Finite Group Databasebeta

← D4×#354 #497301

Subgroups — D4×#354 #497301

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#124C211yesmaximal, characteristic
2C22×#354C212yesmaximal, complemented
3D4×#99C212yesmaximal
4#13224C212yesmaximal, complemented
5#13090C214yesmaximal, complemented
6#12798C211yesmaximal, characteristic, complemented
7#12694C212yesmaximal, complemented
8C4×#354C211yesmaximal, characteristic, complemented
9C2×D4×Q8C411yescharacteristic, complemented
10C2×D42C2211yescharacteristic
11C22×#124C2212yes—
12C22×#99C2214yes—
13#2858C411yescharacteristic, complemented
14#2850C412yescomplemented
15#2548C411yescharacteristic, complemented
16#2547C2211yescharacteristic
17#2491—22no—
18#2472—24no—
19#2436C412yescomplemented
20#2435C2212yes—
21D4×C22⋊C4—22no—
22D4×C22⋊C4—22no—
23C4×#124C2211yescharacteristic
24C42×D4C411yescharacteristic, complemented
25C2×#354—28no—
26C2×#354C2214yes—
27C2×#354C2212yes—
28C2×#354C2211yescharacteristic
29C2×#354C2214yes—
30C2×#354C2211yescharacteristic
31#1964—22no—
32#1898—24no—
33#1799C2212yes—
34#1781C2214yes—
35C4×#99C2212yes—
36#1393C2212yes—
37#1393C2212yes—
38C23×Q8C2×C412yescomplemented
39C23×D4D412yes—
40C23×D4C2312yes—
41D4×Q8—42no—
42D4×Q8—41no—
43D4×Q8—41no—
44D42—41no—
45D42—42no—
46D42—41no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2×C411yescharacteristic, complemented
49C2×#127C2×C411yescharacteristic
50C2×#124—24no—
51C2×#124—24no—
52C2×#124—24no—
53C2×#124—44no—
54C2×#124C2×C412yescomplemented
55C2×#124C2×C412yescomplemented
56C2×#124C2×C412yescomplemented
57C2×#124C2312yes—
58C2×#124C2311yescharacteristic
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8C2×C412yescomplemented
61C2×#121C2×C412yes—
62C2×#121D412yes—
63C2×#120D414yes—
64C2×C4×Q8C2×C411yescharacteristic, complemented
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—42no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C22×C22⋊C4—24no—
71C22×C22⋊C4—24no—
72C22×C42C2×C412yescomplemented
73C2×#99—28no—
74C2×#99—24no—
75C2×#99—24no—
76C2×#99—44no—
77C2×#99C2314yes—
78C2×#99C2312yes—
79C2×#99C2312yes—
80#397—22no—
81#394—22no—
82#394—24no—
83#393—24no—
84#391C2×C412yescomplemented
85#391C2×C411yescharacteristic, complemented
86#390—22no—
87#390—22no—
88#390—22no—
89#390—42no—
90#390C2×C412yescomplemented
91#390C2311yescharacteristic
92#388—24no—
93#388—22no—
94#386—24no—
95#386—24no—
96#386—44no—
97#382C2×C411yescharacteristic, complemented
98#380—22no—
99#380—22no—
100#379—24no—
101#379—24no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C411yescharacteristic, complemented
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4C2×C412yescomplemented
108C43C2×C411yescharacteristic, complemented
109#354—24no—
110#354—22no—
111#354—28no—
112#354—44no—
113#354D414yesdirect product factor
114#342—22no—
115#342—22no—
116C25C2×D414yes—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—82no—
123C22×Q8C22×C412yes—
124C22×Q8C22×C411yescharacteristic
125C22×D4—22no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4—48no—
134C22×D4—82no—
135C22×D4C22⋊C414yes—
136C22×D4C22⋊C412yes—
137C22×D4C22⋊C412yes—
138C22×D4C22⋊C411yescharacteristic
139C22×D4C22×C412yes—
140C22×D4C2×D414yes—
141C22×D4C2411yescharacteristic
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151#127—41no—
152#127—41no—
153#124—22no—
154#124—24no—
155#124—22no—
156#124—22no—
157#124—24no—
158#124—24no—
159#124—42no—
160#124—42no—
161#124—48no—
162#124—48no—
163#124—82no—
164#124C2×D412yes—
165#124#4112yes—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170#121—42no—
171#121—42no—
172#121—42no—
173#121—42no—
174#120—44no—
175#120—44no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×D4—22no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—44no—
184C4×D4—42no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—88no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—48no—
210C2×C22⋊C4—48no—
211C2×C22⋊C4—48no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—84no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C414yes—
218C2×C22⋊C4C2×D412yes—
219C2×C42—24no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42C22×C412yes—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230#99—22no—
231#99—24no—
232#99—22no—
233#99—22no—
234#99—24no—
235#99—48no—
236#99—84no—
237#99C2×D414yes—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—24no—
243#95—42no—
244#95—42no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—24no—
249C24—44no—
250C24—48no—
251C24—84no—
252C24C2×C22⋊C412yes—
253C24C2×C22⋊C414yes—
254C24C22×D412yes—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—82no—
265C2×Q8—84no—
266C2×Q8—84no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8C4×D412yescomplemented
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—84no—
290C2×D4—82no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—84no—
295C2×D4—84no—
296C2×D4—82no—
297C2×D4—88no—
298C2×D4#9912yes—
299C2×D4#9911yescharacteristic
300C2×D4#9911yescharacteristic
301C2×D4C22×D411yescharacteristic
302C2×D4C2×#4111yescharacteristic
303C2×D4H(5,𝔽2)12yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4—164no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C23×C411yescharacteristic
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—84no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—24no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—48no—
355C22⋊C4—48no—
356C22⋊C4—42no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—42no—
366C22⋊C4—84no—
367C22⋊C4—82no—
368C22⋊C4—84no—
369C22⋊C4—88no—
370C22⋊C4—88no—
371C22⋊C4—164no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—22no—
376C42—22no—
377C42—44no—
378C42—41no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—81no—
383C42—84no—
384C42C4×D412yescomplemented
385C23—22no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—22no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—42no—
399C23—42no—
400C23—44no—
401C23—44no—
402C23—48no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—88no—
407C23—88no—
408C23—168no—
409C23C2×#9912yes—
410C23C2×#9912yes—
411C23C22×C22⋊C411yescharacteristic
412C23D4212yes—
413C23#54612yes—
414Q8—42no—
415Q8—42no—
416Q8—82no—
417Q8—82no—
418Q8—88no—
419D4—24no—
420D4—22no—
421D4—41no—
422D4—41no—
423D4—42no—
424D4—48no—
425D4—44no—
426D4—82no—
427D4—82no—
428D4—82no—
429D4—84no—
430D4—84no—
431D4—84no—
432D4—162no—
433D4#35414yesdirect product factor
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—88no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—84no—
477C2×C4—82no—
478C2×C4—84no—
479C2×C4—1616no—
480C2×C4C2×#9911yescharacteristic
481C2×C4C2×#9911yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4#51811yescharacteristic
484C22—21no—
485C22—21no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—44no—
494C22—44no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—82no—
503C22—88no—
504C22—88no—
505C22—84no—
506C22—84no—
507C22—164no—
508C22—164no—
509C22C2×#35414yescomplemented
510C22D4×C22⋊C411yescharacteristic
511C22C22×#9911yescharacteristic
512C22#288311yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—82no—
523C4—84no—
524C4—81no—
525C4—82no—
526C4—162no—
527C4—164no—
528C4C2×#35412yescomplemented
529C2—21no—
530C2—21no—
531C2—24no—
532C2—42no—
533C2—42no—
534C2—42no—
535C2—84no—
536C2D4×#9911yescharacteristic
537C2C22×#35411yescharacteristic
538C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×D42 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#354C2semidirect
4. #13224C2semidirect
5. #13090C2semidirect
6. #12798C2semidirect
7. #12694C2semidirect
8. C4×#354C2semidirect
9. C2×D4×Q8C4semidirect
13. #2858C4semidirect
14. #2850C4semidirect
15. #2548C4semidirect
19. #2436C4semidirect
24. C42×D4C4semidirect
38. C23×Q8C2×C4semidirect
48. C2×C4⋊Q8C2×C4semidirect
54. C2×#124C2×C4semidirect
55. C2×#124C2×C4semidirect
56. C2×#124C2×C4semidirect
60. C2×C22⋊Q8C2×C4semidirect
64. C2×C4×Q8C2×C4semidirect
72. C22×C42C2×C4semidirect
84. #391C2×C4semidirect
85. #391C2×C4semidirect
90. #390C2×C4semidirect
97. #382C2×C4semidirect
103. C4×C4⋊C4C2×C4semidirect
107. C4×C22⋊C4C2×C4semidirect
108. C43C2×C4semidirect
113. #354D4direct (N × H)
269. C2×Q8C4×D4semidirect
384. C42C4×D4semidirect
433. D4#354direct (N × H)
509. C22C2×#354semidirect
528. C4C2×#354semidirect

Maximal subgroups

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

subgroupindex
D4×#1242
C22×#3542
D4×#992
#132242
#130902
#127982
#126942
C4×#3542