Finite Group Databasebeta

← #466964

Subgroups — #466964

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal
2C2×#2024C212yesmaximal
3#15128C211yesmaximal, characteristic
4#14155C211yesmaximal, characteristic
5#11202C211yesmaximal, characteristic
6#11122C212yesmaximal
7#11067C212yesmaximal
8#11010C212yesmaximal
9#10908C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C2×#395—21no—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2066—28no—
25#2024—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32#1446—21no—
33#1445—21no—
34#1445—21no—
35#1444—21no—
36#1441—21no—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×Q8—21no—
42C23×Q8C2312yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—42no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8—41no—
52C2×C4⋊Q8—41no—
53C2×C4⋊Q8—41no—
54C2×C4⋊Q8Q814yescomplemented
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—42no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—41no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85#395—44no—
86#391—24no—
87#391—24no—
88#391—42no—
89#391—42no—
90#389—24no—
91#389—42no—
92C4×C4⋊C4—41no—
93C4×C4⋊C4—41no—
94C4×C4⋊C4—41no—
95C4×C4⋊C4—41no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—24no—
109#340—42no—
110#340D418yes—
111#336—24no—
112#336—24no—
113#336—41no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—44no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×Q8C2×Q814yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×D414yes—
158C4⋊Q8—44no—
159C4⋊Q8—44no—
160C4⋊Q8—44no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8—48no—
164C4⋊Q8—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4SD1614yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4#4114yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C2×Q812yes—
216C2×C42#4112yes—
217#106—48no—
218C4⋊C8—44no—
219#103—48no—
220#103—48no—
221#103—48no—
222#103—48no—
223#95—41no—
224#95—41no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—42no—
231#95—42no—
232#95—44no—
233#95—42no—
234#95—82no—
235C24C22×D411yescharacteristic
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—48no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—88no—
266C2×Q8—84no—
267C2×Q8—88no—
268C2×Q8—84no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4C2×SD1612yes—
325C22×C4C2×SD1612yes—
326C22×C4C2×SD1612yes—
327C22×C4#13712yes—
328C22×C4#13712yes—
329C22×C4C22×D412yes—
330C22×C4C22×Q811yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C4H(5,𝔽2)12yes—
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C42−1+412yes—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—88no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—42no—
376C42—42no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—84no—
382C23C2×C22⋊Q811yescharacteristic
383C23#53411yescharacteristic
384C23#53511yescharacteristic
385C23D4212yes—
386C23#54812yes—
387C23C22×SD1611yescharacteristic
388C23C2×#13711yescharacteristic
389C23C2×#13711yescharacteristic
390C23#57511yescharacteristic
391C23#57712yes—
392C23#57812yes—
393Q8—44no—
394Q8—44no—
395Q8—44no—
396Q8—44no—
397Q8—816no—
398Q8—82no—
399Q8—82no—
400Q8—82no—
401Q8—82no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#47514yes—
462C2×C4#53212yes—
463C2×C4#53512yes—
464C2×C4#53712yes—
465C2×C4#53811yescharacteristic
466C2×C4#53811yescharacteristic
467C2×C4#53812yes—
468C2×C4#54212yes—
469C2×C4#54212yes—
470C2×C4#54311yescharacteristic
471C2×C4#54311yescharacteristic
472C2×C4D4211yescharacteristic
473C2×C4#54612yes—
474C2×C4D4×Q812yes—
475C2×C4#55412yes—
476C2×C4#55511yescharacteristic
477C2×C4#56011yescharacteristic
478C2×C4#56212yes—
479C2×C4#56311yescharacteristic
480C8—44no—
481C8—82no—
482C22#245212yes—
483C22#262611yescharacteristic
484C22#266911yescharacteristic
485C22#267612yes—
486C22#273111yescharacteristic
487C22C2×#47511yescharacteristic
488C22#307811yescharacteristic
489C22#308212yes—
490C22#318711yescharacteristic
491C22#319511yescharacteristic
492C22#320011yescharacteristic
493C22#320211yescharacteristic
494C22#320611yescharacteristic
495C22#321511yescharacteristic
496C22#322512yes—
497C22#326712yes—
498C22#327012yes—
499C22#327612yes—
500C22D4×SD1612yes—
501C22#335912yes—
502C22#336912yes—
503C22#337212yes—
504C22#337712yes—
505C4—24no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—84no—
511C4—81no—
512C4—84no—
513C4—84no—
514C4—84no—
515C4#255511yescharacteristic
516C4#266212yes—
517C4#268311yescharacteristic
518C4#269212yes—
519C4#278411yescharacteristic
520C4#283011yescharacteristic
521C2#1655611yescharacteristic
522C2#2646212yes—
523C2#2647412yes—
524C2#2995111yescharacteristic
525C2#2996011yescharacteristic
526C2#3050011yescharacteristic
527C2#3051711yescharacteristic
528C2#3087812yes—
529C2#3090112yes—
530C2#3263011yescharacteristic
531C2#3264511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
54. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#20242
#151282
#141552
#112022
#111222
#110672
#110102
#109082