Finite Group Databasebeta

← #444554

Subgroups — #444554

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11384C211yesmaximal, characteristic
7#11292C212yesmaximal
8#11207C212yesmaximal
9#10884C212yesmaximal
10#10880C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2551—24no—
17#2551—24no—
18C2×#398—21no—
19C2×#398—21no—
20C2×#395—21no—
21C2×#395—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2099—24no—
31#1990—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×Q8—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—28no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—44no—
60C2×C4⋊Q8—44no—
61C2×C4⋊Q8D414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C4×C8—22no—
84C2×C4×C8—42no—
85#398—44no—
86#398—44no—
87#395—28no—
88#395—28no—
89#389—28no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—24no—
94#389—24no—
95#389D414yes—
96#384—22no—
97#384—22no—
98#384—22no—
99#384—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#340Q818yescomplemented
122#336—22no—
123#336—24no—
124#336—42no—
125C22×Q8—24no—
126C22×Q8—24no—
127C22×Q8—44no—
128C22×Q8—44no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—82no—
134C22×Q8—82no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
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×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×Q814yes—
162C4⋊Q8—28no—
163C4⋊Q8—44no—
164C4⋊Q8—48no—
165C4⋊Q8—48no—
166C4⋊Q8—432no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
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—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4D814yes—
219C2×C4⋊C4SD1614yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×Q814yes—
225C2×C4⋊C4C2×Q814yes—
226C2×C4⋊C4#4114yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42#4114yes—
242C4×C8—44no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—44no—
249#95—42no—
250#95—44no—
251#95—42no—
252#95—44no—
253C24C22×D411yescharacteristic
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—48no—
258C2×Q8—84no—
259C2×Q8—88no—
260C2×Q8—88no—
261C2×Q8—88no—
262C2×Q8—84no—
263C2×Q8—88no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
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—41no—
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#12012yes—
319C22×C4#12012yes—
320C22×C4#12412yes—
321C22×C4#12412yes—
322C22×C4C2×D812yes—
323C22×C4C2×D812yes—
324C22×C4C2×D812yes—
325C22×C4C2×SD1612yes—
326C22×C4C2×SD1612yes—
327C22×C4C2×SD1612yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4#13712yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×Q812yes—
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4H(5,𝔽2)12yes—
336C22×C42−1+412yes—
337C22×C42−1+412yes—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—42no—
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—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—88no—
381C4⋊C4#12314yes—
382C4⋊C4C4⋊Q814yes—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—48no—
389C42—41no—
390C42—41no—
391C42—42no—
392C23C2×#12011yescharacteristic
393C23C2×#12411yescharacteristic
394C23#53611yescharacteristic
395C23#54612yes—
396C23D4×Q812yes—
397C23C22×D811yescharacteristic
398C23C22×SD1611yescharacteristic
399C23C2×AGL(1,ℤ/8)11yescharacteristic
400C23C2×#13711yescharacteristic
401C23#57712yes—
402C23#57812yes—
403Q8—816no—
404Q8—816no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
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—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#44714yes—
469C2×C4#44814yes—
470C2×C4#48614yes—
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×C4⋊Q811yescharacteristic
473C2×C4#53312yes—
474C2×C4#53411yescharacteristic
475C2×C4#53611yescharacteristic
476C2×C4#54112yes—
477C2×C4#54212yes—
478C2×C4#54412yes—
479C2×C4#54612yes—
480C2×C4#54812yes—
481C2×C4#54812yes—
482C2×C4D4×Q812yes—
483C2×C4#55412yes—
484C2×C4#55612yes—
485C2×C4#55712yes—
486C2×C4#56312yes—
487C8—44no—
488C8—82no—
489C22#240511yescharacteristic
490C22#243411yescharacteristic
491C22#243611yescharacteristic
492C22#249212yes—
493C22#271312yes—
494C22C2×#44711yescharacteristic
495C22C2×#44811yescharacteristic
496C22#300911yescharacteristic
497C22#301111yescharacteristic
498C22C2×#48611yescharacteristic
499C22#313712yes—
500C22#314111yescharacteristic
501C22#322611yescharacteristic
502C22#322711yescharacteristic
503C22#324212yes—
504C22#329412yes—
505C22#329812yes—
506C22#331012yes—
507C22#331212yes—
508C22Q8×D812yes—
509C22Q8×SD1612yes—
510C22#338612yes—
511C22#338712yes—
512C4—24no—
513C4—24no—
514C4—24no—
515C4—42no—
516C4—42no—
517C4—44no—
518C4—42no—
519C4—44no—
520C4—84no—
521C4—82no—
522C4—82no—
523C4—162no—
524C4#244612yes—
525C4#255012yes—
526C4#271612yes—
527C4#272912yes—
528C2#1578311yescharacteristic
529C2#2403211yescharacteristic
530C2#2404111yescharacteristic
531C2#2583711yescharacteristic
532C2#2583811yescharacteristic
533C2#2590311yescharacteristic
534C2#2591411yescharacteristic
535C2#2781312yes—
536C2#2782712yes—
537C2#3213012yes—
538C2#3213412yes—
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
121. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#20992
C2×#19902
#151512
#113842
#112922
#112072
#108842
#108802
#108742