes—
41#1744C2211yescharacteristic, complemented
42C4×#97C2211yescharacteristic, 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—48no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#433—44no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—21no—
87C2×#97—22no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90C2×C4×C8C2312yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#387—22no—
100#387—22no—
101#386—22no—
102#386—22no—
103#385—22no—
104#385—22no—
105#384C2×C412yes—
106#382—21no—
107#381—22no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380C2×C412yes—
112#379—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128#376D411yescharacteristic
129#376D411yescharacteristic
130C43C2311yescharacteristic
131C43C2311yescharacteristic
132#336—22no—
133#336—21no—
134#336—21no—
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—44no—
159C23×C4—42no—
160C23×C4—42no—
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—22no—
183#117—22no—
184#117—24no—
185#117—44no—
186#117—48no—
187#117—48no—
188#117—44no—
189#117—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—28no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—48no—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C412yes—
229C2×C42C22×C412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2411yescharacteristic
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—44no—
244C4⋊C8—44no—
245C4⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248#97—42no—
249#97—44no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—22no—
254C4×C8—42no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266C24—21no—
267C24—24no—
268C24—21no—
269C24—44no—
270C24—42no—
271C24—42no—
272C24—82no—
273C24C2×C22⋊C411yescharacteristic
274C2×D4—24no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4#11712yes—
285C2×D4#11712yes—
286C2×D4#13114yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—416no—
307C22×C4—416no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C412yes—
331C22×C4AGL(1,ℤ/8)12yes—
332C22×C4#13712yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—82no—
361C4⋊C4#11712yes—
362C4⋊C4#11712yes—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C42—21no—
377C42—21no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—21no—
385C42—21no—
386C42—22no—
387C42—21no—
388C42—21no—
389C42—42no—
390C42—42no—
391C42—44no—
392C42—42no—
393C42—42no—
394C42—44no—
395C42AGL(1,ℤ/8)14yes—
396C42C2×#4112yes—
397C42C2×#4112yes—
398C23—24no—
399C23—22no—
400C23—21no—
401C23—22no—
402C23—22no—
403C23—21no—
404C23—22no—
405C23—48no—
406C23—48no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—88no—
415C23—84no—
416C23—88no—
417C23#41712yes—
418C23C22×C22⋊C411yescharacteristic
419D4—22no—
420D4—24no—
421D4—24no—
422D4—48no—
423D4—41no—
424D4—41no—
425D4#43314yescomplemented
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—82no—
469C2×C4—88no—
470C2×C4—88no—
471C2×C4#41712yes—
472C2×C4#41814yes—
473C2×C4C2×#11711yescharacteristic
474C2×C4C2×#11711yescharacteristic
475C2×C4#52512yes—
476C2×C4#52512yes—
477C2×C4C2×#13112yes—
478C2×C4#56812yes—
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)12yes—
481C2×C4C2×#13711yescharacteristic
482C8—44no—
483C8—82no—
484C8—82no—
485C8—82no—
486C22—22no—
487C22—24no—
488C22—24no—
489C22—42no—
490C22—41no—
491C22—42no—
492C22—44no—
493C22—42no—
494C22—44no—
495C22—41no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—816no—
500C22—82no—
501C22—82no—
502C22#228011yescharacteristic
503C22#288012yes—
504C22C2×#41811yescharacteristic
505C22C2×#41711yescharacteristic
506C22#290112yes—
507C4—22no—
508C4—22no—
509C4—24no—
510C4—42no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4—82no—
518C4—82no—
519C4C2×#43311yescharacteristic
520C4#292111yescharacteristic
521C4#292412yes—
522C4#310912yes—
523C4#311012yes—
524C2—42no—
525C2—44no—
526C2—82no—
527C2#1751112yes—
528C2#1751212yes—
529C2#2071611yescharacteristic
530C2#2073811yescharacteristic
531C2#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. #21431C2semidirect
3. #20812C2semidirect
4. #8710C2semidirect
5. #8567C2semidirect
6. #8563C2semidirect
8. #8181C2semidirect
9. #8179C2semidirect
10. #8172C2semidirect
11. #8170C2semidirect
32. #1842C22semidirect
37. #1767C22semidirect
38. C4×C4⋊C8C22semidirect
39. C4×C4⋊C8C22semidirect
41. #1744C22semidirect
42. C4×#97C22semidirect
47. #1279C22semidirect
48. #1279C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect
425. D4#433semidirect

Maximal subgroups

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

subgroupindex
D4×#1172
#214312
#208122
#87102
#85672
#85632
#83102
#81812
#81792
#81722
#81702