Finite Group Databasebeta

← #214121

Subgroups — #214121

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#9073C214yesmaximal, complemented
5#8937C214yesmaximal, complemented
6#8706C212yesmaximal, complemented
7#8311C212yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431C2212yes—
12C22×C4⋊C8C2212yes—
13C22×C22⋊C8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8C412yes—
19C4×C22⋊Q8C412yes—
20C4×C22⋊Q8C2212yes—
21C4×C22⋊Q8C2212yes—
22C2×#391C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#387—21no—
25C2×#385—21no—
26C2×#380C411yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—21no—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1779—21no—
38C4×C22⋊C8C2212yes—
39C4×C22⋊C8C2212yes—
40C2×#336—21no—
41C2×Q8⋊C8C2218yescomplemented
42#1279C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C23×C8—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8C818yescomplemented
52C2×C22⋊Q8C2×C412yes—
53C2×C22⋊Q8C2×C412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—44no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42—44no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#431—44no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—44no—
86C2×C4⋊C8C2314yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—44no—
93C2×C4×C8C2314yes—
94#391—21no—
95#391—21no—
96#391C2×C412yes—
97#391C2×C412yes—
98#389—21no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#380—21no—
111#380—21no—
112#380C2×C412yes—
113#380C2×C412yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2312yes—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135Q8⋊C8—48no—
136C25C22⋊C411yescharacteristic
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8C2×C814yescomplemented
142C22×Q8C22×C412yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C2×C814yes—
166C23×C4M1614yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—44no—
179C22×C8—44no—
180C22×C8—44no—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8C2×C818yes—
186C22⋊Q8C2×C818yescomplemented
187C4×Q8—28no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4C2×C814yescomplemented
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—44no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—44no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2411yescharacteristic
251C4⋊C8—28no—
252C4⋊C8—44no—
253C4⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C22⋊C8—44no—
258C22⋊C8—44no—
259C4×C8—28no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24C22⋊C814yes—
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C411yescharacteristic
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—48no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—48no—
288C2×Q8C22×C814yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—416no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C22⋊C814yes—
359C22×C4C22⋊C814yes—
360C22×C4C22⋊C814yes—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C22×C812yes—
371C22×C4C22×C812yes—
372C22×C4C22×C812yes—
373C22×C4C2×M1612yes—
374C22×C4C2×M1612yes—
375C22×C4C2×M1612yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4#13712yes—
378C22×C4#13714yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—88no—
389C4⋊C4—22no—
390C4⋊C4—28no—
391C4⋊C4—28no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4C22×C814yes—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—44no—
422C42—44no—
423C42—42no—
424C42—42no—
425C42—44no—
426C42H(5,𝔽2)14yes—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—28no—
442C23—28no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23C2×C22⋊C812yes—
449C23C2×C22⋊C812yes—
450C23C2×C22⋊C812yes—
451C23#41712yes—
452C23#41912yes—
453C23#42112yes—
454C23#42112yes—
455C23C22×C22⋊C411yescharacteristic
456Q8—28no—
457Q8—48no—
458Q8—44no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4C2×C22⋊C812yes—
524C2×C4C2×C22⋊C812yes—
525C2×C4C2×C22⋊C812yes—
526C2×C4#41712yes—
527C2×C4#41912yes—
528C2×C4#51812yes—
529C2×C4#53412yes—
530C2×C4#53412yes—
531C2×C4C23×C811yescharacteristic
532C2×C4C22×M1611yescharacteristic
533C2×C4#56812yes—
534C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
535C2×C4C2×#13712yes—
536C2×C4C2×#13711yescharacteristic
537C8—84no—
538C8—84no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—416no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22#147714yes—
560C22#147914yes—
561C22#230811yescharacteristic
562C22C22×C22⋊C811yescharacteristic
563C22#287911yescharacteristic
564C22C2×#41911yescharacteristic
565C22C2×#41711yescharacteristic
566C22C2×#42111yescharacteristic
567C22C2×#42111yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—44no—
579C4—44no—
580C4—82no—
581C4—82no—
582C4#297212yes—
583C4#297312yes—
584C4#319312yes—
585C4#319512yes—
586C2—24no—
587C2—24no—
588C2—42no—
589C2C2×#147911yescharacteristic
590C2C2×#147711yescharacteristic
591C2#1749211yescharacteristic
592C2#1749611yescharacteristic
593C2#2173811yescharacteristic
594C2#2176511yescharacteristic
595C2#2177411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9073C2semidirect
5. #8937C2semidirect
6. #8706C2semidirect
7. #8311C2semidirect
41. C2×Q8⋊C8C22semidirect
42. #1279C22semidirect
51. C2×C22⋊Q8C8semidirect
141. C22×Q8C2×C8semidirect
186. C22⋊Q8C2×C8semidirect
206. C2×C4⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#207142
C2×#17972
#90732
#89372
#87062
#83112