Finite Group Databasebeta

← #6489109

Subgroups — #6489109

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#32906C214yesmaximal
4#30042C214yesmaximal, complemented
5#29141C212yesmaximal, complemented
6#28440C212yesmaximal, complemented
7#28161C214yesmaximal, complemented
8#27972C212yesmaximal
9#25722C212yesmaximal
10#23934C214yesmaximal
11#22270C214yesmaximal, complemented
12#20777C211yesmaximal, characteristic
13C2×#533C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C2×#494C2214yes—
17C2×#462C2218yes—
18C2×#451C2218yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2212yes—
22#2859—21no—
23#2859—21no—
24#2857—21no—
25#2857—21no—
26#2857C2212yes—
27#2857C2212yes—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C2×#391C2212yes—
41C2×#389—21no—
42C2×#389—21no—
43#2285—21no—
44#2285—21no—
45C2×#384C2211yescharacteristic
46C2×C4×C4⋊C4—21no—
47C2×C4×C4⋊C4—21no—
48C2×C43C2211yescharacteristic
49#2072C2218yes—
50#2024C2218yes—
51#2016C2218yes—
52#2016C2218yes—
53#1956C2214yes—
54#1956C2214yes—
55#1878C22116yes—
56#1878C22116yes—
57#1872C2218yes—
58#1839C2212yes—
59#1834C2214yes—
60#1831C2212yes—
61#1817C2212yes—
62#1803C2218yes—
63#1797C2212yes—
64C4×C4⋊C8C2212yes—
65C4×C22⋊C8C2212yes—
66#1744C2212yes—
67C23×Q8C2312yes—
68C22×Q16C23116yes—
69#533—48no—
70C2×C4⋊Q8—28no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8C2312yes—
76C2×C4⋊Q8C2312yes—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×#125—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8C2314yes—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—48no—
88C2×C4×Q8C2314yes—
89C2×C4×Q8C2314yes—
90C2×C4×Q8C2314yes—
91C2×#117—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4C2312yes—
98C22×C42—22no—
99C22×C42D411yescharacteristic
100C22×C42D411yescharacteristic
101C22×C42D411yescharacteristic
102C22×C42D411yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#494—216no—
107#462—416no—
108#451—416no—
109C2×C4⋊C8C2314yes—
110C2×#103C23116yes—
111C2×#103C23116yes—
112C2×C22⋊C8C2314yes—
113C2×C4×C8C2314yes—
114#391—22no—
115#391—22no—
116#391—22no—
117#391—22no—
118#391C2314yes—
119#391C2314yes—
120#391C2314yes—
121#391C2314yes—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#384—21no—
139#384—21no—
140#384—21no—
141#384—21no—
142#384—21no—
143#384—21no—
144#384—21no—
145#384—21no—
146#384C2312yes—
147#384C2312yes—
148#384C2312yes—
149#384C2312yes—
150#382—22no—
151#382—22no—
152#380—22no—
153#380—22no—
154#380—22no—
155#380—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C2×#95—22no—
169C2×#95—22no—
170C43D411yescharacteristic
171C43D411yescharacteristic
172C43D411yescharacteristic
173C43D411yescharacteristic
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176C43C2311yescharacteristic
177C43C2311yescharacteristic
178#340C23116yes—
179#336C2314yes—
180#336C2314yes—
181#336C2314yes—
182#336C2314yes—
183C22×Q8—28no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—28no—
187C22×Q8—44no—
188C22×Q8C2414yes—
189C22×Q8C2414yes—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
199C2×Q16—232no—
200C2×Q16—416no—
201C2×Q16—416no—
202C22×C8C2418yes—
203C4⋊Q8—28no—
204C4⋊Q8—28no—
205C4⋊Q8—28no—
206C4⋊Q8—24no—
207C4⋊Q8—28no—
208C4⋊Q8—28no—
209C4⋊Q8—44no—
210C4⋊Q8—44no—
211C4⋊Q8—44no—
212C4⋊Q8—44no—
213C4⋊Q8#4118yes—
214#125—48no—
215C22⋊Q8—216no—
216C22⋊Q8—48no—
217C4×Q8—216no—
218C4×Q8—48no—
219C4×Q8—48no—
220C4×Q8—48no—
221C4×Q8—48no—
222C4×Q8—48no—
223#117—48no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2414yes—
256C2×C22⋊C4—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—42no—
266C2×C42—48no—
267C2×C42Q1618yescomplemented
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C4⋊C8—28no—
297#103—416no—
298#103—416no—
299C22⋊C8—28no—
300C4×C8—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309C24C22×D411yescharacteristic
310C2×Q8—28no—
311C2×Q8—24no—
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—28no—
316C2×Q8—28no—
317C2×Q8—432no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8C2×#4118yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4C2×Q1614yes—
351C22×C4C2×Q1614yes—
352C22×C4C2×Q1614yes—
353C22×C4#13714yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4H(5,𝔽2)14yes—
368C22×C4C2511yescharacteristic
369Q16—832no—
370C2×C8—28no—
371C2×C8—28no—
372C2×C8—28no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C22⋊C4—48no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—41no—
420C42—41no—
421C42—48no—
422C42—48no—
423C42—48no—
424C42C2×Q1614yes—
425C42C2×Q1614yes—
426C42C2×Q1614yes—
427C42C2×Q1614yes—
428C42#13714yes—
429C42C2×#4114yes—
430C42H(5,𝔽2)14yes—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23#53512yes—
436C23C22×Q1612yes—
437C23C2×#13712yes—
438C23C23×D411yescharacteristic
439Q8—216no—
440Q8—416no—
441Q8—48no—
442Q8—416no—
443Q8—88no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—28no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—416no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—88no—
487C2×C4#52512yes—
488C2×C4#52512yes—
489C2×C4#53512yes—
490C2×C4#53512yes—
491C2×C4#53512yes—
492C2×C4C22×Q1612yes—
493C2×C4C22×Q1612yes—
494C2×C4C22×Q1612yes—
495C2×C4C22×Q1612yes—
496C2×C4C22×Q1612yes—
497C2×C4C22×Q1612yes—
498C2×C4C2×#13712yes—
499C2×C4C2×#13712yes—
500C2×C4C2×#13712yes—
501C2×C4C2×#13712yes—
502C2×C4C2×#13712yes—
503C2×C4#57512yes—
504C2×C4C22×#4112yes—
505C2×C4C2×H(5,𝔽2)12yes—
506C2×C4C2×H(5,𝔽2)12yes—
507C8—48no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22#320914yes—
515C22C2×#52511yescharacteristic
516C22C2×#53511yescharacteristic
517C22C2×#53511yescharacteristic
518C22C23×Q1611yescharacteristic
519C22C22×#13711yescharacteristic
520C22C22×#13711yescharacteristic
521C22C2×#57511yescharacteristic
522C4—22no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—41no—
531C4—41no—
532C4—48no—
533C4—84no—
534C4—84no—
535C4—84no—
536C4#310414yes—
537C4#311314yes—
538C4#320914yes—
539C4#351914yes—
540C2—24no—
541C2#3523711yescharacteristic
542C2C2×#310411yescharacteristic
543C2C2×#311311yescharacteristic
544C2C2×#320911yescharacteristic
545C2C2×#320911yescharacteristic
546C2#6080411yescharacteristic
547C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #30042C2semidirect
5. #29141C2semidirect
6. #28440C2semidirect
7. #28161C2semidirect
11. #22270C2semidirect
267. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×C4×C4⋊Q82
#329062
#300422
#291412
#284402
#281612
#279722
#257222
#239342
#222702
#207772