Finite Group Databasebeta

← #432811

Subgroups — #432811

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2451C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11327C212yesmaximal
7#11220C212yesmaximal
8#11067C212yesmaximal
9#10926C211yesmaximal, characteristic
10#10874C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2451—24no—
19#2451—24no—
20C2×#398—21no—
21C2×#398—21no—
22C2×#398—21no—
23C2×#395—21no—
24C2×#391—21no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#387—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2096—24no—
34#1990—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8D414yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—42no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103Q818yescomplemented
86C2×C4×C8—22no—
87C2×C4×C8—42no—
88#398—28no—
89#398—44no—
90#398—44no—
91#395—28no—
92#391—28no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391D414yes—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—42no—
102#389—42no—
103#387—28no—
104#384—22no—
105#384—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—24no—
127#336—22no—
128#336—42no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—44no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—82no—
140C22×Q8C2×D412yes—
141C22×Q8C2×Q814yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C2×Q814yes—
168C22×C8#4114yes—
169C4⋊Q8—28no—
170C4⋊Q8—44no—
171C4⋊Q8—48no—
172C4×Q8—48no—
173C4×Q8—44no—
174C4×Q8—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4SD1614yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×Q814yes—
221C2×C4⋊C4#4114yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—44no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236#106—48no—
237#103—48no—
238#103—48no—
239C4×C8—44no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—44no—
252#95—42no—
253C24C22×D411yescharacteristic
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—48no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—48no—
271C2×Q8—84no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×Q8C4⋊Q814yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4#12012yes—
334C22×C4#12012yes—
335C22×C4C2×SD1612yes—
336C22×C4C2×SD1612yes—
337C22×C4C2×SD1612yes—
338C22×C4#13512yes—
339C22×C4AGL(1,ℤ/8)12yes—
340C22×C4#13712yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×Q812yes—
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4H(5,𝔽2)12yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—88no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4#12314yes—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—48no—
392C42—41no—
393C42—41no—
394C42—42no—
395C42—42no—
396C42—42no—
397C23C2×#12011yescharacteristic
398C23#53211yescharacteristic
399C23#53511yescharacteristic
400C23#54612yes—
401C23D4×Q812yes—
402C23C22×SD1611yescharacteristic
403C23C2×#13511yescharacteristic
404C23C2×AGL(1,ℤ/8)11yescharacteristic
405C23C2×#13711yescharacteristic
406C23#57712yes—
407C23#57812yes—
408Q8—28no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—816no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4#44814yes—
478C2×C4#44914yes—
479C2×C4C2×#12311yescharacteristic
480C2×C4C2×C4⋊Q811yescharacteristic
481C2×C4#53312yes—
482C2×C4#53411yescharacteristic
483C2×C4#53611yescharacteristic
484C2×C4#53912yes—
485C2×C4#54212yes—
486C2×C4#54312yes—
487C2×C4#54612yes—
488C2×C4#54612yes—
489C2×C4D4×Q812yes—
490C2×C4#55012yes—
491C2×C4#55212yes—
492C2×C4#55412yes—
493C2×C4#55412yes—
494C2×C4#56212yes—
495C8—44no—
496C8—44no—
497C22#240511yescharacteristic
498C22#243111yescharacteristic
499C22#243411yescharacteristic
500C22#255612yes—
501C22#269212yes—
502C22C2×#44811yescharacteristic
503C22C2×#44911yescharacteristic
504C22#300911yescharacteristic
505C22#301111yescharacteristic
506C22#316711yescharacteristic
507C22#317812yes—
508C22#318211yescharacteristic
509C22#320811yescharacteristic
510C22#321011yescharacteristic
511C22#322512yes—
512C22#329612yes—
513C22#329812yes—
514C22#330612yes—
515C22#331212yes—
516C22Q8×SD1612yes—
517C22#338312yes—
518C22#338612yes—
519C22#338712yes—
520C4—24no—
521C4—24no—
522C4—44no—
523C4—44no—
524C4—42no—
525C4—42no—
526C4—81no—
527C4—82no—
528C4—84no—
529C4—81no—
530C4—82no—
531C4—162no—
532C4#242312yes—
533C4#254912yes—
534C4#268712yes—
535C4#273212yes—
536C2#1546611yescharacteristic
537C2#2403811yescharacteristic
538C2#2404111yescharacteristic
539C2#2559811yescharacteristic
540C2#2561311yescharacteristic
541C2#2583711yescharacteristic
542C2#2584011yescharacteristic
543C2#2894712yes—
544C2#2897012yes—
545C2#3152012yes—
546C2#3152112yes—
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
85. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24512
C2×#20962
C2×#19902
#151512
#113842
#113272
#112202
#110672
#109262
#108742
#107892