Finite Group Databasebeta

← #403530

Subgroups — #403530

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Q8C211yesmaximal, characteristic
2C2×#2066C212yesmaximal
3C2×#1933C211yesmaximal, characteristic
4#15201C211yesmaximal, characteristic
5#15158C211yesmaximal, characteristic
6#11231C212yesmaximal
7#11227C211yesmaximal, characteristic
8#11077C211yesmaximal, characteristic
9#10926C212yesmaximal
10#10880C212yesmaximal
11#10861C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×C8×Q8—28no—
14C2×C8×Q8—24no—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×C4×C8C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#395—21no—
24C2×#391C2212yes—
25C2×#391C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28#2066—28no—
29#1933—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36#1446—21no—
37#1442—21no—
38#1442—22no—
39#1441—22no—
40#1441—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—24no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C8×Q8—232no—
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8D418yes—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—22no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103Q814yescomplemented
82C2×C4×C8—24no—
83C2×C4×C8—24no—
84C2×C4×C8—24no—
85C2×C4×C8—24no—
86C2×C4×C8D414yes—
87#395—44no—
88#391—24no—
89#391—22no—
90#391—24no—
91#391—22no—
92#389—24no—
93#389—22no—
94#389—24no—
95#389—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—42no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8SD1614yes—
130C22×Q8C2×Q812yes—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C2×D414yes—
154C22×C8C2×D412yes—
155C22×C8C2×D414yes—
156C22×C8C2×D412yes—
157C22×C8C2×Q812yes—
158C22×C8#4112yes—
159C4×Q8—28no—
160C4×Q8—216no—
161C4×Q8—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C2×Q812yes—
198C2×C4⋊C4#4114yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42C2×D414yes—
206C2×C42C2×D412yes—
207C2×C42#4114yes—
208C2×C42#4112yes—
209#106—48no—
210C4⋊C8—28no—
211C4⋊C8—216no—
212#103—28no—
213#103—48no—
214#103—44no—
215C4×C8—216no—
216C4×C8—28no—
217#95—44no—
218#95—42no—
219#95—44no—
220#95—42no—
221#95—44no—
222#95—42no—
223#95—42no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—81no—
228#95—82no—
229#95—81no—
230#95—82no—
231C24C22×D411yescharacteristic
232C2×Q8—28no—
233C2×Q8—28no—
234C2×Q8—216no—
235C2×Q8—28no—
236C2×Q8—216no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4#12112yes—
308C22×C4#12114yes—
309C22×C4#12114yes—
310C22×C4#12112yes—
311C22×C4C2×SD1614yes—
312C22×C4C2×SD1612yes—
313C22×C4#13514yes—
314C22×C4#13512yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D412yes—
317C22×C4C22×Q811yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C42−1+412yes—
324C22×C42−1+411yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8#12014yes—
341C2×C8C22⋊Q814yes—
342C4⋊C4—216no—
343C4⋊C4—28no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—84no—
363C4⋊C4—88no—
364C42—24no—
365C42—22no—
366C42—22no—
367C42—24no—
368C42—24no—
369C42—24no—
370C42—24no—
371C23C2×#12112yes—
372C23C2×#12111yescharacteristic
373C23#54812yes—
374C23D4×Q811yescharacteristic
375C23#55011yescharacteristic
376C23C22×SD1611yescharacteristic
377C23C2×#13512yes—
378C23C2×#13511yescharacteristic
379C23#57711yescharacteristic
380C23#57711yescharacteristic
381C23#57812yes—
382Q8—232no—
383Q8—42no—
384Q8—42no—
385Q8—44no—
386Q8—81no—
387Q8—81no—
388Q8—82no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4#46514yes—
448C2×C4#46518yes—
449C2×C4C2×#12011yescharacteristic
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4#52511yescharacteristic
452C2×C4#52512yes—
453C2×C4#52912yes—
454C2×C4#53211yescharacteristic
455C2×C4#53312yes—
456C2×C4#53611yescharacteristic
457C2×C4#53711yescharacteristic
458C2×C4#54112yes—
459C2×C4#54212yes—
460C2×C4#54612yes—
461C2×C4#54712yes—
462C2×C4#54812yes—
463C2×C4D4×Q812yes—
464C2×C4#55411yescharacteristic
465C2×C4#55612yes—
466C2×C4#55611yescharacteristic
467C8—24no—
468C8—24no—
469C8—44no—
470C22#242612yes—
471C22#242912yes—
472C22#253811yescharacteristic
473C22#254511yescharacteristic
474C22#284711yescharacteristic
475C22C2×#46511yescharacteristic
476C22C2×#46512yes—
477C22#305011yescharacteristic
478C22#305012yes—
479C22#305911yescharacteristic
480C22#306011yescharacteristic
481C22#306312yes—
482C22#306312yes—
483C22#335912yes—
484C22#336012yes—
485C22#336412yes—
486C22#336512yes—
487C22Q8×SD1611yescharacteristic
488C22#338111yescharacteristic
489C22#338312yes—
490C22#339211yescharacteristic
491C22#339312yes—
492C22#339411yescharacteristic
493C4—28no—
494C4—24no—
495C4—42no—
496C4—44no—
497C4—82no—
498C4—82no—
499C4—84no—
500C4—161no—
501C4—162no—
502C4—161no—
503C4#239611yescharacteristic
504C4#240911yescharacteristic
505C4#243011yescharacteristic
506C4#246512yes—
507C4#246912yes—
508C4#263711yescharacteristic
509C2#1487711yescharacteristic
510C2#2529312yes—
511C2#2529712yes—
512C2#2547212yes—
513C2#2547412yes—
514C2#2826411yescharacteristic
515C2#2826811yescharacteristic
516C2#2863911yescharacteristic
517C2#2864211yescharacteristic
518C2#3292911yescharacteristic
519C2#3294011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
81. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Q82
C2×#20662
C2×#19332
#152012
#151582
#112312
#112272
#110772
#109262
#108802
#108612