Finite Group Databasebeta

← #6404549

Subgroups — #6404549

Order 512.

2241 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2991C211yesmaximal, characteristic, complemented
2C2×#2988C211yesmaximal, characteristic, complemented
3C4×#550C211yesmaximal, characteristic
4#23174C213yesmaximal, complemented
5#22989C213yesmaximal, complemented
6#22986C213yesmaximal, complemented
7Q8×C4⋊C8C211yesmaximal, characteristic, complemented
8#22648C213yesmaximal, complemented
9#22530C211yesmaximal, characteristic, complemented
10#22530C211yesmaximal, characteristic, complemented
11#22139C213yesmaximal, complemented
12#22135C213yesmaximal, complemented
13#21682C213yesmaximal, complemented
14#21265C213yesmaximal, complemented
15Q8×#97C211yesmaximal, characteristic, complemented
16C2×#550C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19#2991—28no—
20#2988—28no—
21C2×C8×Q8C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×#436C2213yes—
26C2×#436C2213yes—
27C2×#435C2216yes—
28C2×#435C2216yes—
29C2×C8×D4C2213yes—
30C2×C8×D4C2213yes—
31C2×#431C2213yes—
32C2×#431C2213yes—
33C2×#423C2213yes—
34C22×C4⋊C8C2213yes—
35#2366C413yes—
36#2363C413yes—
37#2363C413yes—
38#2354C413yes—
39Q8×C4⋊C4C411yescharacteristic
40#2341C411yescharacteristic
41#2341C411yescharacteristic
42C4×#127C2213yes—
43C4×#121C2213yes—
44C4×#121C2213yes—
45C42×Q8C2211yescharacteristic
46C42×D4C2213yes—
47#1955C2213yes—
48#1955C2213yes—
49#1955C2216yes—
50#1954C2213yes—
51#1954C2213yes—
52#1954C2213yes—
53#1954C2213yes—
54#1940C2211yescharacteristic
55#1940C2213yes—
56#1939C2213yes—
57#1939C2213yes—
58#1924C2213yes—
59#1924C2213yes—
60#1924C2213yes—
61#1924C2213yes—
62#1870C2216yes—
63#1869C2216yes—
64#1851C2216yes—
65#1850C2216yes—
66#1842C2213yes—
67#1797C2213yes—
68#1797C2213yes—
69#1770C2213yes—
70#1770C2213yes—
71#1767C2213yes—
72C4×C4⋊C8C2213yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75C4×#97C2213yes—
76C22×#41C2×C411yescharacteristic
77C22×#41C2×C411yescharacteristic
78C22×M16C2316yes—
79C23×C8C2316yes—
80#550—28no—
81C2×#127C2×C413yes—
82C2×#121C2×C413yes—
83C2×#121C2×C413yes—
84C4×#41—24no—
85C4×#41—24no—
86C4×#41—24no—
87C4×#41—24no—
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4C2×C413yes—
94C2×C4×D4C2×C413yes—
95C2×C4×D4C2×C413yes—
96C2×C4×D4C2×C413yes—
97C2×C4×D4C2313yes—
98C2×C4×D4C2313yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2316yes—
101C2×#117C2×C413yes—
102C2×#117C2×C413yes—
103C22×C42C2313yes—
104C22×C42C2313yes—
105C8×Q8—24no—
106C8×Q8—24no—
107C8×Q8—24no—
108C8×Q8D418yescomplemented
109#436—212no—
110#436—212no—
111#435—412no—
112#435—412no—
113C8×D4—212no—
114C8×D4—212no—
115#431—212no—
116#431—212no—
117#423—212no—
118C2×C4⋊C8—212no—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2313yes—
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2313yes—
123C2×C4⋊C8C2313yes—
124C2×C4⋊C8C2313yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2316yes—
129C2×C4⋊C8C2313yes—
130C2×C22⋊C8C2313yes—
131C2×C22⋊C8C2313yes—
132C2×C22⋊C8C2313yes—
133C2×C22⋊C8C2313yes—
134C2×C22⋊C8C2313yes—
135C2×C22⋊C8C2313yes—
136C2×C22⋊C8C2316yes—
137C2×C22⋊C8C2316yes—
138C2×C22⋊C8C2313yes—
139C2×C22⋊C8C2313yes—
140C2×#97C2313yes—
141C2×#97C2313yes—
142C2×#97C2311yescharacteristic
143C2×#97C2313yes—
144C2×C4×C8C2313yes—
145C2×C4×C8C2313yes—
146C2×C4×C8C2313yes—
147C2×C4×C8C2313yes—
148#390C2×C413yes—
149#390C2×C413yes—
150#390C2×C416yes—
151#390C2×C416yes—
152#390C2×C413yes—
153#390C2×C413yes—
154#389C2×C413yes—
155#389C2×C413yes—
156#388C2×C416yes—
157#388C2×C416yes—
158#386C2×C416yes—
159#386C2×C416yes—
160#384C2×C413yes—
161#380C2×C413yes—
162#380C2×C413yes—
163C4×C4⋊C4C2×C413yes—
164C4×C4⋊C4C2×C413yes—
165C4×C4⋊C4C2×C413yes—
166C4×C4⋊C4C2×C413yes—
167C4×C4⋊C4C2313yes—
168C4×C4⋊C4C2311yescharacteristic
169C4×C22⋊C4C2×C413yes—
170C4×C22⋊C4C2×C413yes—
171C4×C22⋊C4C2×C413yes—
172C4×C22⋊C4C2×C413yes—
173C4×C22⋊C4C2313yes—
174C4×C22⋊C4C2313yes—
175C43C2313yes—
176#336C2313yes—
177#336C2313yes—
178#336C2316yes—
179#336C2313yes—
180#336C2313yes—
181#336C2311yescharacteristic
182#336C2311yescharacteristic
183C2×#41—24no—
184C2×#41—24no—
185C2×#41—24no—
186C2×#41—24no—
187C2×#41—24no—
188C2×#41M1618yescomplemented
189C22×Q8C22×C411yescharacteristic
190C22×D4C22×C413yes—
191C22×D4C22×C413yes—
192C22×D4C22×C413yes—
193C22×D4C22×C416yes—
194C23×C4C22×C413yes—
195C23×C4C22×C413yes—
196C23×C4C22×C413yes—
197C23×C4C22×C413yes—
198C23×C4C2413yes—
199C23×C4C2413yes—
200C2×M16—212no—
201C2×M16—46no—
202C2×M16—46no—
203C22×C8—212no—
204C22×C8—46no—
205C22×C8—46no—
206C22×C8C2413yes—
207C22×C8C2413yes—
208C22×C8C2411yescharacteristic
209C22×C8C2411yescharacteristic
210C22×C8C2413yes—
211C22×C8C2413yes—
212C22×C8C2411yescharacteristic
213C22×C8C2411yescharacteristic
214#127—212no—
215#121—212no—
216#121—212no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8C2×D414yes—
222C4×Q8#4114yes—
223C4×D4—212no—
224C4×D4—212no—
225C4×D4—212no—
226C4×D4—26no—
227C4×D4—26no—
228C4×D4—26no—
229C4×D4—26no—
230C4×D4—26no—
231C4×D4—26no—
232C4×D4—46no—
233C4×D4—46no—
234C4×D4—46no—
235#117—212no—
236#117—212no—
237C2×C4⋊C4C22×C413yes—
238C2×C4⋊C4C22×C413yes—
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C413yes—
242C2×C4⋊C4C22×C413yes—
243C2×C4⋊C4C22×C413yes—
244C2×C4⋊C4C22×C416yes—
245C2×C4⋊C4C22×C413yes—
246C2×C22⋊C4C22×C416yes—
247C2×C22⋊C4C22×C416yes—
248C2×C22⋊C4C22×C413yes—
249C2×C22⋊C4C22×C413yes—
250C2×C22⋊C4C22×C413yes—
251C2×C22⋊C4C22×C413yes—
252C2×C22⋊C4C22×C413yes—
253C2×C22⋊C4C22×C413yes—
254C2×C22⋊C4C22×C416yes—
255C2×C22⋊C4C22×C416yes—
256C2×C42—26no—
257C2×C42—26no—
258C2×C42—26no—
259C2×C42—26no—
260C2×C42C22×C413yes—
261C2×C42C22×C413yes—
262C2×C42C22×C413yes—
263C2×C42C22×C413yes—
264C2×C42C22×C413yes—
265C2×C42C22×C413yes—
266C2×C42C22×C413yes—
267C2×C42C22×C413yes—
268C2×C42C2413yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2413yes—
272C2×C42C2411yescharacteristic
273C4⋊C8—26no—
274C4⋊C8—26no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—26no—
278C4⋊C8—26no—
279C4⋊C8—26no—
280C4⋊C8—26no—
281C4⋊C8—22no—
282C4⋊C8—46no—
283C4⋊C8C2×D4112yes—
284C22⋊C8—26no—
285C22⋊C8—26no—
286C22⋊C8—26no—
287C22⋊C8—26no—
288C22⋊C8—26no—
289C22⋊C8—26no—
290C22⋊C8—43no—
291C22⋊C8—43no—
292C22⋊C8—46no—
293C22⋊C8—46no—
294#97—26no—
295#97—26no—
296#97—22no—
297#97—43no—
298C4×C8—26no—
299C4×C8—26no—
300C4×C8—26no—
301C4×C8C2×D4112yes—
302#95C22×C411yescharacteristic
303#95C22×C411yescharacteristic
304#95C22×C413yes—
305#95C22×C413yes—
306#95C22×C413yes—
307#95C22×C413yes—
308#95C22×C416yes—
309C24C23×C413yes—
310C24C23×C413yes—
311#41—232no—
312#41—416no—
313C2×Q8C4×D414yes—
314C2×Q8C2×M1614yes—
315C2×Q8#13114yes—
316C2×D4—26no—
317C2×D4—212no—
318C2×D4—26no—
319C2×D4—26no—
320C2×D4—26no—
321C2×D4—26no—
322C2×D4—26no—
323C2×D4—26no—
324C2×D4—43no—
325C2×D4—46no—
326C2×D4—46no—
327C2×D4C2×M16112yes—
328C22×C4—26no—
329C22×C4—26no—
330C22×C4—26no—
331C22×C4—26no—
332C22×C4—26no—
333C22×C4—26no—
334C22×C4—26no—
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—23no—
346C22×C4—23no—
347C22×C4—26no—
348C22×C4—43no—
349C22×C4—43no—
350C22×C4—43no—
351C22×C4C2×M16112yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C23×C413yes—
355C22×C4C23×C413yes—
356C22×C4C23×C413yes—
357C22×C4C23×C413yes—
358C22×C4C23×C413yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C23×C411yescharacteristic
361C22×C4C23×C411yescharacteristic
362C22×C4C23×C411yescharacteristic
363C22×C4C23×C413yes—
364C22×C4C2511yescharacteristic
365M16—812no—
366C2×C8—21no—
367C2×C8—23no—
368C2×C8—23no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—23no—
375C2×C8—23no—
376C2×C8—23no—
377C2×C8—23no—
378C2×C8—23no—
379C2×C8—23no—
380C2×C8—23no—
381C2×C8—23no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—23no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—812no—
388C2×C8C22×D412yes—
389C2×C8C22×D416yes—
390C2×C8H(5,𝔽2)12yes—
391C4⋊C4—26no—
392C4⋊C4—26no—
393C4⋊C4—26no—
394C4⋊C4—22no—
395C4⋊C4—26no—
396C4⋊C4—26no—
397C4⋊C4—22no—
398C4⋊C4—46no—
399C4⋊C4C4×D4112yes—
400C22⋊C4—26no—
401C22⋊C4—212no—
402C22⋊C4—212no—
403C22⋊C4—26no—
404C22⋊C4—26no—
405C22⋊C4—26no—
406C22⋊C4—26no—
407C22⋊C4—46no—
408C22⋊C4—46no—
409C22⋊C4—43no—
410C42—26no—
411C42—26no—
412C42—26no—
413C42—26no—
414C42—26no—
415C42—26no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—23no—
421C42—23no—
422C42—43no—
423C42—43no—
424C42C22×D416yes—
425C42C2×#4116yes—
426C42H(5,𝔽2)12yes—
427C422−1+412yes—
428C23—23no—
429C23—23no—
430C23—23no—
431C23—23no—
432C23—23no—
433C23—23no—
434C23—23no—
435C23—23no—
436C23—23no—
437C23—23no—
438C23—23no—
439C23—23no—
440C23—23no—
441C23C22×M1616yes—
442C23C24×C411yescharacteristic
443Q8#435116yescomplemented
444D4—224no—
445D4—412no—
446D4—412no—
447D4—812no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—23no—
455C2×C4—21no—
456C2×C4—23no—
457C2×C4—21no—
458C2×C4—224no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—23no—
462C2×C4—23no—
463C2×C4—23no—
464C2×C4—23no—
465C2×C4—23no—
466C2×C4—23no—
467C2×C4—23no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—21no—
471C2×C4—23no—
472C2×C4—21no—
473C2×C4—23no—
474C2×C4—21no—
475C2×C4—23no—
476C2×C4—21no—
477C2×C4—412no—
478C2×C4—412no—
479C2×C4—43no—
480C2×C4—43no—
481C2×C4—43no—
482C2×C4—43no—
483C2×C4—43no—
484C2×C4—43no—
485C2×C4—43no—
486C2×C4—43no—
487C2×C4—86no—
488C2×C4C2×C4×D416yes—
489C2×C4C2×C4×D416yes—
490C2×C4#52012yes—
491C2×C4#52012yes—
492C2×C4C22×M1612yes—
493C2×C4C22×M1616yes—
494C2×C4C2×#13116yes—
495C2×C4#56812yes—
496C2×C4C23×D411yescharacteristic
497C2×C4C22×#4111yescharacteristic
498C2×C4C2×H(5,𝔽2)11yescharacteristic
499C2×C4C2×H(5,𝔽2)11yescharacteristic
500C2×C4C2×2−1+411yescharacteristic
501C2×C4#58511yescharacteristic
502C8—22no—
503C8—26no—
504C8—22no—
505C8—41no—
506C8—41no—
507C8—43no—
508C8—43no—
509C8—43no—
510C22—26no—
511C22—26no—
512C22—26no—
513C22—43no—
514C22—43no—
515C22—43no—
516C22—43no—
517C22—43no—
518C22—43no—
519C22—43no—
520C22—43no—
521C22—43no—
522C22—43no—
523C22—43no—
524C22C22×C4×D411yescharacteristic
525C22C2×#52011yescharacteristic
526C22C4×H(5,𝔽2)11yescharacteristic
527C22C23×M1611yescharacteristic
528C22C22×#13111yescharacteristic
529C22C2×#56811yescharacteristic
530C22#357311yescharacteristic
531C4—22no—
532C4—22no—
533C4—22no—
534C4—22no—
535C4—26no—
536C4—43no—
537C4—43no—
538C4—43no—
539C4—43no—
540C4—41no—
541C4—41no—
542C4C2×#435112yes—
543C4#349711yescharacteristic
544C4#349811yescharacteristic
545C4#351711yescharacteristic
546C4#352211yescharacteristic
547C2—46no—
548C2—83no—
549C2#3431311yescharacteristic
550C2C22×#43511yescharacteristic
551C2#6014611yescharacteristic
552C2#6026311yescharacteristic
553C2#6027411yescharacteristic
554C2#6032511yescharacteristic
555C2#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×#2988C2semidirect
4. #23174C2semidirect
5. #22989C2semidirect
6. #22986C2semidirect
7. Q8×C4⋊C8C2semidirect
8. #22648C2semidirect
9. #22530C2semidirect
10. #22530C2semidirect
11. #22139C2semidirect
12. #22135C2semidirect
13. #21682C2semidirect
14. #21265C2semidirect
15. Q8×#97C2semidirect
108. C8×Q8D4semidirect
188. C2×#41M16semidirect
443. Q8#435semidirect

Maximal subgroups

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

subgroupindex
C2×#29912
C2×#29882
C4×#5502
#231742
#229892
#229862
Q8×C4⋊C82
#226482
#225302
#221392
#221352
#216822
#212652
Q8×#972