Finite Group Databasebeta

← #190874

Subgroups — #190874

Order 512.

1246 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
1D4×#117C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#8709C212yesmaximal, complemented
5#8567C212yesmaximal, complemented
6#8563C212yesmaximal, 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—
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#1923—22no—
31#1916—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
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
79#433—44no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8C2312yes—
91C2×C4×C8C2312yes—
92#390C2×C412yes—
93#390C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#386—22no—
103#386—22no—
104#385—22no—
105#385—22no—
106#384C2×C412yes—
107#382—21no—
108#381—22no—
109#380—21no—
110#380—22no—
111#380—22no—
112#380C2×C412yes—
113#379—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129#376D411yescharacteristic
130#376D411yescharacteristic
131C43C2311yescharacteristic
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139D4⋊C8—24no—
140D4⋊C8—24no—
141D4⋊C8—24no—
142D4⋊C8—24no—
143C25—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—42no—
149C22×D4C22×C411yescharacteristic
150C22×D4C22×C411yescharacteristic
151C23×C4—28no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—44no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C4×D4—28no—
170C4×D4—28no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4#4114yes—
182C4×D4#4114yes—
183#117—42no—
184#117—42no—
185#117—42no—
186#117—48no—
187#117—48no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—42no—
192#117D814yes—
193#117SD1614yes—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C22⋊C4—28no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
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—22no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—44no—
250C4⋊C8—44no—
251C4⋊C8—44no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254#97—42no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—44no—
260C4×C8—42no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95—21no—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272C24—21no—
273C24—24no—
274C24—21no—
275C24—44no—
276C24—42no—
277C24—42no—
278C24—82no—
279C24C2×C22⋊C411yescharacteristic
280C2×D4—24no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4#11712yes—
291C2×D4#11712yes—
292C2×D4#13114yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—416no—
313C22×C4—416no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×D812yes—
338C22×C4C2×SD1612yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—82no—
366C4⋊C4#10218yes—
367C4⋊C4#11712yes—
368C4⋊C4#11712yes—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—42no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4#10218yes—
382C42—21no—
383C42—21no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—42no—
393C42—42no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—44no—
399C42—44no—
400C42C2×D812yes—
401C42C2×D812yes—
402C42C2×SD1612yes—
403C42C2×SD1612yes—
404C42C2×#4112yes—
405C42C2×#4112yes—
406C23—24no—
407C23—22no—
408C23—21no—
409C23—22no—
410C23—22no—
411C23—21no—
412C23—22no—
413C23—48no—
414C23—48no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—88no—
423C23—84no—
424C23—88no—
425C23C2×#10212yes—
426C23C22×C22⋊C411yescharacteristic
427D4—22no—
428D4—24no—
429D4—24no—
430D4—48no—
431D4—41no—
432D4—41no—
433D4#43314yescomplemented
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—88no—
477C2×C4—88no—
478C2×C4C2×#10214yes—
479C2×C4C2×#10214yes—
480C2×C4C2×#10212yes—
481C2×C4C2×#11711yescharacteristic
482C2×C4C2×#11711yescharacteristic
483C2×C4#52512yes—
484C2×C4#52512yes—
485C2×C4C2×#13112yes—
486C2×C4#56812yes—
487C2×C4C22×D811yescharacteristic
488C2×C4C22×SD1611yescharacteristic
489C2×C4#57512yes—
490C8—44no—
491C8—82no—
492C8—82no—
493C8—82no—
494C22—24no—
495C22—24no—
496C22—22no—
497C22—41no—
498C22—42no—
499C22—44no—
500C22—42no—
501C22—44no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—816no—
508C22—82no—
509C22—82no—
510C22#228011yescharacteristic
511C22#288012yes—
512C22C22×#10211yescharacteristic
513C22#289611yescharacteristic
514C22#290112yes—
515C4—22no—
516C4—24no—
517C4—24no—
518C4—22no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—41no—
523C4—42no—
524C4—42no—
525C4—82no—
526C4—82no—
527C4C2×#43311yescharacteristic
528C4#292111yescharacteristic
529C4#292412yes—
530C4#310012yes—
531C4#310112yes—
532C2—42no—
533C2—44no—
534C2—82no—
535C2#1750712yes—
536C2#1751312yes—
537C2#2071611yescharacteristic
538C2#2073211yescharacteristic
539C2#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. #20812C2semidirect
4. #8709C2semidirect
5. #8567C2semidirect
6. #8563C2semidirect
8. #8183C2semidirect
9. #8177C2semidirect
10. #8174C2semidirect
11. C4×D4⋊C8C2semidirect
32. #1842C22semidirect
37. #1767C22semidirect
38. C4×C4⋊C8C22semidirect
39. C4×C4⋊C8C22semidirect
41. #1744C22semidirect
42. C42×C8C22semidirect
47. #1279C22semidirect
48. #1279C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect
433. D4#433semidirect

Maximal subgroups

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

subgroupindex
D4×#1172
#214302
#208122
#87092
#85672
#85632
#83102
#81832
#81772
#81742
C4×D4⋊C82