Finite Group Databasebeta

← #6404356

Subgroups — #6404356

Order 512.

2043 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×#2991C211yesmaximal, characteristic, complemented
2C2×#2984C211yesmaximal, characteristic, complemented
3C4×#548C211yesmaximal, characteristic
4#23208C213yesmaximal, complemented
5#23088C213yesmaximal, complemented
6#22957C213yesmaximal, complemented
7#22751C213yesmaximal, complemented
8#22734C211yesmaximal, characteristic, complemented
9#22530C211yesmaximal, characteristic, complemented
10Q8×C22⋊C8C211yesmaximal, characteristic, complemented
11#22137C213yesmaximal, complemented
12#22102C213yesmaximal, complemented
13#21631C213yesmaximal, complemented
14#21264C213yesmaximal, complemented
15C4×C8×Q8C211yesmaximal, characteristic, complemented
16C2×#548C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2991—28no—
20#2984—28no—
21C2×#446C2213yes—
22C2×#446C2213yes—
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×#436C2213yes—
28C2×#436C2213yes—
29C2×#432C2213yes—
30C2×#431C2213yes—
31C2×#431C2213yes—
32C2×#423C2213yes—
33#2367C413yes—
34#2362C413yes—
35#2361C413yes—
36#2352C413yes—
37#2351C411yescharacteristic
38#2341C411yescharacteristic
39Q8×C22⋊C4C411yescharacteristic
40C4×#124C2213yes—
41C4×C22⋊Q8C2213yes—
42C4×#121C2213yes—
43C42×Q8C2211yescharacteristic
44C42×D4C2213yes—
45#1984C2213yes—
46#1984C2213yes—
47#1954C2213yes—
48#1954C2213yes—
49#1954C2213yes—
50#1954C2213yes—
51#1939C2213yes—
52#1939C2213yes—
53#1939C2213yes—
54#1939C2213yes—
55#1924C2216yes—
56#1923C2211yescharacteristic
57#1923C2213yes—
58#1923C2213yes—
59#1923C2213yes—
60#1923C2216yes—
61#1917C2216yes—
62#1870C2216yes—
63#1851C2216yes—
64#1831C2213yes—
65#1797C2213yes—
66#1797C2213yes—
67C8×C4⋊C4C2213yes—
68C8×C4⋊C4C2213yes—
69C4×C4⋊C8C2213yes—
70C4×C4⋊C8C2213yes—
71#1753C2216yes—
72#1753C2216yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75C42×C8C2213yes—
76C22×#41C2×C411yescharacteristic
77C23×Q8C2×C411yescharacteristic
78C22×M16C2316yes—
79#548—28no—
80C2×#124C2×C413yes—
81C2×C22⋊Q8C2×C413yes—
82C2×#121C2×C413yes—
83C4×#41—24no—
84C4×#41—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8C2×C413yes—
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2313yes—
93C2×C4×Q8C2311yescharacteristic
94C2×C4×D4C2×C413yes—
95C2×C4×D4C2×C413yes—
96C2×C4×D4C2×C413yes—
97C2×C4×D4C2313yes—
98C2×C4×D4C2313yes—
99C2×#117C2×C413yes—
100C22×C4⋊C4C2×C413yes—
101C22×C42C2313yes—
102C22×C42C2313yes—
103#446—212no—
104#446—212no—
105C8×Q8—24no—
106C8×Q8—24no—
107C8×Q8—24no—
108C8×Q8D418yescomplemented
109#436—212no—
110#436—212no—
111#432—212no—
112#431—212no—
113#431—212no—
114#423—212no—
115C2×C4⋊C8C2313yes—
116C2×C4⋊C8C2313yes—
117C2×C4⋊C8C2313yes—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2313yes—
123C2×C4⋊C8C2313yes—
124C2×C4⋊C8C2313yes—
125C2×C22⋊C8C2313yes—
126C2×C22⋊C8C2311yescharacteristic
127C2×C22⋊C8C2313yes—
128C2×C22⋊C8C2313yes—
129C2×C22⋊C8C2311yescharacteristic
130C2×C22⋊C8C2313yes—
131C2×C22⋊C8C2313yes—
132C2×C22⋊C8C2316yes—
133C2×C22⋊C8C2313yes—
134C2×#97C2316yes—
135C2×C4×C8C2313yes—
136C2×C4×C8C2313yes—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2313yes—
139C2×C4×C8C2313yes—
140C2×C4×C8C2313yes—
141C2×C4×C8C2313yes—
142C2×C4×C8C2313yes—
143#391C2×C413yes—
144#391C2×C413yes—
145#390C2×C413yes—
146#390C2×C413yes—
147#389C2×C413yes—
148#389C2×C413yes—
149#388C2×C416yes—
150#388C2×C416yes—
151#388C2×C416yes—
152#387C2×C416yes—
153#386C2×C416yes—
154#385C2×C416yes—
155#382C2×C413yes—
156#380C2×C413yes—
157#380C2×C413yes—
158C4×C4⋊C4C2×C413yes—
159C4×C4⋊C4C2×C413yes—
160C4×C4⋊C4C2×C413yes—
161C4×C4⋊C4C2×C413yes—
162C4×C4⋊C4C2313yes—
163C4×C4⋊C4C2313yes—
164C4×C22⋊C4C2×C413yes—
165C4×C22⋊C4C2×C413yes—
166C4×C22⋊C4C2×C413yes—
167C4×C22⋊C4C2×C413yes—
168C4×C22⋊C4C2316yes—
169C4×C22⋊C4C2311yescharacteristic
170C4×C22⋊C4C2313yes—
171C43C2313yes—
172#336C2313yes—
173#336C2316yes—
174#336C2313yes—
175#336C2316yes—
176#336C2316yes—
177#336C2311yescharacteristic
178#336C2313yes—
179#336C2311yescharacteristic
180#336C2313yes—
181C2×#41—24no—
182C2×#41—24no—
183C2×#41—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8C22×C413yes—
188C22×Q8C22×C411yescharacteristic
189C22×D4C22×C413yes—
190C22×D4C22×C413yes—
191C23×C4C22×C413yes—
192C23×C4C22×C413yes—
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C2413yes—
196C23×C4C2411yescharacteristic
197C2×M16—212no—
198C2×M16—46no—
199C2×M16—46no—
200C22×C8C2413yes—
201C22×C8C2413yes—
202C22×C8C2411yescharacteristic
203C22×C8C2413yes—
204C22×C8C2411yescharacteristic
205C22×C8C2413yes—
206C22×C8C2411yescharacteristic
207C22×C8C2411yescharacteristic
208#124—212no—
209C22⋊Q8—212no—
210#121—212no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—212no—
215C4×Q8—26no—
216C4×Q8—26no—
217C4×Q8M1618yescomplemented
218C4×Q8C2×D414yes—
219C4×Q8#4114yes—
220C4×D4—212no—
221C4×D4—212no—
222C4×D4—212no—
223C4×D4—26no—
224C4×D4—26no—
225C4×D4—43no—
226C4×D4—43no—
227#117—212no—
228C2×C4⋊C4—212no—
229C2×C4⋊C4C22×C413yes—
230C2×C4⋊C4C22×C416yes—
231C2×C4⋊C4C22×C413yes—
232C2×C4⋊C4C22×C413yes—
233C2×C4⋊C4C22×C413yes—
234C2×C4⋊C4C22×C413yes—
235C2×C4⋊C4C22×C413yes—
236C2×C4⋊C4C22×C413yes—
237C2×C4⋊C4C22×C413yes—
238C2×C22⋊C4C22×C416yes—
239C2×C22⋊C4C22×C411yescharacteristic
240C2×C22⋊C4C22×C413yes—
241C2×C22⋊C4C22×C413yes—
242C2×C22⋊C4C22×C416yes—
243C2×C22⋊C4C22×C413yes—
244C2×C22⋊C4C22×C413yes—
245C2×C22⋊C4C22×C416yes—
246C2×C22⋊C4C22×C411yescharacteristic
247C2×C42—26no—
248C2×C42—26no—
249C2×C42—26no—
250C2×C42—26no—
251C2×C42C22×C413yes—
252C2×C42C22×C413yes—
253C2×C42C22×C413yes—
254C2×C42C22×C413yes—
255C2×C42C22×C413yes—
256C2×C42C22×C413yes—
257C2×C42C22×C413yes—
258C2×C42C22×C413yes—
259C2×C42C2413yes—
260C2×C42C2413yes—
261C2×C42C2411yescharacteristic
262C2×C42C2411yescharacteristic
263C2×C42C2413yes—
264C4⋊C8—26no—
265C4⋊C8—22no—
266C4⋊C8—26no—
267C4⋊C8—26no—
268C4⋊C8—26no—
269C4⋊C8—26no—
270C4⋊C8—26no—
271C4⋊C8—43no—
272C4⋊C8—43no—
273C4⋊C8C2×D4112yes—
274C22⋊C8—22no—
275C22⋊C8—26no—
276C22⋊C8—26no—
277C22⋊C8—26no—
278C22⋊C8—22no—
279C22⋊C8—26no—
280C22⋊C8—43no—
281C22⋊C8—46no—
282C22⋊C8—43no—
283#97—212no—
284C4×C8—22no—
285C4×C8—26no—
286C4×C8—26no—
287C4×C8—26no—
288C4×C8—43no—
289C4×C8—43no—
290C4×C8—43no—
291C4×C8C2×D4112yes—
292#95C22×C411yescharacteristic
293#95C22×C411yescharacteristic
294#95C22×C413yes—
295#95C22×C413yes—
296#95C22×C413yes—
297#95C22×C413yes—
298#95C22×C416yes—
299#95C22×C416yes—
300#95C22×C416yes—
301C24C23×C413yes—
302C24C23×C411yescharacteristic
303#41—416no—
304C2×Q8—26no—
305C2×Q8—26no—
306C2×Q8—26no—
307C2×Q8—416no—
308C2×Q8C4×D414yes—
309C2×Q8C2×M1614yes—
310C2×Q8#13114yes—
311C2×D4—26no—
312C2×D4—26no—
313C2×D4—26no—
314C2×D4—26no—
315C2×D4—26no—
316C2×D4—43no—
317C22×C4—26no—
318C22×C4—26no—
319C22×C4—26no—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—26no—
323C22×C4—26no—
324C22×C4—26no—
325C22×C4—26no—
326C22×C4—26no—
327C22×C4—21no—
328C22×C4—23no—
329C22×C4—21no—
330C22×C4—23no—
331C22×C4—21no—
332C22×C4—23no—
333C22×C4—21no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—23no—
339C22×C4—43no—
340C22×C4—43no—
341C22×C4C23×C413yes—
342C22×C4C23×C413yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C23×C413yes—
345C22×C4C23×C413yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C23×C413yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C23×C413yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C23×C413yes—
353C22×C4C2511yescharacteristic
354M16—812no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—23no—
360C2×C8—23no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—23no—
364C2×C8—23no—
365C2×C8—23no—
366C2×C8—21no—
367C2×C8—23no—
368C2×C8—23no—
369C2×C8—23no—
370C2×C8—21no—
371C2×C8—23no—
372C2×C8—23no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—23no—
376C2×C8C22×D412yes—
377C2×C8C22×D416yes—
378C2×C82−1+412yes—
379C4⋊C4—26no—
380C4⋊C4—212no—
381C4⋊C4—26no—
382C4⋊C4—26no—
383C4⋊C4—26no—
384C4⋊C4—26no—
385C4⋊C4—43no—
386C4⋊C4C4×D4112yes—
387C4⋊C4C2×M16112yes—
388C22⋊C4—212no—
389C22⋊C4—22no—
390C22⋊C4—26no—
391C22⋊C4—212no—
392C22⋊C4—26no—
393C22⋊C4—46no—
394C22⋊C4—43no—
395C22⋊C4—43no—
396C22⋊C4#13114yes—
397C42—26no—
398C42—26no—
399C42—26no—
400C42—26no—
401C42—26no—
402C42—26no—
403C42—23no—
404C42—21no—
405C42—21no—
406C42—23no—
407C42—21no—
408C42—21no—
409C42—23no—
410C42—23no—
411C42—43no—
412C42C2×M16112yes—
413C42C22×D416yes—
414C42C2×#4116yes—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—23no—
422C23—23no—
423C23—23no—
424C23—23no—
425C23—23no—
426C23—23no—
427C23—23no—
428C23C2×#13112yes—
429C23C24×C411yescharacteristic
430Q8—412no—
431Q8#436116yes—
432D4—412no—
433D4—412no—
434C2×C4—21no—
435C2×C4—23no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—23no—
440C2×C4—23no—
441C2×C4—21no—
442C2×C4—23no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—21no—
451C2×C4—23no—
452C2×C4—21no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—21no—
456C2×C4—23no—
457C2×C4—23no—
458C2×C4—21no—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—412no—
463C2×C4—412no—
464C2×C4—412no—
465C2×C4—412no—
466C2×C4—41no—
467C2×C4—43no—
468C2×C4—43no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—43no—
472C2×C4—41no—
473C2×C4—43no—
474C2×C4C2×C4×D416yes—
475C2×C4C2×C4×D416yes—
476C2×C4C22×M1612yes—
477C2×C4C22×M1616yes—
478C2×C4C22×M1616yes—
479C2×C4C2×#13112yes—
480C2×C4C2×#13112yes—
481C2×C4C2×#13116yes—
482C2×C4C23×D411yescharacteristic
483C2×C4C22×#4111yescharacteristic
484C2×C4C2×2−1+411yescharacteristic
485C2×C4#58511yescharacteristic
486C2×C4#58511yescharacteristic
487C2×C4#58511yescharacteristic
488C8—22no—
489C8—26no—
490C8—22no—
491C8—41no—
492C8—43no—
493C8—41no—
494C8—43no—
495C8—43no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—43no—
504C22—43no—
505C22—43no—
506C22—43no—
507C22—43no—
508C22—43no—
509C22—43no—
510C22C22×C4×D411yescharacteristic
511C22#342811yescharacteristic
512C22C4×2−1+411yescharacteristic
513C22C23×M1611yescharacteristic
514C22C22×#13111yescharacteristic
515C22C22×#13111yescharacteristic
516C22#357311yescharacteristic
517C4—26no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—26no—
522C4—41no—
523C4—43no—
524C4—43no—
525C4—41no—
526C4—43no—
527C4—43no—
528C4C2×#436112yes—
529C4#349711yescharacteristic
530C4#350011yescharacteristic
531C4#351611yescharacteristic
532C4#352411yescharacteristic
533C2—42no—
534C2—83no—
535C2#3431211yescharacteristic
536C2C22×#43611yescharacteristic
537C2D4×#13111yescharacteristic
538C2#6027811yescharacteristic
539C2#6030111yescharacteristic
540C2#6033111yescharacteristic
541C2#6033311yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC2×#548C4
LCSG ▷ C23 ▷ 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×#2991C2semidirect
2. C2×#2984C2semidirect
4. #23208C2semidirect
5. #23088C2semidirect
6. #22957C2semidirect
7. #22751C2semidirect
8. #22734C2semidirect
9. #22530C2semidirect
10. Q8×C22⋊C8C2semidirect
11. #22137C2semidirect
12. #22102C2semidirect
13. #21631C2semidirect
14. #21264C2semidirect
15. C4×C8×Q8C2semidirect
108. C8×Q8D4semidirect
217. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29912
C2×#29842
C4×#5482
#232082
#230882
#229572
#227512
#227342
#225302
Q8×C22⋊C82
#221372
#221022
#216312
#212642
C4×C8×Q82