Finite Group Databasebeta

← #6400877

Subgroups — #6400877

Order 512.

2152 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2991C211yesmaximal, characteristic, complemented
2C2×#2971C211yesmaximal, characteristic, complemented
3C4×#550C211yesmaximal, characteristic
4#23176C213yesmaximal, complemented
5#23088C213yesmaximal, complemented
6#22990C213yesmaximal, complemented
7#22751C213yesmaximal, complemented
8#22734C211yesmaximal, characteristic, complemented
9#22530C211yesmaximal, characteristic, complemented
10#22530C211yesmaximal, characteristic, complemented
11#22264C213yesmaximal, complemented
12#22137C213yesmaximal, complemented
13#21488C213yesmaximal, complemented
14#21102C213yesmaximal, complemented
15#20569C211yesmaximal, characteristic, complemented
16C2×#550C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19#2991—28no—
20#2971—28no—
21C2×C8×Q8C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×#436C2213yes—
26C2×#436C2213yes—
27C2×#436C2213yes—
28C2×#436C2213yes—
29C2×#436C2216yes—
30C2×#432C2213yes—
31C2×#431C2213yes—
32C2×#423C2213yes—
33C2×C4×M16C2213yes—
34#2366C413yes—
35#2363C413yes—
36#2363C413yes—
37#2354C413yes—
38Q8×C4⋊C4C411yescharacteristic
39#2341C411yescharacteristic
40#2341C411yescharacteristic
41C4×#127C2213yes—
42C4×#121C2213yes—
43C4×#121C2213yes—
44C42×Q8C2211yescharacteristic
45C42×D4C2213yes—
46#1954C2213yes—
47#1954C2213yes—
48#1954C2213yes—
49#1954C2213yes—
50#1954C2216yes—
51#1939C2213yes—
52#1939C2213yes—
53#1924C2213yes—
54#1924C2213yes—
55#1923C2216yes—
56#1917C2211yescharacteristic
57#1917C2213yes—
58#1870C2216yes—
59#1870C2213yes—
60#1870C2213yes—
61#1870C2216yes—
62#1870C2216yes—
63#1851C2213yes—
64#1851C2213yes—
65#1831C2213yes—
66#1797C2213yes—
67#1797C2213yes—
68C8×C4⋊C4C2213yes—
69C8×C4⋊C4C2213yes—
70C4×C4⋊C8C2213yes—
71C4×C4⋊C8C2213yes—
72#1753C2216yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75#1744C2213yes—
76C22×#41C2×C411yescharacteristic
77C22×#41C2×C411yescharacteristic
78C22×M16C2313yes—
79C22×M16C2313yes—
80C22×M16C2316yes—
81#550—28no—
82C2×#127C2×C413yes—
83C2×#121C2×C413yes—
84C2×#121C2×C413yes—
85C4×#41—24no—
86C4×#41—24no—
87C4×#41—24no—
88C4×#41—24no—
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8C2311yescharacteristic
94C2×C4×D4C2×C413yes—
95C2×C4×D4C2×C413yes—
96C2×C4×D4C2×C413yes—
97C2×C4×D4C2×C413yes—
98C2×C4×D4C2313yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×C4×D4C2316yes—
102C2×#117C2×C413yes—
103C2×#117C2×C413yes—
104C22×C42C2313yes—
105C22×C42C2313yes—
106C8×Q8—24no—
107C8×Q8—24no—
108C8×Q8—24no—
109C8×Q8D418yescomplemented
110#436—212no—
111#436—212no—
112#436—212no—
113#436—212no—
114#436—412no—
115#432—212no—
116#431—212no—
117#423—212no—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2313yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2313yes—
123C2×C4⋊C8C2313yes—
124C2×C4⋊C8C2313yes—
125C2×C4⋊C8C2313yes—
126C2×C22⋊C8C2313yes—
127C2×C22⋊C8C2316yes—
128C2×C22⋊C8C2313yes—
129C2×C22⋊C8C2313yes—
130C2×C22⋊C8C2316yes—
131C2×C22⋊C8C2316yes—
132C2×C22⋊C8C2313yes—
133C2×C22⋊C8C2313yes—
134C2×C22⋊C8C2313yes—
135C4×M16—212no—
136C2×#97C2311yescharacteristic
137C2×#97C2313yes—
138C2×C4×C8C2313yes—
139C2×C4×C8C2313yes—
140C2×C4×C8C2313yes—
141C2×C4×C8C2316yes—
142C2×C4×C8C2313yes—
143#390C2×C413yes—
144#390C2×C413yes—
145#390C2×C416yes—
146#390C2×C416yes—
147#390C2×C413yes—
148#390C2×C413yes—
149#389C2×C413yes—
150#389C2×C413yes—
151#388C2×C416yes—
152#388C2×C416yes—
153#386C2×C416yes—
154#386C2×C416yes—
155#384C2×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⋊C4C2311yescharacteristic
163C4×C4⋊C4C2313yes—
164C4×C22⋊C4C2×C413yes—
165C4×C22⋊C4C2×C413yes—
166C4×C22⋊C4C2×C413yes—
167C4×C22⋊C4C2×C413yes—
168C4×C22⋊C4C2313yes—
169C4×C22⋊C4C2313yes—
170C43C2313yes—
171#336C2313yes—
172#336C2313yes—
173#336C2313yes—
174#336C2316yes—
175#336C2311yescharacteristic
176#336C2313yes—
177#336C2313yes—
178#336C2311yescharacteristic
179#336C2313yes—
180#336C2311yescharacteristic
181#336C2311yescharacteristic
182C2×#41—24no—
183C2×#41—24no—
184C2×#41—24no—
185C2×#41—24no—
186C2×#41—24no—
187C2×#41—24no—
188C22×Q8C22×C411yescharacteristic
189C22×D4C22×C413yes—
190C22×D4C22×C413yes—
191C22×D4C22×C413yes—
192C22×D4C22×C416yes—
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C22×C413yes—
196C23×C4C22×C413yes—
197C23×C4C2413yes—
198C23×C4C2413yes—
199C2×M16—26no—
200C2×M16—26no—
201C2×M16—212no—
202C2×M16—26no—
203C2×M16—43no—
204C2×M16—43no—
205C2×M16—43no—
206C2×M16—46no—
207C2×M16—46no—
208C22×C8C2413yes—
209C22×C8C2413yes—
210C22×C8C2411yescharacteristic
211C22×C8C2413yes—
212C22×C8C2411yescharacteristic
213C22×C8C2411yescharacteristic
214C22×C8C2413yes—
215C22×C8C2411yescharacteristic
216#127—212no—
217#121—212no—
218#121—212no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8M1618yescomplemented
223C4×Q8C2×D414yes—
224C4×Q8#4114yes—
225C4×D4—212no—
226C4×D4—212no—
227C4×D4—212no—
228C4×D4—26no—
229C4×D4—26no—
230C4×D4—26no—
231C4×D4—26no—
232C4×D4—46no—
233C4×D4—43no—
234C4×D4—43no—
235C4×D4—46no—
236C4×D4—46no—
237#117—212no—
238#117—212no—
239C2×C4⋊C4C22×C413yes—
240C2×C4⋊C4C22×C413yes—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C413yes—
244C2×C4⋊C4C22×C413yes—
245C2×C4⋊C4C22×C416yes—
246C2×C4⋊C4C22×C413yes—
247C2×C4⋊C4C22×C413yes—
248C2×C22⋊C4C22×C416yes—
249C2×C22⋊C4C22×C416yes—
250C2×C22⋊C4C22×C413yes—
251C2×C22⋊C4C22×C413yes—
252C2×C22⋊C4C22×C413yes—
253C2×C22⋊C4C22×C413yes—
254C2×C22⋊C4C22×C413yes—
255C2×C22⋊C4C22×C413yes—
256C2×C22⋊C4C22×C416yes—
257C2×C22⋊C4C22×C416yes—
258C2×C42—26no—
259C2×C42—26no—
260C2×C42—26no—
261C2×C42—26no—
262C2×C42C22×C413yes—
263C2×C42C22×C413yes—
264C2×C42C22×C413yes—
265C2×C42C22×C413yes—
266C2×C42C22×C413yes—
267C2×C42C22×C413yes—
268C2×C42C22×C413yes—
269C2×C42C22×C413yes—
270C2×C42C2411yescharacteristic
271C2×C42C2413yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2413yes—
275C4⋊C8—26no—
276C4⋊C8—22no—
277C4⋊C8—26no—
278C4⋊C8—26no—
279C4⋊C8—26no—
280C4⋊C8—26no—
281C4⋊C8—43no—
282C4⋊C8C2×D4112yes—
283C22⋊C8—26no—
284C22⋊C8—26no—
285C22⋊C8—212no—
286C22⋊C8—26no—
287C22⋊C8—43no—
288C22⋊C8—46no—
289C22⋊C8—43no—
290C22⋊C8—43no—
291C22⋊C8—46no—
292#97—22no—
293#97—26no—
294C4×C8—26no—
295C4×C8—26no—
296C4×C8—26no—
297C4×C8—46no—
298C4×C8C2×D4112yes—
299#95C22×C411yescharacteristic
300#95C22×C411yescharacteristic
301#95C22×C413yes—
302#95C22×C413yes—
303#95C22×C413yes—
304#95C22×C413yes—
305#95C22×C416yes—
306C24C23×C413yes—
307C24C23×C413yes—
308#41—416no—
309#41—416no—
310C2×Q8C4×D414yes—
311C2×Q8C2×M1614yes—
312C2×Q8#13114yes—
313C2×D4—26no—
314C2×D4—26no—
315C2×D4—26no—
316C2×D4—26no—
317C2×D4—26no—
318C2×D4—212no—
319C2×D4—26no—
320C2×D4—26no—
321C2×D4—43no—
322C2×D4—43no—
323C2×D4—46no—
324C2×D4—46no—
325C22×C4—26no—
326C22×C4—26no—
327C22×C4—26no—
328C22×C4—26no—
329C22×C4—26no—
330C22×C4—26no—
331C22×C4—26no—
332C22×C4—26no—
333C22×C4—23no—
334C22×C4—23no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—23no—
338C22×C4—23no—
339C22×C4—23no—
340C22×C4—23no—
341C22×C4—23no—
342C22×C4—23no—
343C22×C4—23no—
344C22×C4—23no—
345C22×C4—26no—
346C22×C4—43no—
347C22×C4—43no—
348C22×C4—43no—
349C22×C4C23×C411yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C23×C413yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C23×C413yes—
355C22×C4C23×C413yes—
356C22×C4C23×C413yes—
357C22×C4C23×C413yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C23×C411yescharacteristic
360C22×C4C23×C413yes—
361C22×C4C2511yescharacteristic
362M16—412no—
363M16—412no—
364M16—812no—
365C2×C8—21no—
366C2×C8—23no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—23no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—23no—
373C2×C8—23no—
374C2×C8—21no—
375C2×C8—23no—
376C2×C8—23no—
377C2×C8—23no—
378C2×C8—21no—
379C2×C8—23no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—23no—
383C2×C8—23no—
384C2×C8—23no—
385C2×C8—21no—
386C2×C8C22×D412yes—
387C2×C8C22×D416yes—
388C2×C8H(5,𝔽2)12yes—
389C4⋊C4—26no—
390C4⋊C4—26no—
391C4⋊C4—22no—
392C4⋊C4—26no—
393C4⋊C4—22no—
394C4⋊C4—43no—
395C4⋊C4—46no—
396C4⋊C4C4×D4112yes—
397C4⋊C4C2×M16112yes—
398C22⋊C4—212no—
399C22⋊C4—212no—
400C22⋊C4—26no—
401C22⋊C4—26no—
402C22⋊C4—26no—
403C22⋊C4—43no—
404C22⋊C4—43no—
405C22⋊C4—46no—
406C22⋊C4—46no—
407C22⋊C4—43no—
408C42—26no—
409C42—26no—
410C42—26no—
411C42—26no—
412C42—26no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—23no—
418C42—23no—
419C42—21no—
420C42—21no—
421C42—43no—
422C42—43no—
423C42C2×M16112yes—
424C42C22×D416yes—
425C42C2×#4116yes—
426C23—23no—
427C23—23no—
428C23—23no—
429C23—23no—
430C23—23no—
431C23—23no—
432C23—23no—
433C23—23no—
434C23—23no—
435C23—23no—
436C23—23no—
437C23—23no—
438C23—23no—
439C23—23no—
440C23C24×C411yescharacteristic
441Q8#436116yescomplemented
442D4—412no—
443D4—412no—
444D4—412no—
445D4—812no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—23no—
449C2×C4—21no—
450C2×C4—23no—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—21no—
460C2×C4—23no—
461C2×C4—23no—
462C2×C4—23no—
463C2×C4—23no—
464C2×C4—23no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—23no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—23no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—412no—
476C2×C4—412no—
477C2×C4—412no—
478C2×C4—43no—
479C2×C4—43no—
480C2×C4—43no—
481C2×C4—43no—
482C2×C4—43no—
483C2×C4—43no—
484C2×C4—43no—
485C2×C4—43no—
486C2×C4—86no—
487C2×C4C2×C4×D416yes—
488C2×C4C2×C4×D416yes—
489C2×C4C22×M1612yes—
490C2×C4C22×M1616yes—
491C2×C4C22×M1616yes—
492C2×C4C2×#13116yes—
493C2×C4#56812yes—
494C2×C4C23×D411yescharacteristic
495C2×C4C22×#4111yescharacteristic
496C2×C4C2×H(5,𝔽2)11yescharacteristic
497C2×C4#58511yescharacteristic
498C2×C4#58511yescharacteristic
499C2×C4#58511yescharacteristic
500C8—22no—
501C8—26no—
502C8—22no—
503C8—41no—
504C8—43no—
505C8—43no—
506C8—43no—
507C8—41no—
508C22—43no—
509C22—43no—
510C22—43no—
511C22—43no—
512C22—43no—
513C22—43no—
514C22—43no—
515C22—43no—
516C22—43no—
517C22—43no—
518C22—43no—
519C22—43no—
520C22—43no—
521C22—43no—
522C22C22×C4×D411yescharacteristic
523C22#342811yescharacteristic
524C22C4×H(5,𝔽2)11yescharacteristic
525C22C23×M1611yescharacteristic
526C22C22×#13111yescharacteristic
527C22C2×#56811yescharacteristic
528C22#357311yescharacteristic
529C4—22no—
530C4—26no—
531C4—22no—
532C4—26no—
533C4—41no—
534C4—43no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—43no—
539C4—43no—
540C4C2×#436112yes—
541C4#348711yescharacteristic
542C4#348811yescharacteristic
543C4#351711yescharacteristic
544C4#352411yescharacteristic
545C2—83no—
546C2—83no—
547C2#3429511yescharacteristic
548C2C22×#43611yescharacteristic
549C2#6014611yescharacteristic
550C2#6028411yescharacteristic
551C2#6030111yescharacteristic
552C2#6032511yescharacteristic
553C2#6032811yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC2×#550C4
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×#2971C2semidirect
4. #23176C2semidirect
5. #23088C2semidirect
6. #22990C2semidirect
7. #22751C2semidirect
8. #22734C2semidirect
9. #22530C2semidirect
10. #22530C2semidirect
11. #22264C2semidirect
12. #22137C2semidirect
13. #21488C2semidirect
14. #21102C2semidirect
15. #20569C2semidirect
109. C8×Q8D4semidirect
222. C4×Q8M16semidirect
441. Q8#436semidirect

Maximal subgroups

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

subgroupindex
C2×#29912
C2×#29712
C4×#5502
#231762
#230882
#229902
#227512
#227342
#225302
#222642
#221372
#214882
#211022
#205692