Finite Group Databasebeta

← #6385030

Subgroups — #6385030

Order 512.

2313 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2971C211yesmaximal, characteristic, complemented
2C2×C8×#41C211yesmaximal, characteristic, complemented
3C4×#550C211yesmaximal, characteristic
4#23174C213yesmaximal, complemented
5#23087C213yesmaximal, complemented
6#23084C213yesmaximal, complemented
7#22797C213yesmaximal, complemented
8Q8×C4⋊C8C211yesmaximal, characteristic, complemented
9#22530C211yesmaximal, characteristic, complemented
10#22530C211yesmaximal, characteristic, complemented
11#21682C213yesmaximal, complemented
12#21487C213yesmaximal, complemented
13#21486C213yesmaximal, complemented
14Q8×#97C211yesmaximal, characteristic, complemented
15C4×#435C213yesmaximal, complemented
16C2×#550C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19#2971—28no—
20C8×#41—28no—
21C2×C8×Q8C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×#436C2213yes—
26C2×#436C2213yes—
27C2×#435C2213yes—
28C2×#435C2213yes—
29C2×#435C2216yes—
30C2×C8×D4C2213yes—
31C2×C8×D4C2213yes—
32C2×#432C2213yes—
33C2×#431C2213yes—
34C2×C4×M16C2213yes—
35C22×C4×C8C2213yes—
36#2366C413yes—
37#2363C413yes—
38#2363C413yes—
39#2354C413yes—
40Q8×C4⋊C4C411yescharacteristic
41#2341C411yescharacteristic
42#2341C411yescharacteristic
43C4×#127C2213yes—
44C4×#121C2213yes—
45C4×#121C2213yes—
46C42×Q8C2211yescharacteristic
47C42×D4C2213yes—
48#1955C2216yes—
49#1955C2216yes—
50#1954C2213yes—
51#1954C2213yes—
52#1954C2213yes—
53#1954C2213yes—
54#1939C2213yes—
55#1939C2213yes—
56#1924C2216yes—
57#1923C2216yes—
58#1870C2216yes—
59#1869C2216yes—
60#1842C2213yes—
61#1797C2213yes—
62#1797C2213yes—
63#1771C2211yescharacteristic
64#1771C2213yes—
65#1770C2213yes—
66#1770C2213yes—
67#1767C2213yes—
68C4×C4⋊C8C2213yes—
69#1753C2213yes—
70#1753C2213yes—
71#1752C2213yes—
72#1752C2213yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75C4×#97C2213yes—
76C22×#41C2×C411yescharacteristic
77C22×#41C2×C411yescharacteristic
78C22×M16C2313yes—
79C22×M16C2313yes—
80C23×C8C2313yes—
81C23×C8C2313yes—
82#550—28no—
83C2×#127C2×C413yes—
84C2×#121C2×C413yes—
85C2×#121C2×C413yes—
86C4×#41—24no—
87C4×#41—24no—
88C4×#41—24no—
89C4×#41—24no—
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4C2×C413yes—
96C2×C4×D4C2×C413yes—
97C2×C4×D4C2×C413yes—
98C2×C4×D4C2×C413yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×C4×D4C2313yes—
102C2×C4×D4C2316yes—
103C2×#117C2×C413yes—
104C2×#117C2×C413yes—
105C22×C42C2313yes—
106C22×C42C2313yes—
107C8×Q8—24no—
108C8×Q8—24no—
109C8×Q8—24no—
110C8×Q8D418yescomplemented
111#436—212no—
112#436—212no—
113#435—212no—
114#435—212no—
115#435—412no—
116C8×D4—212no—
117C8×D4—212no—
118#432—212no—
119#431—212no—
120C2×C4⋊C8C2313yes—
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2313yes—
123C2×C4⋊C8C2313yes—
124C2×C4⋊C8C2316yes—
125C2×C4⋊C8C2313yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C22⋊C8C2313yes—
128C2×C22⋊C8C2316yes—
129C2×C22⋊C8C2313yes—
130C2×C22⋊C8C2316yes—
131C2×C22⋊C8C2313yes—
132C2×C22⋊C8C2313yes—
133C2×C22⋊C8C2316yes—
134C2×C22⋊C8C2316yes—
135C4×M16—212no—
136C2×#97C2311yescharacteristic
137C2×#97C2313yes—
138C2×#97C2313yes—
139C2×#97C2313yes—
140C2×#97C2313yes—
141C2×#97C2311yescharacteristic
142C2×C4×C8—212no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2313yes—
145C2×C4×C8C2313yes—
146C2×C4×C8C2313yes—
147C2×C4×C8C2313yes—
148C2×C4×C8C2313yes—
149#390C2×C413yes—
150#390C2×C413yes—
151#390C2×C416yes—
152#390C2×C416yes—
153#390C2×C413yes—
154#390C2×C413yes—
155#389C2×C413yes—
156#389C2×C413yes—
157#388C2×C416yes—
158#388C2×C416yes—
159#386C2×C416yes—
160#386C2×C416yes—
161#384C2×C413yes—
162#380C2×C413yes—
163#380C2×C413yes—
164C4×C4⋊C4C2×C413yes—
165C4×C4⋊C4C2×C413yes—
166C4×C4⋊C4C2×C413yes—
167C4×C4⋊C4C2×C413yes—
168C4×C4⋊C4C2313yes—
169C4×C4⋊C4C2311yescharacteristic
170C4×C22⋊C4C2×C413yes—
171C4×C22⋊C4C2×C413yes—
172C4×C22⋊C4C2×C413yes—
173C4×C22⋊C4C2×C413yes—
174C4×C22⋊C4C2313yes—
175C4×C22⋊C4C2313yes—
176C43C2313yes—
177#336C2313yes—
178#336C2313yes—
179#336C2313yes—
180#336C2316yes—
181#336C2313yes—
182#336C2311yescharacteristic
183#336C2311yescharacteristic
184C2×#41—24no—
185C2×#41—24no—
186C2×#41—24no—
187C2×#41—24no—
188C2×#41—24no—
189C2×#41M1618yescomplemented
190C22×Q8C22×C411yescharacteristic
191C22×D4C22×C413yes—
192C22×D4C22×C413yes—
193C22×D4C22×C413yes—
194C22×D4C22×C416yes—
195C23×C4C22×C413yes—
196C23×C4C22×C413yes—
197C23×C4C22×C413yes—
198C23×C4C22×C413yes—
199C23×C4C2413yes—
200C23×C4C2413yes—
201C2×M16—26no—
202C2×M16—26no—
203C2×M16—26no—
204C2×M16—26no—
205C2×M16—26no—
206C2×M16—43no—
207C22×C8—26no—
208C22×C8—26no—
209C22×C8—26no—
210C22×C8—26no—
211C22×C8—26no—
212C22×C8—43no—
213C22×C8C2413yes—
214C22×C8C2413yes—
215C22×C8C2413yes—
216C22×C8C2413yes—
217C22×C8C2411yescharacteristic
218C22×C8C2411yescharacteristic
219C22×C8C2411yescharacteristic
220C22×C8C2411yescharacteristic
221#127—212no—
222#121—212no—
223#121—212no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8C2×D414yes—
229C4×Q8#4114yes—
230C4×D4—212no—
231C4×D4—212no—
232C4×D4—212no—
233C4×D4—26no—
234C4×D4—26no—
235C4×D4—26no—
236C4×D4—26no—
237C4×D4—26no—
238C4×D4—26no—
239C4×D4—46no—
240C4×D4—46no—
241C4×D4—46no—
242#117—212no—
243#117—212no—
244C2×C4⋊C4C22×C413yes—
245C2×C4⋊C4C22×C413yes—
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C413yes—
249C2×C4⋊C4C22×C413yes—
250C2×C4⋊C4C22×C413yes—
251C2×C4⋊C4C22×C416yes—
252C2×C4⋊C4C22×C413yes—
253C2×C22⋊C4C22×C416yes—
254C2×C22⋊C4C22×C416yes—
255C2×C22⋊C4C22×C413yes—
256C2×C22⋊C4C22×C413yes—
257C2×C22⋊C4C22×C413yes—
258C2×C22⋊C4C22×C413yes—
259C2×C22⋊C4C22×C413yes—
260C2×C22⋊C4C22×C413yes—
261C2×C22⋊C4C22×C416yes—
262C2×C22⋊C4C22×C416yes—
263C2×C42—26no—
264C2×C42—26no—
265C2×C42—26no—
266C2×C42—26no—
267C2×C42C22×C413yes—
268C2×C42C22×C413yes—
269C2×C42C22×C413yes—
270C2×C42C22×C413yes—
271C2×C42C22×C413yes—
272C2×C42C22×C413yes—
273C2×C42C22×C413yes—
274C2×C42C22×C413yes—
275C2×C42C2413yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2413yes—
279C2×C42C2411yescharacteristic
280C4⋊C8—26no—
281C4⋊C8—26no—
282C4⋊C8—26no—
283C4⋊C8—26no—
284C4⋊C8—46no—
285C4⋊C8C2×D4112yes—
286C4⋊C8#4114yescomplemented
287C22⋊C8—212no—
288C22⋊C8—212no—
289C22⋊C8—26no—
290C22⋊C8—26no—
291C22⋊C8—26no—
292C22⋊C8—26no—
293C22⋊C8—46no—
294C22⋊C8—46no—
295#97—22no—
296#97—26no—
297#97—26no—
298#97—26no—
299#97—43no—
300#97#4114yescomplemented
301C4×C8—22no—
302C4×C8—26no—
303C4×C8—26no—
304C4×C8—26no—
305C4×C8—26no—
306C4×C8C2×D4112yes—
307#95C22×C411yescharacteristic
308#95C22×C411yescharacteristic
309#95C22×C413yes—
310#95C22×C413yes—
311#95C22×C413yes—
312#95C22×C413yes—
313#95C22×C416yes—
314C24C23×C413yes—
315C24C23×C413yes—
316#41—232no—
317#41—416no—
318C2×Q8C4×D414yes—
319C2×Q8C2×M1614yes—
320C2×Q8#13114yes—
321C2×D4—26no—
322C2×D4—212no—
323C2×D4—26no—
324C2×D4—26no—
325C2×D4—26no—
326C2×D4—26no—
327C2×D4—26no—
328C2×D4—26no—
329C2×D4—43no—
330C2×D4—46no—
331C2×D4—46no—
332C2×D4C2×M16112yes—
333C22×C4—26no—
334C22×C4—26no—
335C22×C4—26no—
336C22×C4—26no—
337C22×C4—26no—
338C22×C4—26no—
339C22×C4—26no—
340C22×C4—23no—
341C22×C4—23no—
342C22×C4—23no—
343C22×C4—23no—
344C22×C4—23no—
345C22×C4—23no—
346C22×C4—23no—
347C22×C4—23no—
348C22×C4—23no—
349C22×C4—23no—
350C22×C4—23no—
351C22×C4—23no—
352C22×C4—26no—
353C22×C4—43no—
354C22×C4—43no—
355C22×C4—43no—
356C22×C4C2×M16112yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C23×C411yescharacteristic
359C22×C4C23×C413yes—
360C22×C4C23×C413yes—
361C22×C4C23×C413yes—
362C22×C4C23×C413yes—
363C22×C4C23×C413yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C23×C411yescharacteristic
366C22×C4C23×C411yescharacteristic
367C22×C4C23×C411yescharacteristic
368C22×C4C23×C413yes—
369C22×C4C2511yescharacteristic
370M16—412no—
371M16—412no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—23no—
376C2×C8—21no—
377C2×C8—23no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—23no—
381C2×C8—23no—
382C2×C8—23no—
383C2×C8—23no—
384C2×C8—23no—
385C2×C8—23no—
386C2×C8—23no—
387C2×C8—23no—
388C2×C8—23no—
389C2×C8—412no—
390C2×C8—412no—
391C2×C8C22×D416yes—
392C2×C8C22×D412yes—
393C2×C8C2×#4112yes—
394C2×C8C2×#4112yes—
395C2×C8C2×#4112yes—
396C2×C8C2×#4112yes—
397C2×C8H(5,𝔽2)12yes—
398C4⋊C4—26no—
399C4⋊C4—26no—
400C4⋊C4—26no—
401C4⋊C4—22no—
402C4⋊C4—26no—
403C4⋊C4—26no—
404C4⋊C4—22no—
405C4⋊C4—46no—
406C4⋊C4C4×D4112yes—
407C22⋊C4—26no—
408C22⋊C4—212no—
409C22⋊C4—212no—
410C22⋊C4—26no—
411C22⋊C4—26no—
412C22⋊C4—26no—
413C22⋊C4—26no—
414C22⋊C4—46no—
415C22⋊C4—46no—
416C22⋊C4—43no—
417C42—26no—
418C42—26no—
419C42—26no—
420C42—26no—
421C42—26no—
422C42—26no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—23no—
428C42—23no—
429C42—43no—
430C42—43no—
431C42C22×D416yes—
432C42C2×#4116yes—
433C42C2×#4112yes—
434C42C2×#4112yes—
435C23—23no—
436C23—23no—
437C23—23no—
438C23—23no—
439C23—23no—
440C23—23no—
441C23—23no—
442C23—23no—
443C23—23no—
444C23—23no—
445C23—23no—
446C23—23no—
447C23—23no—
448C23C22×M1616yes—
449C23C24×C411yescharacteristic
450Q8#435116yescomplemented
451D4—224no—
452D4—412no—
453D4—412no—
454D4—812no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—23no—
462C2×C4—21no—
463C2×C4—23no—
464C2×C4—21no—
465C2×C4—224no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—23no—
471C2×C4—23no—
472C2×C4—23no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—23no—
477C2×C4—21no—
478C2×C4—23no—
479C2×C4—21no—
480C2×C4—23no—
481C2×C4—21no—
482C2×C4—23no—
483C2×C4—21no—
484C2×C4—412no—
485C2×C4—412no—
486C2×C4—43no—
487C2×C4—43no—
488C2×C4—43no—
489C2×C4—43no—
490C2×C4—43no—
491C2×C4—43no—
492C2×C4—43no—
493C2×C4—43no—
494C2×C4—86no—
495C2×C4C2×C4×D416yes—
496C2×C4C2×C4×D416yes—
497C2×C4C4×#4112yes—
498C2×C4C4×#4112yes—
499C2×C4C22×M1612yes—
500C2×C4C22×M1616yes—
501C2×C4C2×#13116yes—
502C2×C4#56812yes—
503C2×C4C23×D411yescharacteristic
504C2×C4C22×#4111yescharacteristic
505C2×C4C22×#4111yescharacteristic
506C2×C4C22×#4111yescharacteristic
507C2×C4C2×H(5,𝔽2)11yescharacteristic
508C2×C4#58511yescharacteristic
509C8—22no—
510C8—26no—
511C8—22no—
512C8—22no—
513C8—43no—
514C8—43no—
515C8—43no—
516C8#55014yescomplemented
517C22—26no—
518C22—26no—
519C22—26no—
520C22—43no—
521C22—43no—
522C22—43no—
523C22—43no—
524C22—43no—
525C22—43no—
526C22—43no—
527C22—43no—
528C22—43no—
529C22—43no—
530C22—43no—
531C22C22×C4×D411yescharacteristic
532C22C2×C4×#4111yescharacteristic
533C22C4×H(5,𝔽2)11yescharacteristic
534C22C23×M1611yescharacteristic
535C22C22×#13111yescharacteristic
536C22C2×#56811yescharacteristic
537C22#357311yescharacteristic
538C4—22no—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—26no—
543C4—43no—
544C4—43no—
545C4—43no—
546C4—43no—
547C4—41no—
548C4—41no—
549C4C2×#435112yes—
550C4C2×#55011yescharacteristic
551C4D4×#4111yescharacteristic
552C4#348011yescharacteristic
553C4#348311yescharacteristic
554C2—46no—
555C2—83no—
556C2C4×#55011yescharacteristic
557C2C22×#43511yescharacteristic
558C2#6014611yescharacteristic
559C2M16×#4111yescharacteristic
560C2#6021811yescharacteristic
561C2#6032511yescharacteristic
562C2#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×#2971C2semidirect
2. C2×C8×#41C2semidirect
4. #23174C2semidirect
5. #23087C2semidirect
6. #23084C2semidirect
7. #22797C2semidirect
8. Q8×C4⋊C8C2semidirect
9. #22530C2semidirect
10. #22530C2semidirect
11. #21682C2semidirect
12. #21487C2semidirect
13. #21486C2semidirect
14. Q8×#97C2semidirect
15. C4×#435C2semidirect
110. C8×Q8D4semidirect
189. C2×#41M16semidirect
286. C4⋊C8#41semidirect
300. #97#41semidirect
450. Q8#435semidirect
516. C8#550semidirect

Maximal subgroups

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

subgroupindex
C2×#29712
C2×C8×#412
C4×#5502
#231742
#230872
#230842
#227972
Q8×C4⋊C82
#225302
#216822
#214872
#214862
Q8×#972
C4×#4352