Finite Group Databasebeta

← #444575

Subgroups — #444575

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic
3C2×#2084C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11380C211yesmaximal, characteristic
7#11293C212yesmaximal
8#11220C212yesmaximal
9#10908C212yesmaximal
10#10884C211yesmaximal, characteristic
11#10884C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#106C2212yes—
16#2546—24no—
17#2546—24no—
18C2×#398—21no—
19C2×#398—21no—
20C2×#391—21no—
21C2×#391—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2100—24no—
30#2084—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1441—21no—
43#1441—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8D411yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—28no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—44no—
57C2×C4⋊Q8—44no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—24no—
61C2×C4×Q8—42no—
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—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86#398—28no—
87#398—28no—
88#391—28no—
89#391—44no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389D414yes—
97#389D414yes—
98#384—22no—
99#384—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—42no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340Q818yescomplemented
120#336—22no—
121#336—22no—
122#336—24no—
123#336—42no—
124C22×Q8—24no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8SD1614yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
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×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×Q814yes—
165C4⋊Q8—48no—
166C4⋊Q8—48no—
167C4⋊Q8—432no—
168C4×Q8—48no—
169C4×Q8—48no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
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—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
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—22no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
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×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42#4114yes—
245#106—48no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—42no—
252#95—44no—
253#95—44no—
254#95—82no—
255#95—82no—
256C24C22×D411yescharacteristic
257C2×Q8—28no—
258C2×Q8—28no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—84no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4#12012yes—
337C22×C4#12012yes—
338C22×C4#12312yes—
339C22×C4#12312yes—
340C22×C4C2×SD1612yes—
341C22×C4C2×SD1612yes—
342C22×C4C2×SD1612yes—
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4AGL(1,ℤ/8)12yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×Q812yes—
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352C22×C42−1+412yes—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4#12414yes—
394C4⋊C4C4⋊Q814yes—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—48no—
400C42—42no—
401C42—42no—
402C42—84no—
403C23C2×#12011yescharacteristic
404C23C2×#12311yescharacteristic
405C23#53411yescharacteristic
406C23#54812yes—
407C23D4×Q812yes—
408C23C22×SD1611yescharacteristic
409C23C2×AGL(1,ℤ/8)11yescharacteristic
410C23C2×AGL(1,ℤ/8)11yescharacteristic
411C23#57511yescharacteristic
412C23#57712yes—
413C23#57812yes—
414Q8—416no—
415Q8—816no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4#45014yes—
489C2×C4#48114yes—
490C2×C4C2×#12411yescharacteristic
491C2×C4C2×C4⋊Q811yescharacteristic
492C2×C4#53312yes—
493C2×C4#53611yescharacteristic
494C2×C4#53611yescharacteristic
495C2×C4#54112yes—
496C2×C4#54212yes—
497C2×C4#54412yes—
498C2×C4#54612yes—
499C2×C4#54612yes—
500C2×C4#54812yes—
501C2×C4D4×Q812yes—
502C2×C4#55412yes—
503C2×C4#55612yes—
504C2×C4#55712yes—
505C2×C4#56312yes—
506C8—44no—
507C8—82no—
508C22#240511yescharacteristic
509C22#243411yescharacteristic
510C22#243611yescharacteristic
511C22#244612yes—
512C22#272912yes—
513C22C2×#45011yescharacteristic
514C22#300611yescharacteristic
515C22#301011yescharacteristic
516C22#301111yescharacteristic
517C22C2×#48111yescharacteristic
518C22#309411yescharacteristic
519C22#309712yes—
520C22#318711yescharacteristic
521C22#319211yescharacteristic
522C22#320112yes—
523C22#335912yes—
524C22#336712yes—
525C22#336812yes—
526C22#337112yes—
527C22Q8×SD1612yes—
528C22#338412yes—
529C22#338612yes—
530C22#339012yes—
531C4—24no—
532C4—24no—
533C4—44no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—44no—
538C4—84no—
539C4—82no—
540C4—82no—
541C4—81no—
542C4—162no—
543C4#249212yes—
544C4#255012yes—
545C4#271312yes—
546C4#271612yes—
547C2#1578311yescharacteristic
548C2#2400511yescharacteristic
549C2#2401411yescharacteristic
550C2#2583411yescharacteristic
551C2#2584911yescharacteristic
552C2#2592911yescharacteristic
553C2#2594411yescharacteristic
554C2#2627812yes—
555C2#2629412yes—
556C2#3256612yes—
557C2#3258912yes—
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
119. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#21002
C2×#20842
#139262
#113842
#113802
#112932
#112202
#109082
#108842