Finite Group Databasebeta

← D4×Q8⋊C8 #219798

Subgroups — D4×Q8⋊C8 #219798

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×Q8C211yesmaximal, characteristic
2D4×C4⋊C8C211yesmaximal, characteristic
3C4×C8×D4C211yesmaximal, characteristic
4C22×Q8⋊C8C212yesmaximal, complemented
5#9673C211yesmaximal, characteristic, complemented
6#9637C211yesmaximal, characteristic, complemented
7#9071C212yesmaximal, complemented
8#8937C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8311C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×D4×Q8C411yescharacteristic
13C22×C4×Q8C2212yes—
14C2×C8×D4—21no—
15C2×C8×D4—21no—
16C22×C4⋊C8C2212yes—
17C22×C4×C8C2212yes—
18#2365—21no—
19#2365C411yescharacteristic
20#2360—22no—
21#2360C412yes—
22Q8×C4⋊C4C411yescharacteristic
23D4×C4⋊C4—21no—
24D4×C4⋊C4C411yescharacteristic
25Q8×C22⋊C4C412yes—
26C4×C4⋊Q8—21no—
27C4×C4⋊Q8C2211yescharacteristic
28C4×C22⋊Q8—22no—
29C4×C22⋊Q8C2212yes—
30C42×Q8C2211yescharacteristic
31C42×D4—21no—
32C42×D4C2211yescharacteristic
33#1939—21no—
34#1850—22no—
35#1842C2211yescharacteristic
36#1797C2212yes—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C8×C22⋊C4—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8—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—
55C23×C8—22no—
56C23×C8—22no—
57D4×Q8—24no—
58D4×Q8C818yescomplemented
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8C2×C412yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—42no—
68C2×C4×Q8—42no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C412yes—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2312yes—
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2×C412yes—
82C22×C42—22no—
83C22×C42C2312yes—
84C8×D4—28no—
85C8×D4—44no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—42no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2311yescharacteristic
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C4×C8—21no—
98C2×C4×C8—24no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—42no—
103C2×C4×C8C2312yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—22no—
107#391—22no—
108#391C2×C412yes—
109#391C2×C412yes—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—22no—
115#389—22no—
116#389C2×C412yes—
117#387—22no—
118#387—22no—
119#387—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#384—21no—
124#384C2×C411yescharacteristic
125#380—22no—
126#380C2×C412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4C2×C412yes—
145C4×C22⋊C4D412yes—
146C4×C22⋊C4D412yes—
147C43—21no—
148C43C2311yescharacteristic
149#336—22no—
150#336—21no—
151#336—22no—
152#336—21no—
153Q8⋊C8—24no—
154Q8⋊C8—22no—
155Q8⋊C8—24no—
156Q8⋊C8—24no—
157Q8⋊C8—42no—
158Q8⋊C8—42no—
159Q8⋊C8D414yesdirect product factor
160C22×Q8—22no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8C2×C818yescomplemented
167C22×Q8C22×C412yes—
168C22×Q8C22×C411yescharacteristic
169C22×D4C22⋊C411yescharacteristic
170C23×C4—22no—
171C23×C4—22no—
172C23×C4C22×C412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C22×C8—21no—
176C22×C8—28no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—44no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8—44no—
186C22×C8—82no—
187C4⋊Q8—28no—
188C4⋊Q8—22no—
189C4⋊Q8C2×C814yescomplemented
190C22⋊Q8—24no—
191C22⋊Q8—216no—
192C22⋊Q8C2×C818yescomplemented
193C4×Q8—22no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—24no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—44no—
204C4×Q8—44no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—41no—
208C4×Q8—44no—
209C4×Q8—44no—
210C4×Q8C2×C814yescomplemented
211C4×Q8C2×D412yes—
212C4×Q8#4112yes—
213C4×D4—28no—
214C4×D4—24no—
215C4×D4—44no—
216C4×D4C2×C814yes—
217C4×D4M1614yes—
218C4×D4SD1614yes—
219C4×D4Q1614yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4C22⋊C412yes—
247C2×C22⋊C4C22⋊C412yes—
248C2×C22⋊C4C22⋊C412yes—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42C22×C412yes—
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2411yescharacteristic
274C4⋊C8—24no—
275C4⋊C8—22no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—22no—
280C4⋊C8—44no—
281C4⋊C8—42no—
282C4⋊C8—81no—
283C4⋊C8C2×D412yes—
284C22⋊C8—28no—
285C22⋊C8—44no—
286C4×C8—24no—
287C4×C8—22no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—22no—
292C4×C8—44no—
293C4×C8—42no—
294C4×C8—42no—
295C4×C8C2×D412yes—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—21no—
300#95—21no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—21no—
305#95C22⋊C412yes—
306#95C22⋊C412yes—
307#95C22⋊C412yes—
308C24C2×C22⋊C412yes—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—21no—
313C2×Q8—28no—
314C2×Q8—216no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8C4×D412yes—
325C2×Q8C22×C814yes—
326C2×Q8C22×C812yes—
327C2×Q8#13112yes—
328C2×D4C22⋊C814yes—
329C2×D4#10314yes—
330C2×D4#10414yes—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—82no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C22×C814yes—
363C22×C4C2×M1614yes—
364C22×C4C2×SD1614yes—
365C22×C4C2×Q1614yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—416no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—88no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—28no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—81no—
410C4⋊C4—82no—
411C4⋊C4C22⋊C814yes—
412C4⋊C4#10314yes—
413C4⋊C4C4×D412yes—
414C4⋊C4C22×C812yes—
415C22⋊C4—28no—
416C22⋊C4—24no—
417C22⋊C4—44no—
418C22⋊C4C22⋊C818yes—
419C22⋊C4#10318yes—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—24no—
433C42—22no—
434C42—41no—
435C42—44no—
436C42—44no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42—81no—
444C42C22×C812yes—
445C42C2×M1612yes—
446C42C2×SD1612yes—
447C42C2×Q1612yes—
448C42C22×D411yescharacteristic
449C42C2×#4111yescharacteristic
450C42H(5,𝔽2)11yescharacteristic
451C42H(5,𝔽2)11yescharacteristic
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23C2×C22⋊C814yes—
457C23C2×#10314yes—
458C23C2×#10414yes—
459C23C22×C22⋊C411yescharacteristic
460Q8—22no—
461Q8—22no—
462Q8—22no—
463Q8—24no—
464Q8—24no—
465Q8—416no—
466Q8—42no—
467Q8—42no—
468Q8C8×D414yescomplemented
469D4Q8⋊C8116yesdirect product factor
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—416no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—88no—
516C2×C4C2×C22⋊C812yes—
517C2×C4C2×C22⋊C814yes—
518C2×C4C2×C22⋊C812yes—
519C2×C4C2×#10312yes—
520C2×C4C2×#10314yes—
521C2×C4C2×#10312yes—
522C2×C4C2×#10412yes—
523C2×C4C2×C4×D411yescharacteristic
524C2×C4#51811yescharacteristic
525C2×C4D4211yescharacteristic
526C2×C4D4211yescharacteristic
527C2×C4#54611yescharacteristic
528C2×C4#54611yescharacteristic
529C2×C4C23×C811yescharacteristic
530C2×C4C22×M1611yescharacteristic
531C2×C4C2×#13111yescharacteristic
532C2×C4#56811yescharacteristic
533C2×C4C22×SD1611yescharacteristic
534C2×C4C22×Q1611yescharacteristic
535C2×C4#57711yescharacteristic
536C2×C4#57811yescharacteristic
537C8—24no—
538C8—42no—
539C8—44no—
540C8—42no—
541C8—82no—
542C8—81no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22C2×Q8⋊C818yescomplemented
550C22D4×C22⋊C411yescharacteristic
551C22C22×C22⋊C811yescharacteristic
552C22#288011yescharacteristic
553C22C22×#10311yescharacteristic
554C22#289811yescharacteristic
555C22C22×#10411yescharacteristic
556C22#290111yescharacteristic
557C4—22no—
558C4—22no—
559C4—24no—
560C4—24no—
561C4—24no—
562C4—21no—
563C4—21no—
564C4—42no—
565C4—41no—
566C4—44no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—82no—
571C4C2×Q8⋊C814yescomplemented
572C4C2×C8×D411yescharacteristic
573C4D4×M1611yescharacteristic
574C4#297211yescharacteristic
575C4#297411yescharacteristic
576C4D4×SD1611yescharacteristic
577C4D4×Q1611yescharacteristic
578C4#329811yescharacteristic
579C4#329911yescharacteristic
580C2—28no—
581C2C22×Q8⋊C811yescharacteristic
582C2#1743911yescharacteristic
583C2#1750911yescharacteristic
584C2#1751911yescharacteristic
585C2D4×C22⋊C811yescharacteristic
586C2D4×#10311yescharacteristic
587C2D4×#10411yescharacteristic
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. C22×Q8⋊C8C2semidirect
5. #9673C2semidirect
6. #9637C2semidirect
7. #9071C2semidirect
8. #8937C2semidirect
9. #8564C2semidirect
10. #8311C2semidirect
11. C4×Q8⋊C8C2semidirect
58. D4×Q8C8semidirect
159. Q8⋊C8D4direct (N × H)
166. C22×Q8C2×C8semidirect
189. C4⋊Q8C2×C8semidirect
192. C22⋊Q8C2×C8semidirect
210. C4×Q8C2×C8semidirect
468. Q8C8×D4semidirect
469. D4Q8⋊C8direct (N × H)
549. C22C2×Q8⋊C8semidirect
571. C4C2×Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Q82
D4×C4⋊C82
C4×C8×D42
C22×Q8⋊C82
#96732
#96372
#90712
#89372
#85642
#83112
C4×Q8⋊C82