Finite Group Databasebeta

← #6490748

Subgroups — #6490748

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 544 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#33108C214yesmaximal
4#30060C214yesmaximal, complemented
5#29255C214yesmaximal, complemented
6#29206C212yesmaximal, complemented
7#28748C212yesmaximal, complemented
8#27905C212yesmaximal
9#25902C212yesmaximal
10#24063C214yesmaximal
11#22218C214yesmaximal, complemented
12#20777C211yesmaximal, characteristic
13C2×#533C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8—21no—
16C2×#487C2214yes—
17C2×#483C2218yes—
18#2868C2211yescharacteristic
19#2868C2211yescharacteristic
20#2862C2212yes—
21#2859—21no—
22#2859—21no—
23#2857—21no—
24#2857—21no—
25#2857C2212yes—
26#2857C2212yes—
27C4×C4⋊Q8—21no—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8C2212yes—
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C2×#391—21no—
40C2×#389—21no—
41C2×#389C2212yes—
42#2285—21no—
43#2285—21no—
44C2×#384C2211yescharacteristic
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4C2212yes—
47C2×C43C2211yescharacteristic
48#2096C2218yes—
49#2058C2218yes—
50#2043C2218yes—
51#2024C2218yes—
52#1990C2214yes—
53#1981C2218yes—
54#1956C2214yes—
55#1948C2218yes—
56#1927C22116yes—
57#1872C2218yes—
58#1865C22116yes—
59#1844C2214yes—
60#1839C2212yes—
61#1831C2212yes—
62#1817C2212yes—
63#1797C2212yes—
64C4×C4⋊C8C2212yes—
65C4×C22⋊C8C2212yes—
66#1744C2212yes—
67C23×Q8—21no—
68#533—48no—
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8C2312yes—
75C2×C4⋊Q8C2312yes—
76C2×C4⋊Q8C2312yes—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×#125—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8C2314yes—
83C2×C4×Q8—22no—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×Q8—48no—
89C2×C4×Q8C2314yes—
90C2×#117—22no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C42—22no—
98C22×C42D411yescharacteristic
99C22×C42D411yescharacteristic
100C22×C42D411yescharacteristic
101C22×C42D411yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#487—216no—
106#483—416no—
107C2×#106C23116yes—
108C2×C4⋊C8C2314yes—
109C2×#103C23116yes—
110C2×#103C23116yes—
111C2×C22⋊C8C2314yes—
112C2×C4×C8C2314yes—
113#391—22no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391—22no—
118#391—22no—
119#391C2314yes—
120#391C2314yes—
121#389—22no—
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#389C2314yes—
136#389C2314yes—
137#384—21no—
138#384—21no—
139#384—21no—
140#384—21no—
141#384—21no—
142#384—21no—
143#384—21no—
144#384—21no—
145#384C2312yes—
146#384C2312yes—
147#384C2312yes—
148#384C2312yes—
149#382—22no—
150#382—22no—
151#380—22no—
152#380—22no—
153#380—22no—
154#380—22no—
155C4×C4⋊C4—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⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C2×#95—22no—
168C2×#95—22no—
169C43D411yescharacteristic
170C43D411yescharacteristic
171C43D411yescharacteristic
172C43D411yescharacteristic
173C43C2311yescharacteristic
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176C43C2311yescharacteristic
177#340C23116yes—
178#336C2314yes—
179#336C2314yes—
180#336C2314yes—
181#336C2314yes—
182C22×Q8—28no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8C2414yes—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2411yescharacteristic
198C22×C8C2418yes—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—28no—
202C4⋊Q8—24no—
203C4⋊Q8—28no—
204C4⋊Q8—28no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209C4⋊Q8#4118yes—
210#125—48no—
211C22⋊Q8—216no—
212C22⋊Q8—48no—
213C4×Q8—48no—
214C4×Q8—48no—
215C4×Q8—48no—
216C4×Q8—48no—
217C4×Q8—48no—
218C4×Q8—48no—
219#117—48no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2414yes—
252C2×C22⋊C4—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—42no—
262C2×C42—48no—
263C2×C42SD1618yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292#106—416no—
293C4⋊C8—28no—
294#103—232no—
295#103—416no—
296C22⋊C8—28no—
297C4×C8—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306C24C22×D411yescharacteristic
307C2×Q8—28no—
308C2×Q8—28no—
309C2×Q8—28no—
310C2×Q8—28no—
311C2×Q8—28no—
312C2×Q8—44no—
313C2×Q8—432no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4C2×SD1614yes—
347C22×C4C2×SD1614yes—
348C22×C4C2×SD1614yes—
349C22×C4#13714yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
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×C4H(5,𝔽2)14yes—
364C22×C4C2511yescharacteristic
365C2×C8—28no—
366C2×C8—28no—
367C2×C8—28no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4C2×#4118yes—
395C22⋊C4—48no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
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—41no—
416C42—41no—
417C42—48no—
418C42—48no—
419C42—48no—
420C42C2×SD1614yes—
421C42C2×SD1614yes—
422C42C2×SD1614yes—
423C42C2×SD1614yes—
424C42#13714yes—
425C42C2×#4114yes—
426C422−1+414yes—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23#53512yes—
432C23C22×SD1612yes—
433C23C2×#13712yes—
434C23C23×D411yescharacteristic
435Q8—416no—
436Q8—416no—
437Q8—416no—
438Q8—88no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—28no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—416no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—88no—
482C2×C4#52512yes—
483C2×C4#52512yes—
484C2×C4#53512yes—
485C2×C4#53612yes—
486C2×C4#53612yes—
487C2×C4C22×SD1612yes—
488C2×C4C22×SD1612yes—
489C2×C4C22×SD1612yes—
490C2×C4C22×SD1612yes—
491C2×C4C22×SD1612yes—
492C2×C4C22×SD1612yes—
493C2×C4C2×#13712yes—
494C2×C4C2×#13712yes—
495C2×C4C2×#13712yes—
496C2×C4C2×#13712yes—
497C2×C4C2×#13712yes—
498C2×C4#57512yes—
499C2×C4C22×#4112yes—
500C2×C4C2×H(5,𝔽2)12yes—
501C2×C4C2×2−1+412yes—
502C8—48no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22#320614yes—
510C22C2×#52511yescharacteristic
511C22C2×#53511yescharacteristic
512C22C2×#53611yescharacteristic
513C22C23×SD1611yescharacteristic
514C22C22×#13711yescharacteristic
515C22C22×#13711yescharacteristic
516C22C2×#57511yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—48no—
526C4—41no—
527C4—41no—
528C4—84no—
529C4—84no—
530C4—84no—
531C4#310114yes—
532C4#311314yes—
533C4#322914yes—
534C4#352214yes—
535C2—24no—
536C2#3524911yescharacteristic
537C2C2×#310111yescharacteristic
538C2C2×#311311yescharacteristic
539C2C2×#320611yescharacteristic
540C2#6080411yescharacteristic
541C2C2×#322911yescharacteristic
542C2#6086511yescharacteristic
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. #30060C2semidirect
5. #29255C2semidirect
6. #29206C2semidirect
7. #28748C2semidirect
11. #22218C2semidirect

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
#331082
#300602
#292552
#292062
#287482
#279052
#259022
#240632
#222182
#207772