Finite Group Databasebeta

← #401944

Subgroups — #401944

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1862C211yesmaximal, characteristic
2C2×#1802C211yesmaximal, characteristic
3#15153C211yesmaximal, characteristic, complemented
4D4×#95C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11315C212yesmaximal, complemented
7#11256C212yesmaximal, complemented
8#11255C211yesmaximal, characteristic, complemented
9#11227C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C2×#400—21no—
20C2×#400—21no—
21C2×#389C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C22×#95—21no—
24#1862—24no—
25#1802—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1445—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C22⋊#95—21no—
41C22⋊#95—22no—
42C4×#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C24×C4—22no—
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4—22no—
55C2×C4×D4—42no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—42no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—42no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—24no—
78C2×#102—22no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102Q814yes—
83#400—44no—
84#400—44no—
85#389—22no—
86#389—22no—
87#384—22no—
88#384—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—44no—
99C2×#95D411yescharacteristic
100C2×#95D412yes—
101C2×#95D411yescharacteristic
102#340—24no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#336—22no—
109#336—42no—
110#336—42no—
111C25—21no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4C2×D412yes—
118C22×D4C2×Q812yes—
119C22×D4#4112yes—
120C22×D4#4114yes—
121C22×D4#4114yes—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—28no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8C2×Q812yes—
161C4×D4—28no—
162C4×D4—44no—
163C4×D4—48no—
164C4×D4—48no—
165C4×D4—48no—
166C4×D4—48no—
167C4×D4—48no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
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—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42C2×D412yes—
216C2×C42#4112yes—
217#107—44no—
218C4⋊C8—44no—
219#102—28no—
220#102—44no—
221#102—48no—
222#95—22no—
223#95—42no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—44no—
228#95—42no—
229#95—44no—
230#95—44no—
231#95—81no—
232#95—82no—
233#95D814yes—
234C24—22no—
235C24—24no—
236C24—24no—
237C24—22no—
238C24—22no—
239C24—41no—
240C24—41no—
241C24—42no—
242C24—42no—
243C24—41no—
244C24—41no—
245C24C22×D411yescharacteristic
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4C22⋊Q814yes—
260C2×D4#12314yes—
261C2×D4#12414yes—
262C2×D4#12518yes—
263C2×D4#12618yes—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—48no—
287C22×C4—416no—
288C22×C4—416no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—48no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—84no—
319C22×C4—88no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4C2×D814yes—
329C22×C4C2×D812yes—
330C22×C4AGL(1,ℤ/8)12yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×Q811yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4112yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4H(5,𝔽2)12yes—
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C42−1+411yescharacteristic
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C22⋊C4—48no—
380C22⋊C4—416no—
381C42—22no—
382C42—22no—
383C42—24no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—28no—
392C23—28no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—42no—
398C23—44no—
399C23—42no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23C2×#12111yescharacteristic
412C23#52512yes—
413C23#54611yescharacteristic
414C23#54612yes—
415C23D4×Q811yescharacteristic
416C23C22×D811yescharacteristic
417C23C2×AGL(1,ℤ/8)11yescharacteristic
418C23#57512yes—
419C23#57712yes—
420C23#57711yescharacteristic
421C23#57811yescharacteristic
422D4—24no—
423D4—24no—
424D4—28no—
425D4—24no—
426D4—28no—
427D4#40018yescomplemented
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—816no—
461C2×C4—832no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
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—81no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4#45914yes—
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12411yescharacteristic
488C2×C4C2×#12512yes—
489C2×C4C2×#12612yes—
490C2×C4#52912yes—
491C2×C4#53211yescharacteristic
492C2×C4#53411yescharacteristic
493C2×C4#53711yescharacteristic
494C2×C4#53812yes—
495C2×C4#53912yes—
496C2×C4#55112yes—
497C2×C4#55112yes—
498C2×C4#55112yes—
499C2×C4#55211yescharacteristic
500C2×C4#55212yes—
501C2×C4#55412yes—
502C2×C4#55411yescharacteristic
503C8—42no—
504C8—82no—
505C8—81no—
506C22—28no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—48no—
511C22—48no—
512C22—48no—
513C22—48no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22#241211yescharacteristic
518C22#241711yescharacteristic
519C22#247312yes—
520C22#248012yes—
521C22#285011yescharacteristic
522C22C2×#45911yescharacteristic
523C22#304011yescharacteristic
524C22#304211yescharacteristic
525C22#304611yescharacteristic
526C22#310012yes—
527C22#311512yes—
528C22#312012yes—
529C22#312312yes—
530C22#329411yescharacteristic
531C22#329412yes—
532C22#330612yes—
533C22#331112yes—
534C22#331512yes—
535C22#332312yes—
536C22#333011yescharacteristic
537C22Q8×D811yescharacteristic
538C22#338411yescharacteristic
539C22#338812yes—
540C4—24no—
541C4—44no—
542C4—42no—
543C4—84no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—162no—
548C4—161no—
549C4C2×#40011yescharacteristic
550C4#249312yes—
551C4#249811yescharacteristic
552C4#250512yes—
553C4#256411yescharacteristic
554C4#263911yescharacteristic
555C2—48no—
556C2#1482611yescharacteristic
557C2#2452611yescharacteristic
558C2#2454511yescharacteristic
559C2#2481111yescharacteristic
560C2#2483411yescharacteristic
561C2#2729412yes—
562C2#2731812yes—
563C2#2751812yes—
564C2#2754212yes—
565C2#3302411yescharacteristic
566C2#3304011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15153C2semidirect
5. #11357C2semidirect
6. #11315C2semidirect
7. #11256C2semidirect
8. #11255C2semidirect
9. #11227C2semidirect
427. D4#400semidirect

Maximal subgroups

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

subgroupindex
C2×#18622
C2×#18022
#151532
D4×#952
#113572
#113152
#112562
#112552
#112272
#107882