Finite Group Databasebeta

← #6400682

Subgroups — #6400682

Order 512.

2332 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2988C211yesmaximal, characteristic, complemented
2C22×C8×Q8C211yesmaximal, characteristic, complemented
3C4×#548C211yesmaximal, characteristic
4#23204C213yesmaximal, complemented
5#22985C213yesmaximal, complemented
6#22952C213yesmaximal, complemented
7Q8×C4⋊C8C211yesmaximal, characteristic, complemented
8D4×C4⋊C8C213yesmaximal, complemented
9#22530C211yesmaximal, characteristic, complemented
10Q8×C22⋊C8C211yesmaximal, characteristic, complemented
11#22208C213yesmaximal, complemented
12#22098C213yesmaximal, complemented
13C8×#121C213yesmaximal, complemented
14#21101C213yesmaximal, complemented
15#20569C211yesmaximal, characteristic, complemented
16C2×#548C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2988—28no—
20C2×C8×Q8—28no—
21C2×C8×Q8C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23C2×C8×Q8C2213yes—
24C2×C8×Q8C2213yes—
25C2×C8×Q8C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×C8×D4C2213yes—
28C2×C8×D4C2213yes—
29C2×C8×D4C2216yes—
30C2×#431C2213yes—
31C22×C4⋊C8C2213yes—
32C22×C4⋊C8C2213yes—
33C22×C4×C8C2213yes—
34#2367C413yes—
35#2362C413yes—
36#2361C413yes—
37#2352C413yes—
38#2351C411yescharacteristic
39#2341C411yescharacteristic
40Q8×C22⋊C4C411yescharacteristic
41C4×#124C2213yes—
42C4×C22⋊Q8C2213yes—
43C4×#121C2213yes—
44C42×Q8C2211yescharacteristic
45C42×D4C2213yes—
46#1984C2213yes—
47#1984C2213yes—
48#1954C2213yes—
49#1954C2213yes—
50#1939C2216yes—
51#1939C2213yes—
52#1939C2213yes—
53#1923C2216yes—
54#1923C2216yes—
55#1923C2213yes—
56#1923C2213yes—
57#1916C2213yes—
58#1916C2213yes—
59#1869C2213yes—
60#1869C2213yes—
61#1850C2211yescharacteristic
62#1850C2213yes—
63#1850C2216yes—
64#1850C2216yes—
65#1842C2213yes—
66#1797C2213yes—
67#1797C2213yes—
68C8×C4⋊C4C2213yes—
69C8×C4⋊C4C2213yes—
70C4×C4⋊C8C2213yes—
71C4×C4⋊C8C2213yes—
72C8×C22⋊C4C2216yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75#1744C2213yes—
76C22×#41C2×C411yescharacteristic
77C23×Q8C2×C411yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2313yes—
80C23×C8C2316yes—
81#548C8116yescomplemented
82C2×#124C2×C413yes—
83C2×C22⋊Q8C2×C413yes—
84C2×#121C2×C413yes—
85C4×#41—24no—
86C4×#41—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C413yes—
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2313yes—
95C2×C4×Q8C2311yescharacteristic
96C2×C4×D4C2×C413yes—
97C2×C4×D4C2×C413yes—
98C2×C4×D4C2×C413yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×#117C2×C413yes—
102C22×C4⋊C4C2×C413yes—
103C22×C42C2313yes—
104C22×C42C2313yes—
105C8×Q8—24no—
106C8×Q8—24no—
107C8×Q8—24no—
108C8×Q8—212no—
109C8×Q8—212no—
110C8×Q8D418yescomplemented
111C8×D4—212no—
112C8×D4—212no—
113C8×D4—412no—
114#431—212no—
115C2×C4⋊C8—212no—
116C2×C4⋊C8—212no—
117C2×C4⋊C8C2313yes—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2316yes—
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2313yes—
125C2×C4⋊C8C2316yes—
126C2×C4⋊C8C2313yes—
127C2×C22⋊C8C2313yes—
128C2×C22⋊C8C2311yescharacteristic
129C2×C22⋊C8C2313yes—
130C2×C22⋊C8C2313yes—
131C2×C22⋊C8C2313yes—
132C2×C22⋊C8C2316yes—
133C2×C22⋊C8C2313yes—
134C2×C4×C8—212no—
135C2×C4×C8C2313yes—
136C2×C4×C8C2313yes—
137C2×C4×C8C2313yes—
138C2×C4×C8C2313yes—
139C2×C4×C8C2313yes—
140C2×C4×C8C2316yes—
141C2×C4×C8C2313yes—
142#391C2×C413yes—
143#391C2×C413yes—
144#390C2×C413yes—
145#390C2×C413yes—
146#389C2×C413yes—
147#389C2×C413yes—
148#388C2×C416yes—
149#388C2×C416yes—
150#388C2×C416yes—
151#387C2×C416yes—
152#386C2×C416yes—
153#385C2×C416yes—
154#382C2×C413yes—
155#380C2×C413yes—
156#380C2×C413yes—
157C4×C4⋊C4C2×C413yes—
158C4×C4⋊C4C2×C413yes—
159C4×C4⋊C4C2×C413yes—
160C4×C4⋊C4C2×C413yes—
161C4×C4⋊C4C2313yes—
162C4×C4⋊C4C2313yes—
163C4×C22⋊C4C2×C413yes—
164C4×C22⋊C4C2×C413yes—
165C4×C22⋊C4C2×C413yes—
166C4×C22⋊C4C2×C413yes—
167C4×C22⋊C4C2311yescharacteristic
168C4×C22⋊C4C2313yes—
169C4×C22⋊C4C2316yes—
170C43C2313yes—
171#336C2313yes—
172#336C2316yes—
173#336C2313yes—
174#336C2313yes—
175#336C2313yes—
176#336C2311yescharacteristic
177#336C2311yescharacteristic
178#336C2313yes—
179#336C2313yes—
180#336C2313yes—
181#336C2313yes—
182C2×#41—24no—
183C2×#41—24no—
184C2×#41C2×C818yescomplemented
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8C2×C818yescomplemented
188C22×Q8C22×C413yes—
189C22×Q8C22×C411yescharacteristic
190C22×D4C22×C413yes—
191C22×D4C22×C413yes—
192C23×C4C22×C413yes—
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C22×C413yes—
196C23×C4C2413yes—
197C23×C4C2411yescharacteristic
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—26no—
202C22×C8—26no—
203C22×C8—212no—
204C22×C8—43no—
205C22×C8—46no—
206C22×C8—46no—
207C22×C8C2413yes—
208C22×C8C2413yes—
209C22×C8C2411yescharacteristic
210C22×C8C2411yescharacteristic
211C22×C8C2413yes—
212C22×C8C2413yes—
213C22×C8C2411yescharacteristic
214C22×C8C2411yescharacteristic
215#124C2×C8124yescomplemented
216C22⋊Q8C2×C8124yescomplemented
217#121C2×C8124yescomplemented
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—212no—
222C4×Q8—26no—
223C4×Q8—26no—
224C4×Q8C2×C818yescomplemented
225C4×Q8C2×D414yes—
226C4×Q8#4114yes—
227C4×D4—26no—
228C4×D4—26no—
229C4×D4—212no—
230C4×D4—26no—
231C4×D4—26no—
232C4×D4—46no—
233C4×D4C2×C8124yescomplemented
234#117—212no—
235C2×C4⋊C4—212no—
236C2×C4⋊C4C22×C413yes—
237C2×C4⋊C4C22×C416yes—
238C2×C4⋊C4C22×C413yes—
239C2×C4⋊C4C22×C413yes—
240C2×C4⋊C4C22×C413yes—
241C2×C4⋊C4C22×C413yes—
242C2×C4⋊C4C22×C413yes—
243C2×C4⋊C4C22×C413yes—
244C2×C4⋊C4C22×C413yes—
245C2×C22⋊C4C22×C416yes—
246C2×C22⋊C4C22×C411yescharacteristic
247C2×C22⋊C4C22×C413yes—
248C2×C22⋊C4C22×C413yes—
249C2×C22⋊C4C22×C413yes—
250C2×C22⋊C4C22×C416yes—
251C2×C22⋊C4C22×C411yescharacteristic
252C2×C22⋊C4C22×C413yes—
253C2×C22⋊C4C22×C416yes—
254C2×C42—26no—
255C2×C42—26no—
256C2×C42—26no—
257C2×C42—26no—
258C2×C42C22×C413yes—
259C2×C42C22×C413yes—
260C2×C42C22×C413yes—
261C2×C42C22×C413yes—
262C2×C42C22×C413yes—
263C2×C42C22×C413yes—
264C2×C42C22×C413yes—
265C2×C42C22×C413yes—
266C2×C42C2413yes—
267C2×C42C2413yes—
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2413yes—
271C4⋊C8—26no—
272C4⋊C8—212no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—26no—
276C4⋊C8—26no—
277C4⋊C8—26no—
278C4⋊C8—43no—
279C4⋊C8—46no—
280C4⋊C8C2×D4112yes—
281C22⋊C8—26no—
282C22⋊C8—22no—
283C22⋊C8—26no—
284C22⋊C8—26no—
285C22⋊C8—26no—
286C22⋊C8—43no—
287C22⋊C8—46no—
288C4×C8—26no—
289C4×C8—26no—
290C4×C8—26no—
291C4×C8—26no—
292C4×C8—26no—
293C4×C8—46no—
294C4×C8C2×D4112yes—
295#95C22×C411yescharacteristic
296#95C22×C411yescharacteristic
297#95C22×C413yes—
298#95C22×C413yes—
299#95C22×C413yes—
300#95C22×C413yes—
301#95C22×C416yes—
302#95C22×C416yes—
303#95C22×C416yes—
304C24C23×C413yes—
305C24C23×C411yescharacteristic
306#41—232no—
307C2×Q8—232no—
308C2×Q8—26no—
309C2×Q8—26no—
310C2×Q8C4×D414yes—
311C2×Q8C22×C8112yes—
312C2×Q8C22×C814yes—
313C2×Q8#13114yes—
314C2×D4—26no—
315C2×D4—26no—
316C2×D4—26no—
317C2×D4—26no—
318C2×D4C22×C8112yes—
319C2×D4C22×C8112yes—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—26no—
323C22×C4—26no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—26no—
329C22×C4—26no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—23no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—26no—
337C22×C4—43no—
338C22×C4—43no—
339C22×C4—43no—
340C22×C4C22×C8112yes—
341C22×C4C22×C8112yes—
342C22×C4C23×C413yes—
343C22×C4C23×C413yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C23×C411yescharacteristic
346C22×C4C23×C413yes—
347C22×C4C23×C413yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C23×C411yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C23×C413yes—
352C22×C4C23×C413yes—
353C22×C4C23×C413yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C42−1+412yes—
356C22×C4C2511yescharacteristic
357C2×C8—28no—
358C2×C8—21no—
359C2×C8—23no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—23no—
364C2×C8—21no—
365C2×C8—23no—
366C2×C8—23no—
367C2×C8—21no—
368C2×C8—23no—
369C2×C8—23no—
370C2×C8—23no—
371C2×C8—23no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—23no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—23no—
378C2×C8—23no—
379C2×C8—412no—
380C2×C8—812no—
381C2×C8C22×D412yes—
382C2×C8C22×D416yes—
383C2×C82−1+412yes—
384C4⋊C4—212no—
385C4⋊C4—26no—
386C4⋊C4—26no—
387C4⋊C4—26no—
388C4⋊C4—26no—
389C4⋊C4—43no—
390C4⋊C4C4×D4112yes—
391C4⋊C4C22×C8112yes—
392C4⋊C4C22×C8112yes—
393C22⋊C4—26no—
394C22⋊C4—22no—
395C22⋊C4—212no—
396C22⋊C4—26no—
397C22⋊C4—46no—
398C22⋊C4—43no—
399C22⋊C4C22×C814yes—
400C22⋊C4C22×C8112yes—
401C22⋊C4C22×C8124yes—
402C42—26no—
403C42—26no—
404C42—26no—
405C42—26no—
406C42—26no—
407C42—23no—
408C42—21no—
409C42—21no—
410C42—23no—
411C42—23no—
412C42—21no—
413C42—23no—
414C42—21no—
415C42—43no—
416C42—43no—
417C42C22×C8112yes—
418C42C22×D416yes—
419C42C2×#4116yes—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—23no—
425C23—23no—
426C23—23no—
427C23—23no—
428C23—23no—
429C23—23no—
430C23#52012yes—
431C23C23×C816yes—
432C23C23×C812yes—
433C23#56812yes—
434C23C24×C411yescharacteristic
435Q8—224no—
436Q8C8×D4116yescomplemented
437D4—224no—
438D4—224no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—23no—
442C2×C4—23no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—21no—
450C2×C4—23no—
451C2×C4—224no—
452C2×C4—224no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—21no—
463C2×C4—23no—
464C2×C4—21no—
465C2×C4—23no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—23no—
471C2×C4—21no—
472C2×C4—412no—
473C2×C4—43no—
474C2×C4—43no—
475C2×C4—43no—
476C2×C4—43no—
477C2×C4—86no—
478C2×C4C2×C4×D416yes—
479C2×C4C2×C4×D416yes—
480C2×C4#52012yes—
481C2×C4C23×C812yes—
482C2×C4C23×C816yes—
483C2×C4C23×C816yes—
484C2×C4C23×C812yes—
485C2×C4C23×C816yes—
486C2×C4C2×#13116yes—
487C2×C4C23×D411yescharacteristic
488C2×C4C22×#4111yescharacteristic
489C2×C4C2×H(5,𝔽2)11yescharacteristic
490C2×C4C2×2−1+411yescharacteristic
491C2×C4C2×2−1+411yescharacteristic
492C2×C4#58511yescharacteristic
493C8—22no—
494C8—22no—
495C8—26no—
496C8—41no—
497C8—43no—
498C8—43no—
499C8—41no—
500C8—43no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—26no—
508C22—26no—
509C22—26no—
510C22—43no—
511C22—43no—
512C22—43no—
513C22—43no—
514C22#298814yescomplemented
515C22C22×C4×D411yescharacteristic
516C22C2×#52011yescharacteristic
517C22C4×2−1+411yescharacteristic
518C22C24×C811yescharacteristic
519C22C22×#13111yescharacteristic
520C22C2×#56811yescharacteristic
521C22#357311yescharacteristic
522C4—26no—
523C4—22no—
524C4—22no—
525C4—26no—
526C4—26no—
527C4—22no—
528C4—43no—
529C4—41no—
530C4—43no—
531C4—41no—
532C4—43no—
533C4C2×C8×D4112yes—
534C4#348611yescharacteristic
535C4#348811yescharacteristic
536C4#351611yescharacteristic
537C4#352211yescharacteristic
538C2—24no—
539C2—46no—
540C2#3429411yescharacteristic
541C2C22×C8×D411yescharacteristic
542C2#6014611yescharacteristic
543C2C2×#298811yescharacteristic
544C2#6027411yescharacteristic
545C2C8×2−1+411yescharacteristic
546C2#6033511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutatorC22C24×C8
socleC23C24×C4
prime radicalC2×#548C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2988C2semidirect
2. C22×C8×Q8C2semidirect
4. #23204C2semidirect
5. #22985C2semidirect
6. #22952C2semidirect
7. Q8×C4⋊C8C2semidirect
8. D4×C4⋊C8C2semidirect
9. #22530C2semidirect
10. Q8×C22⋊C8C2semidirect
11. #22208C2semidirect
12. #22098C2semidirect
13. C8×#121C2semidirect
14. #21101C2semidirect
15. #20569C2semidirect
81. #548C8semidirect
110. C8×Q8D4semidirect
184. C2×#41C2×C8semidirect
187. C22×Q8C2×C8semidirect
215. #124C2×C8semidirect
216. C22⋊Q8C2×C8semidirect
217. #121C2×C8semidirect
224. C4×Q8C2×C8semidirect
233. C4×D4C2×C8semidirect
436. Q8C8×D4semidirect
514. C22#2988semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
C22×C8×Q82
C4×#5482
#232042
#229852
#229522
Q8×C4⋊C82
D4×C4⋊C82
#225302
Q8×C22⋊C82
#222082
#220982
C8×#1212
#211012
#205692