Finite Group Databasebeta

← C4×#3358 #6384497

Subgroups — C4×#3358 #6384497

Order 512.

2410 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3358C211yesmaximal, characteristic
2C4×D4×Q8C211yesmaximal, characteristic, complemented
3C4×#550C211yesmaximal, characteristic, complemented
4#23216C213yesmaximal, complemented
5#23060C213yesmaximal, complemented
6#22995C213yesmaximal, complemented
7#22766C213yesmaximal, complemented
8#22740C211yesmaximal, characteristic, complemented
9#22540C211yesmaximal, characteristic, complemented
10Q8×#102C211yesmaximal, characteristic, complemented
11C4×#486C213yesmaximal, complemented
12C4×#474C213yesmaximal, complemented
13C4×#459C213yesmaximal, complemented
14C4×C8×Q8C211yesmaximal, characteristic, complemented
15C42×D8C213yesmaximal, complemented
16C2×#550C2211yescharacteristic
17C2×D4×Q8C2211yescharacteristic
18C2×C4×#41—21no—
19C22×C4×Q8—21no—
20#3358C4116yesdirect product factor
21C2×#486C2213yes—
22C2×#474C2213yes—
23C2×#459C2213yes—
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C4×D8C2213yes—
27C2×C4×D8C2213yes—
28C2×C4×D8C2213yes—
29#2366C2213yes—
30#2365C2213yes—
31#2363—23no—
32#2360—23no—
33#2354C2213yes—
34Q8×C4⋊C4C2211yescharacteristic
35Q8×C4⋊C4C2211yescharacteristic
36D4×C4⋊C4C2213yes—
37#2341—21no—
38Q8×C22⋊C4—21no—
39C4×C4⋊Q8C2213yes—
40C4×#127C2213yes—
41C4×C22⋊Q8—23no—
42C4×#121—23no—
43C42×Q8C2211yescharacteristic
44C42×D4C2213yes—
45C42×D4C2213yes—
46#1990C2213yes—
47#1988C2213yes—
48#1982C2213yes—
49#1957C2213yes—
50#1947C2213yes—
51#1947C2213yes—
52#1934C2216yes—
53#1934C2216yes—
54#1925C2216yes—
55#1918C2216yes—
56#1879C2216yes—
57#1862C2216yes—
58#1834C2213yes—
59#1802C2213yes—
60#1802C2213yes—
61C4×#107C2213yes—
62C4×#107C2213yes—
63C4×#107C2213yes—
64C8×C4⋊C4C2213yes—
65C8×C4⋊C4C2213yes—
66C4×C4⋊C8C2213yes—
67C4×#102C2213yes—
68C4×#102C2211yescharacteristic
69C4×#102C2213yes—
70C4×#102C2213yes—
71C4×#102C2213yes—
72C4×#102C2213yes—
73C4×#102C2216yes—
74C42×C8C2213yes—
75C22×#41—21no—
76C23×Q8—21no—
77C22×D8C2313yes—
78#550—24no—
79#550C2×C418yescomplemented
80D4×Q8—24no—
81D4×Q8C2×C418yescomplemented
82C2×C4⋊Q8C2313yes—
83C2×#127C2313yes—
84C2×C22⋊Q8—23no—
85C2×#121—23no—
86C4×#41—48no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—23no—
90C2×C4×Q8—23no—
91C2×C4×Q8—48no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4—23no—
96C2×C4×D4—23no—
97C2×C4×D4—23no—
98C2×C4×D4C2313yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×C4×D4C2313yes—
102C2×C4×D4C2313yes—
103C2×C4×D4C2311yescharacteristic
104C2×#117—23no—
105C22×C4⋊C4—23no—
106C22×C42—23no—
107C22×C42—23no—
108#486C2×C4124yescomplemented
109#474C2×C4124yescomplemented
110#459C2×C4124yescomplemented
111C8×Q8—24no—
112C8×Q8C2×C418yescomplemented
113C4×D8—212no—
114C4×D8—212no—
115C4×D8C2×C4124yescomplemented
116C2×#107C2313yes—
117C2×#107C2313yes—
118C2×#107C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2313yes—
121C2×#102C2313yes—
122C2×#102C2313yes—
123C2×#102C2313yes—
124C2×#102C2316yes—
125C2×#102C2313yes—
126C2×#102C2316yes—
127C2×#102C2311yescharacteristic
128C2×#102C2311yescharacteristic
129C2×C4×C8C2313yes—
130C2×C4×C8C2313yes—
131C2×C4×C8C2313yes—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2313yes—
134C2×C4×C8C2313yes—
135#391—23no—
136#391—23no—
137#390—23no—
138#390—23no—
139#390—23no—
140#390C2316yes—
141#389—23no—
142#389—23no—
143#389—23no—
144#389C2316yes—
145#388—26no—
146#387—26no—
147#386—26no—
148#385—26no—
149#384C2313yes—
150#384C2313yes—
151#380—23no—
152#380—23no—
153C4×C4⋊C4—23no—
154C4×C4⋊C4D413yes—
155C4×C4⋊C4D413yes—
156C4×C4⋊C4C2313yes—
157C4×C4⋊C4C2313yes—
158C4×C4⋊C4C2313yes—
159C4×C4⋊C4C2313yes—
160C4×C4⋊C4C2313yes—
161C4×C4⋊C4C2311yescharacteristic
162C4×C4⋊C4C2313yes—
163C4×C22⋊C4—23no—
164C4×C22⋊C4—23no—
165C4×C22⋊C4—23no—
166C4×C22⋊C4—23no—
167C4×C22⋊C4—23no—
168C4×C22⋊C4—23no—
169C43C2313yes—
170#340C2313yes—
171#340C2316yes—
172#340C2313yes—
173#340C2316yes—
174#340C2313yes—
175#340C2313yes—
176#340C2311yescharacteristic
177#340C2311yescharacteristic
178#336C2316yes—
179C2×#41—28no—
180C2×#41—44no—
181C2×#41—44no—
182C22×Q8—28no—
183C22×Q8—23no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8C2×D411yescharacteristic
187C22×D4—23no—
188C22×D4—23no—
189C22×D4C2413yes—
190C22×D4C2411yescharacteristic
191C23×C4—23no—
192C23×C4—23no—
193C23×C4—21no—
194C23×C4—23no—
195C23×C4—23no—
196C23×C4—23no—
197C2×D8—26no—
198C2×D8—26no—
199C2×D8C22×C4112yes—
200C22×C8C2413yes—
201C22×C8C2413yes—
202C22×C8C2411yescharacteristic
203C22×C8C2411yescharacteristic
204C4⋊Q8—26no—
205C4⋊Q8C22×C4112yes—
206#127—26no—
207#127C22×C4112yes—
208C22⋊Q8—224no—
209#121—224no—
210C4×Q8—24no—
211C4×Q8—44no—
212C4×Q8—44no—
213C4×Q8—412no—
214C4×Q8—412no—
215C4×Q8D818yescomplemented
216C4×Q8C22×C414yes—
217C4×Q8#4114yes—
218C4×D4—26no—
219C4×D4—26no—
220C4×D4—26no—
221C4×D4—26no—
222C4×D4—26no—
223C4×D4—22no—
224C4×D4—26no—
225C4×D4—43no—
226C4×D4—43no—
227C4×D4—412no—
228C4×D4—412no—
229C4×D4—86no—
230C4×D4C22×C4112yes—
231C4×D4C22×C4112yes—
232C4×D4#4114yescomplemented
233#117—412no—
234C2×C4⋊C4—23no—
235C2×C4⋊C4—23no—
236C2×C4⋊C4—26no—
237C2×C4⋊C4—23no—
238C2×C4⋊C4—23no—
239C2×C4⋊C4—23no—
240C2×C4⋊C4—412no—
241C2×C4⋊C4C2×D413yes—
242C2×C4⋊C4C2×D413yes—
243C2×C4⋊C4C2×D413yes—
244C2×C4⋊C4C2413yes—
245C2×C4⋊C4C2413yes—
246C2×C4⋊C4C2411yescharacteristic
247C2×C4⋊C4C2413yes—
248C2×C4⋊C4C2411yescharacteristic
249C2×C4⋊C4C2411yescharacteristic
250C2×C22⋊C4—23no—
251C2×C22⋊C4—23no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—26no—
254C2×C22⋊C4—23no—
255C2×C22⋊C4—23no—
256C2×C22⋊C4—23no—
257C2×C22⋊C4—26no—
258C2×C42—23no—
259C2×C42—23no—
260C2×C42—23no—
261C2×C42—23no—
262C2×C42—23no—
263C2×C42—21no—
264C2×C42—412no—
265C2×C42—412no—
266C2×C42C2×D413yes—
267C2×C42C2×D413yes—
268C2×C42C2×D413yes—
269C2×C42C2×D413yes—
270C2×C42C2413yes—
271C2×C42C2413yes—
272C2×C42C2411yescharacteristic
273#107—26no—
274#107—26no—
275#107C22×C4112yes—
276C4⋊C8—26no—
277C4⋊C8C22×C4112yes—
278#102—26no—
279#102—26no—
280#102—26no—
281#102—46no—
282#102C22×C414yes—
283#102C22×C4112yes—
284#102C22×C4124yes—
285#102#4114yescomplemented
286C4×C8—26no—
287C4×C8—26no—
288C4×C8—26no—
289C4×C8—22no—
290C4×C8—43no—
291C4×C8C22×C4112yes—
292#95—21no—
293#95—23no—
294#95—23no—
295#95—23no—
296#95—23no—
297#95—23no—
298#95—26no—
299#95C2×D416yes—
300C24—21no—
301C24—23no—
302#41—432no—
303C2×Q8—212no—
304C2×Q8—432no—
305C2×Q8—46no—
306C2×Q8—46no—
307C2×Q8C4×D414yes—
308C2×Q8C2×D814yes—
309C2×Q8#13514yes—
310C2×D4—212no—
311C2×D4—212no—
312C2×D4—21no—
313C2×D4—23no—
314C2×D4—23no—
315C2×D4—23no—
316C2×D4—21no—
317C2×D4—23no—
318C2×D4—21no—
319C2×D4—23no—
320C2×D4—46no—
321C2×D4—46no—
322C2×D4—83no—
323C2×D4—83no—
324C2×D4C23×C416yes—
325C2×D4C23×C412yes—
326C2×D4C2×#4112yes—
327C2×D4C2×#4112yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—23no—
332C22×C4—212no—
333C22×C4—212no—
334C22×C4—24no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—46no—
338C22×C4—46no—
339C22×C4—42no—
340C22×C4—46no—
341C22×C4—46no—
342C22×C4—46no—
343C22×C4—46no—
344C22×C4—42no—
345C22×C4—46no—
346C22×C4—46no—
347C22×C4—46no—
348C22×C4—46no—
349C22×C4—46no—
350C22×C4—83no—
351C22×C4—83no—
352C22×C4C22×D413yes—
353C22×C4C22×D413yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D413yes—
356C22×C4C22×D413yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C2511yescharacteristic
359D8—224no—
360C2×C8—21no—
361C2×C8—23no—
362C2×C8—23no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—23no—
367C2×C8—23no—
368C2×C8—23no—
369C2×C8C23×C412yes—
370C2×C8C23×C416yes—
371C2×C8C2×#4112yes—
372C4⋊C4—212no—
373C4⋊C4—26no—
374C4⋊C4—21no—
375C4⋊C4—23no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—23no—
379C4⋊C4—23no—
380C4⋊C4—23no—
381C4⋊C4—23no—
382C4⋊C4—412no—
383C4⋊C4—46no—
384C4⋊C4—46no—
385C4⋊C4—83no—
386C4⋊C4—83no—
387C4⋊C4C4×D4112yes—
388C4⋊C4C2×D8112yes—
389C4⋊C4C23×C416yes—
390C4⋊C4C23×C412yes—
391C4⋊C4C2×#4112yes—
392C4⋊C4C2×#4112yes—
393C22⋊C4—212no—
394C22⋊C4—212no—
395C22⋊C4—24no—
396C22⋊C4—46no—
397C22⋊C4—46no—
398C22⋊C4—46no—
399C22⋊C4—412no—
400C22⋊C4—86no—
401C42—26no—
402C42—26no—
403C42—23no—
404C42—23no—
405C42—43no—
406C42—42no—
407C42—46no—
408C42—46no—
409C42—46no—
410C42—46no—
411C42—46no—
412C42C2×D8112yes—
413C42C23×C416yes—
414C42C2×#4116yes—
415C42C2×#4112yes—
416C422−1+412yes—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—26no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—43no—
426C23—43no—
427C23—43no—
428C23—43no—
429C23—43no—
430C23—43no—
431C23C23×D411yescharacteristic
432Q8—424no—
433Q8C4×D8116yescomplemented
434D4—26no—
435D4—22no—
436D4—26no—
437D4—22no—
438D4—22no—
439D4—424no—
440D4—43no—
441D4—43no—
442D4—812no—
443D4C4×#4114yescomplemented
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—21no—
449C2×C4—23no—
450C2×C4—22no—
451C2×C4—26no—
452C2×C4—22no—
453C2×C4—23no—
454C2×C4—22no—
455C2×C4—23no—
456C2×C4—23no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—43no—
462C2×C4—41no—
463C2×C4—43no—
464C2×C4—41no—
465C2×C4—48no—
466C2×C4—424no—
467C2×C4—424no—
468C2×C4—43no—
469C2×C4—43no—
470C2×C4—43no—
471C2×C4—43no—
472C2×C4—43no—
473C2×C4—43no—
474C2×C4—812no—
475C2×C4—812no—
476C2×C4—812no—
477C2×C4C2×C4×D416yes—
478C2×C4C2×C4×D416yes—
479C2×C4#54812yes—
480C2×C4#54812yes—
481C2×C4C22×D816yes—
482C2×C4C22×D812yes—
483C2×C4C22×D816yes—
484C2×C4C2×#13516yes—
485C2×C4#57712yes—
486C2×C4C24×C411yescharacteristic
487C2×C4C22×#4111yescharacteristic
488C2×C4C22×#4111yescharacteristic
489C2×C4C22×#4111yescharacteristic
490C2×C4C2×2−1+411yescharacteristic
491C2×C4#58511yescharacteristic
492C8—26no—
493C8—22no—
494C8—22no—
495C8—43no—
496C22—24no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—46no—
504C22—46no—
505C22—46no—
506C22—83no—
507C22—83no—
508C22—83no—
509C22—83no—
510C22C22×C4×D411yescharacteristic
511C22C2×#54811yescharacteristic
512C22D4×#4111yescharacteristic
513C22C23×D811yescharacteristic
514C22C22×#13511yescharacteristic
515C22C2×#57711yescharacteristic
516C22#358411yescharacteristic
517C4—26no—
518C4—22no—
519C4—26no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—46no—
524C4—43no—
525C4—83no—
526C4—83no—
527C4C2×C4×D8112yes—
528C4#335814yesdirect product factor
529C4C2×C4×#4111yescharacteristic
530C4C4×2−1+411yescharacteristic
531C4#348011yescharacteristic
532C4#348211yescharacteristic
533C2—44no—
534C2—86no—
535C2C4×#54811yescharacteristic
536C2C22×C4×D811yescharacteristic
537C2C4×#57711yescharacteristic
538C2C2×#335811yescharacteristic
539C2#6130311yescharacteristic
540C2D8×#4111yescharacteristic
541C2#6143411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#3358C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×D4×Q8C2semidirect
3. C4×#550C2semidirect
4. #23216C2semidirect
5. #23060C2semidirect
6. #22995C2semidirect
7. #22766C2semidirect
8. #22740C2semidirect
9. #22540C2semidirect
10. Q8×#102C2semidirect
11. C4×#486C2semidirect
12. C4×#474C2semidirect
13. C4×#459C2semidirect
14. C4×C8×Q8C2semidirect
15. C42×D8C2semidirect
20. #3358C4direct (N × H)
79. #550C2×C4semidirect
81. D4×Q8C2×C4semidirect
108. #486C2×C4semidirect
109. #474C2×C4semidirect
110. #459C2×C4semidirect
112. C8×Q8C2×C4semidirect
115. C4×D8C2×C4semidirect
215. C4×Q8D8semidirect
232. C4×D4#41semidirect
285. #102#41semidirect
433. Q8C4×D8semidirect
443. D4C4×#41semidirect
528. C4#3358direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33582
C4×D4×Q82
C4×#5502
#232162
#230602
#229952
#227662
#227402
#225402
Q8×#1022
C4×#4862
C4×#4742
C4×#4592
C4×C8×Q82
C42×D82