Finite Group Databasebeta

← #219805

Subgroups — #219805

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×Q8C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic
3#21264C211yesmaximal, characteristic
4C2×#1479C212yesmaximal, complemented
5#9673C211yesmaximal, characteristic, complemented
6#9637C211yesmaximal, characteristic, complemented
7#9073C212yesmaximal, complemented
8#8938C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8311C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×D4×Q8C411yescharacteristic
13C22×C4×Q8C2212yes—
14C2×#431C2212yes—
15C2×#431C2212yes—
16#2365—21no—
17#2365C411yescharacteristic
18#2360—22no—
19#2360C412yes—
20Q8×C4⋊C4C411yescharacteristic
21D4×C4⋊C4—21no—
22D4×C4⋊C4C411yescharacteristic
23Q8×C22⋊C4C412yes—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C22⋊Q8—22no—
27C4×C22⋊Q8C2212yes—
28C42×Q8C2211yescharacteristic
29C42×D4—21no—
30C42×D4C2211yescharacteristic
31#1954—21no—
32#1954—21no—
33#1939—21no—
34#1923—22no—
35#1923—22no—
36#1842C2211yescharacteristic
37#1797C2212yes—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1479—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2211yescharacteristic
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2211yescharacteristic
51#1279C2211yescharacteristic
52#1279C2212yes—
53#1279C2211yescharacteristic
54C23×Q8C2×C412yes—
55D4×Q8—24no—
56D4×Q8C818yescomplemented
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8C2×C412yes—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—42no—
66C2×C4×Q8—42no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2312yes—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C22×C4⋊C4—22no—
79C22×C4⋊C4C2×C412yes—
80C22×C42—22no—
81C22×C42C2312yes—
82#431—24no—
83#431—24no—
84#431—42no—
85#431—42no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2311yescharacteristic
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2312yes—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#391—22no—
105#391—22no—
106#391C2×C412yes—
107#391C2×C412yes—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—22no—
113#389—22no—
114#389C2×C412yes—
115#387—22no—
116#387—22no—
117#387—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#384—21no—
122#384C2×C411yescharacteristic
123#380—22no—
124#380C2×C412yes—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4C2×C412yes—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4D412yes—
145C43—21no—
146C43C2311yescharacteristic
147#336—22no—
148#336—21no—
149#336—21no—
150#336—22no—
151Q8⋊C8—24no—
152Q8⋊C8—22no—
153Q8⋊C8—24no—
154Q8⋊C8—24no—
155Q8⋊C8—42no—
156Q8⋊C8—42no—
157Q8⋊C8D414yescomplemented
158C22×Q8—22no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8C2×C818yes—
165C22×Q8C22×C412yes—
166C22×Q8C22×C411yescharacteristic
167C22×D4C22⋊C411yescharacteristic
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C22×C412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C4⋊Q8—28no—
180C4⋊Q8—22no—
181C4⋊Q8C2×C814yes—
182C22⋊Q8—24no—
183C22⋊Q8—216no—
184C22⋊Q8C2×C818yes—
185C4×Q8—22no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—24no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—41no—
200C4×Q8—44no—
201C4×Q8—44no—
202C4×Q8C2×C814yes—
203C4×Q8C2×D412yes—
204C4×Q8#4112yes—
205C4×D4—28no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—42no—
209C4×D4—44no—
210C4×D4C2×C814yes—
211C4×D4M1614yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C412yes—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42C22×C412yes—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—22no—
271C4⋊C8—44no—
272C4⋊C8—42no—
273C4⋊C8—42no—
274C4⋊C8—81no—
275C4⋊C8C2×D412yes—
276C22⋊C8—44no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279C22⋊C8—82no—
280C4×C8—22no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—22no—
285C4×C8—44no—
286C4×C8—42no—
287C4×C8—42no—
288C4×C8—42no—
289C4×C8C2×D412yes—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—21no—
294#95—21no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95C22⋊C412yes—
300#95C22⋊C412yes—
301#95C22⋊C412yes—
302C24C2×C22⋊C412yes—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—22no—
306C2×Q8—21no—
307C2×Q8—28no—
308C2×Q8—216no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8C4×D412yes—
319C2×Q8C22×C814yes—
320C2×Q8C22×C812yes—
321C2×Q8#13112yes—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4C22⋊C814yes—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—82no—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C22×C814yes—
359C22×C4C2×M1614yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—42no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—81no—
403C4⋊C4—82no—
404C4⋊C4C22⋊C814yes—
405C4⋊C4C4×D412yes—
406C4⋊C4C22×C812yes—
407C22⋊C4—28no—
408C22⋊C4—24no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4C22⋊C818yes—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—24no—
425C42—22no—
426C42—41no—
427C42—44no—
428C42—44no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—41no—
433C42—41no—
434C42—42no—
435C42—81no—
436C42C22×C812yes—
437C42C2×M1612yes—
438C42#13712yes—
439C42#13712yes—
440C42C22×D411yescharacteristic
441C42C2×#4111yescharacteristic
442C42H(5,𝔽2)11yescharacteristic
443C42H(5,𝔽2)11yescharacteristic
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23C2×C22⋊C814yes—
451C23C22×C22⋊C411yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—22no—
455Q8—24no—
456Q8—24no—
457Q8—416no—
458Q8—42no—
459Q8—42no—
460Q8C8×D414yes—
461D4—28no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—416no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—88no—
509C2×C4C2×C22⋊C812yes—
510C2×C4C2×C22⋊C814yes—
511C2×C4C2×C22⋊C812yes—
512C2×C4#41912yes—
513C2×C4#41912yes—
514C2×C4#42112yes—
515C2×C4C2×C4×D411yescharacteristic
516C2×C4#51811yescharacteristic
517C2×C4D4211yescharacteristic
518C2×C4D4211yescharacteristic
519C2×C4#54611yescharacteristic
520C2×C4#54611yescharacteristic
521C2×C4C23×C811yescharacteristic
522C2×C4C22×M1611yescharacteristic
523C2×C4C2×#13111yescharacteristic
524C2×C4#56811yescharacteristic
525C2×C4C2×#13711yescharacteristic
526C2×C4C2×#13711yescharacteristic
527C2×C4#57711yescharacteristic
528C2×C4#57811yescharacteristic
529C8—42no—
530C8—42no—
531C8—42no—
532C8—82no—
533C8—81no—
534C8—82no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22D4×C22⋊C411yescharacteristic
543C22C22×C22⋊C811yescharacteristic
544C22#288011yescharacteristic
545C22C2×#41911yescharacteristic
546C22#289811yescharacteristic
547C22C2×#42111yescharacteristic
548C22#290111yescharacteristic
549C4—22no—
550C4—22no—
551C4—24no—
552C4—24no—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—41no—
557C4—44no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—81no—
562C4—82no—
563C4#147914yescomplemented
564C4C2×C8×D411yescharacteristic
565C4D4×M1611yescharacteristic
566C4#297211yescharacteristic
567C4#297411yescharacteristic
568C4#327611yescharacteristic
569C4#327811yescharacteristic
570C4#332811yescharacteristic
571C4#332911yescharacteristic
572C2—44no—
573C2C2×#147911yescharacteristic
574C2#1743911yescharacteristic
575C2#1751411yescharacteristic
576C2#1752511yescharacteristic
577C2D4×C22⋊C811yescharacteristic
578C2#2243011yescharacteristic
579C2#2243511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1479C2semidirect
5. #9673C2semidirect
6. #9637C2semidirect
7. #9073C2semidirect
8. #8938C2semidirect
9. #8564C2semidirect
10. #8311C2semidirect
11. C4×Q8⋊C8C2semidirect
56. D4×Q8C8semidirect
157. Q8⋊C8D4semidirect
563. C4#1479semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Q82
#227952
#212642
C2×#14792
#96732
#96372
#90732
#89382
#85642
#83112
C4×Q8⋊C82