Finite Group Databasebeta

← #6404238

Subgroups — #6404238

Order 512.

1994 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic, complemented
2C2×#2982C211yesmaximal, characteristic, complemented
3C4×D4×Q8C211yesmaximal, characteristic
4#23150C213yesmaximal, complemented
5#23054C213yesmaximal, complemented
6#23051C213yesmaximal, complemented
7#22797C213yesmaximal
8Q8×C4⋊C8C211yesmaximal, characteristic, complemented
9Q8×C22⋊C8C211yesmaximal, characteristic, complemented
10Q8×C22⋊C8C211yesmaximal, characteristic, complemented
11#22104C213yesmaximal, complemented
12#22100C213yesmaximal, complemented
13#21714C213yesmaximal, complemented
14#21265C213yesmaximal
15Q8×#97C211yesmaximal, characteristic, complemented
16C2×D4×Q8C411yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2984—28no—
20#2982—28no—
21C2×#446C2213yes—
22C2×#446C2213yes—
23C2×C8×Q8C2213yes—
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2213yes—
26C2×C8×Q8C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×C8×Q8C2211yescharacteristic
29C2×#432C2213yes—
30C2×#431C2213yes—
31C2×#431C2213yes—
32C2×#431C2213yes—
33#2365C413yes—
34#2360C413yes—
35#2360C413yes—
36Q8×C4⋊C4C411yescharacteristic
37D4×C4⋊C4C413yes—
38Q8×C22⋊C4C411yescharacteristic
39Q8×C22⋊C4C411yescharacteristic
40C4×C4⋊Q8C2213yes—
41C4×C22⋊Q8C2213yes—
42C4×C22⋊Q8C2213yes—
43C42×Q8C2211yescharacteristic
44C42×D4C2213yes—
45#1984C2213yes—
46#1984C2213yes—
47#1984C2213yes—
48#1984C2213yes—
49#1955C2213yes—
50#1955C2211yescharacteristic
51#1940C2213yes—
52#1940C2213yes—
53#1940C2216yes—
54#1939C2213yes—
55#1939C2213yes—
56#1924C2213yes—
57#1924C2216yes—
58#1924C2213yes—
59#1924C2213yes—
60#1924C2213yes—
61#1923C2216yes—
62#1917C2216yes—
63#1916C2216yes—
64#1842C2213yes—
65#1797C2213yes—
66#1797C2213yes—
67#1771C2216yes—
68#1771C2216yes—
69#1770C2213yes—
70#1770C2213yes—
71#1767C2213yes—
72C4×C4⋊C8C2213yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75C4×#97C2213yes—
76C23×Q8C2×C411yescharacteristic
77C23×Q8C2×C411yescharacteristic
78D4×Q8—28no—
79C2×C4⋊Q8C2×C413yes—
80C2×C22⋊Q8C2×C413yes—
81C2×C22⋊Q8C2×C413yes—
82C2×C4×Q8—24no—
83C2×C4×Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×Q8C2×C411yescharacteristic
90C2×C4×Q8C2×C413yes—
91C2×C4×Q8C2×C413yes—
92C2×C4×Q8C2313yes—
93C2×C4×Q8C2313yes—
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4C2×C413yes—
96C2×C4×D4C2×C413yes—
97C2×C4×D4C2311yescharacteristic
98C22×C4⋊C4C2×C413yes—
99C22×C4⋊C4C2×C413yes—
100C22×C42C2313yes—
101C22×C42C2313yes—
102#446—212no—
103#446—212no—
104C8×Q8—24no—
105C8×Q8—24no—
106C8×Q8—24no—
107C8×Q8—212no—
108C8×Q8—212no—
109C8×Q8D418yescomplemented
110#432—212no—
111#431—212no—
112#431—212no—
113#431—212no—
114C2×C4⋊C8C2313yes—
115C2×C4⋊C8C2313yes—
116C2×C4⋊C8C2313yes—
117C2×C4⋊C8C2313yes—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2313yes—
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2316yes—
124C2×C4⋊C8C2313yes—
125C2×C22⋊C8C2311yescharacteristic
126C2×C22⋊C8C2311yescharacteristic
127C2×C22⋊C8C2313yes—
128C2×C22⋊C8C2313yes—
129C2×C22⋊C8C2311yescharacteristic
130C2×C22⋊C8C2311yescharacteristic
131C2×C22⋊C8C2313yes—
132C2×C22⋊C8C2313yes—
133C2×#97C2316yes—
134C2×#97C2313yes—
135C2×#97C2311yescharacteristic
136C2×#97C2313yes—
137C2×#97C2313yes—
138C2×C4×C8C2316yes—
139C2×C4×C8C2313yes—
140C2×C4×C8C2313yes—
141C2×C4×C8C2313yes—
142C2×C4×C8C2313yes—
143#391C2×C413yes—
144#391C2×C413yes—
145#391C2×C413yes—
146#391C2×C413yes—
147#389C2×C416yes—
148#389C2×C416yes—
149#389C2×C413yes—
150#389C2×C413yes—
151#387C2×C416yes—
152#387C2×C416yes—
153#385C2×C416yes—
154#385C2×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⋊C4C2313yes—
163C4×C4⋊C4C2316yes—
164C4×C4⋊C4C2313yes—
165C4×C22⋊C4C2×C413yes—
166C4×C22⋊C4C2×C413yes—
167C4×C22⋊C4C2×C413yes—
168C4×C22⋊C4C2×C413yes—
169C4×C22⋊C4C2313yes—
170C4×C22⋊C4C2313yes—
171C43C2313yes—
172#336C2316yes—
173#336C2316yes—
174#336C2313yes—
175#336C2313yes—
176#336C2316yes—
177#336C2313yes—
178#336C2313yes—
179#336C2313yes—
180#336C2313yes—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8M1618yes—
187C22×Q8C22×C413yes—
188C22×Q8C22×C413yes—
189C22×Q8C22×C411yescharacteristic
190C22×D4C22×C411yescharacteristic
191C23×C4C22×C413yes—
192C23×C4C22×C413yes—
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C22×C8C2413yes—
198C22×C8C2411yescharacteristic
199C22×C8C2411yescharacteristic
200C22×C8C2413yes—
201C22×C8C2413yes—
202C22×C8C2413yes—
203C22×C8C2411yescharacteristic
204C22×C8C2411yescharacteristic
205C4⋊Q8—212no—
206C22⋊Q8—212no—
207C22⋊Q8—212no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—212no—
213C4×Q8—212no—
214C4×Q8—26no—
215C4×Q8—26no—
216C4×Q8—26no—
217C4×Q8—26no—
218C4×Q8C2×D414yes—
219C4×Q8#4114yes—
220C4×D4—212no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—46no—
224C2×C4⋊C4—212no—
225C2×C4⋊C4—212no—
226C2×C4⋊C4C22×C413yes—
227C2×C4⋊C4C22×C413yes—
228C2×C4⋊C4C22×C416yes—
229C2×C4⋊C4C22×C416yes—
230C2×C4⋊C4C22×C413yes—
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C413yes—
233C2×C4⋊C4C22×C413yes—
234C2×C4⋊C4C22×C416yes—
235C2×C4⋊C4C22×C413yes—
236C2×C4⋊C4C22×C416yes—
237C2×C4⋊C4C22×C413yes—
238C2×C22⋊C4C22×C413yes—
239C2×C22⋊C4C22×C413yes—
240C2×C22⋊C4C22×C413yes—
241C2×C22⋊C4C22×C411yescharacteristic
242C2×C22⋊C4C22×C411yescharacteristic
243C2×C22⋊C4C22×C413yes—
244C2×C42—26no—
245C2×C42—26no—
246C2×C42—26no—
247C2×C42—26no—
248C2×C42C22×C413yes—
249C2×C42C22×C413yes—
250C2×C42C22×C413yes—
251C2×C42C22×C413yes—
252C2×C42C22×C413yes—
253C2×C42C22×C413yes—
254C2×C42C22×C413yes—
255C2×C42C22×C413yes—
256C2×C42C2413yes—
257C2×C42C2413yes—
258C2×C42C2413yes—
259C2×C42C2413yes—
260C2×C42C2411yescharacteristic
261C4⋊C8—26no—
262C4⋊C8—26no—
263C4⋊C8—26no—
264C4⋊C8—26no—
265C4⋊C8—26no—
266C4⋊C8—26no—
267C4⋊C8—26no—
268C4⋊C8—22no—
269C4⋊C8—26no—
270C4⋊C8—46no—
271C4⋊C8C2×D4112yes—
272C22⋊C8—26no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—26no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—43no—
279C22⋊C8—43no—
280#97—212no—
281#97—26no—
282#97—26no—
283#97—22no—
284#97—43no—
285C4×C8—212no—
286C4×C8—26no—
287C4×C8—26no—
288C4×C8—26no—
289C4×C8C2×D4112yes—
290#95C22×C413yes—
291#95C22×C413yes—
292#95C22×C413yes—
293#95C22×C413yes—
294#95C22×C413yes—
295#95C22×C413yes—
296#95C22×C416yes—
297#95C22×C416yes—
298#95C22×C416yes—
299C24C23×C411yescharacteristic
300C24C23×C411yescharacteristic
301C2×Q8—232no—
302C2×Q8—26no—
303C2×Q8—26no—
304C2×Q8—26no—
305C2×Q8—26no—
306C2×Q8—26no—
307C2×Q8—416no—
308C2×Q8C4×D414yes—
309C2×Q8C2×M16112yes—
310C2×Q8C2×M1614yes—
311C2×Q8#13114yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C22×C4—26no—
316C22×C4—26no—
317C22×C4—26no—
318C22×C4—26no—
319C22×C4—26no—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—26no—
335C22×C4—43no—
336C22×C4—43no—
337C22×C4—43no—
338C22×C4C2×M16112yes—
339C22×C4C23×C413yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C23×C411yescharacteristic
342C22×C4C23×C413yes—
343C22×C4C23×C413yes—
344C22×C4C23×C413yes—
345C22×C4C23×C413yes—
346C22×C4C23×C413yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C23×C411yescharacteristic
349C22×C4C23×C413yes—
350C22×C4C23×C413yes—
351C22×C4C2511yescharacteristic
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—23no—
355C2×C8—23no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—23no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—23no—
362C2×C8—23no—
363C2×C8—23no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—23no—
367C2×C8—23no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—23no—
371C2×C8—23no—
372C2×C8—23no—
373C2×C8C22×D416yes—
374C2×C8C22×D412yes—
375C2×C82−1+412yes—
376C4⋊C4—212no—
377C4⋊C4—26no—
378C4⋊C4—26no—
379C4⋊C4—212no—
380C4⋊C4—22no—
381C4⋊C4—26no—
382C4⋊C4—26no—
383C4⋊C4—212no—
384C4⋊C4—26no—
385C4⋊C4—46no—
386C4⋊C4—43no—
387C4⋊C4C4×D4112yes—
388C22⋊C4—26no—
389C22⋊C4—26no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—26no—
393C22⋊C4—43no—
394C42—26no—
395C42—26no—
396C42—26no—
397C42—26no—
398C42—26no—
399C42—26no—
400C42—23no—
401C42—23no—
402C42—23no—
403C42—23no—
404C42—23no—
405C42—23no—
406C42—43no—
407C42—43no—
408C42C22×D416yes—
409C42C2×#4116yes—
410C422−1+412yes—
411C422−1+412yes—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23C22×M1612yes—
425C23#56812yes—
426C23C24×C411yescharacteristic
427Q8—224no—
428Q8—412no—
429Q8#435116yes—
430D4—28no—
431C2×C4—23no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—23no—
435C2×C4—21no—
436C2×C4—23no—
437C2×C4—21no—
438C2×C4—23no—
439C2×C4—224no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—23no—
443C2×C4—23no—
444C2×C4—23no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—23no—
457C2×C4—23no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—412no—
462C2×C4—412no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—86no—
472C2×C4C2×C4×D416yes—
473C2×C4C2×C4×D416yes—
474C2×C4#51912yes—
475C2×C4#51912yes—
476C2×C4C22×M1616yes—
477C2×C4C22×M1616yes—
478C2×C4C2×#13116yes—
479C2×C4C23×D411yescharacteristic
480C2×C4C22×#4111yescharacteristic
481C2×C4C2×2−1+411yescharacteristic
482C2×C4C2×2−1+411yescharacteristic
483C2×C4C2×2−1+411yescharacteristic
484C2×C4#58511yescharacteristic
485C8—22no—
486C8—26no—
487C8—22no—
488C8—43no—
489C8—43no—
490C8—41no—
491C8—41no—
492C8—43no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22C22×C4×D411yescharacteristic
508C22C2×#51911yescharacteristic
509C22C4×2−1+411yescharacteristic
510C22C23×M1611yescharacteristic
511C22C22×#13111yescharacteristic
512C22C2×#56811yescharacteristic
513C22#357311yescharacteristic
514C4—22no—
515C4—22no—
516C4—26no—
517C4—26no—
518C4—41no—
519C4—41no—
520C4—43no—
521C4—43no—
522C4—43no—
523C4—43no—
524C4—43no—
525C4C2×#435112yes—
526C4#349711yescharacteristic
527C4#349911yescharacteristic
528C4#351811yescharacteristic
529C4#352311yescharacteristic
530C2—42no—
531C2—42no—
532C2#3431111yescharacteristic
533C2C22×#43511yescharacteristic
534C2#6014611yescharacteristic
535C2#6024811yescharacteristic
536C2#6025311yescharacteristic
537C2#6033311yescharacteristic
538C2#6033511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC22×D4C22×C4
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×#2984C2semidirect
2. C2×#2982C2semidirect
4. #23150C2semidirect
5. #23054C2semidirect
6. #23051C2semidirect
8. Q8×C4⋊C8C2semidirect
9. Q8×C22⋊C8C2semidirect
10. Q8×C22⋊C8C2semidirect
11. #22104C2semidirect
12. #22100C2semidirect
13. #21714C2semidirect
15. Q8×#97C2semidirect
109. C8×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
C2×#29822
C4×D4×Q82
#231502
#230542
#230512
#227972
Q8×C4⋊C82
Q8×C22⋊C82
#221042
#221002
#217142
#212652
Q8×#972