Finite Group Databasebeta

← #409373

Subgroups — #409373

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1807C211yesmaximal, characteristic
2C22×#336C211yesmaximal, characteristic
3#14685C211yesmaximal, characteristic
4#14685C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11316C212yesmaximal, complemented
7#11315C212yesmaximal, complemented
8#11257C212yesmaximal, complemented
9#10721C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#381—21no—
18C2×#381—21no—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1807—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336—24no—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39#1434—22no—
40#1434—22no—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8D414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42D412yes—
76C22×C42C2312yes—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—24no—
80C2×#106—42no—
81#381—44no—
82#381—44no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—24no—
112#336—24no—
113C25C2×D411yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—42no—
135C23×C4SD1614yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—28no—
150C22×C8—28no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—28no—
162C22×C8—44no—
163C22×C8—48no—
164C22×C8—48no—
165C22×C8—48no—
166C22×C8—48no—
167C22×C8—44no—
168C22×C8—44no—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8#4114yes—
172C22×C8#4114yes—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
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—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
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—22no—
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×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42—44no—
226#106—44no—
227#106—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—42no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24#12112yes—
245C24#12312yes—
246C24#12312yes—
247C24C2×SD1612yes—
248C24#13512yes—
249C24AGL(1,ℤ/8)12yes—
250C24#13712yes—
251C24C22×D411yescharacteristic
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4#12112yes—
327C22×C4#12314yes—
328C22×C4#12312yes—
329C22×C4#12312yes—
330C22×C4#12414yes—
331C22×C4#12614yes—
332C22×C4#12614yes—
333C22×C4C2×SD1614yes—
334C22×C4#13514yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C42−1+412yes—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—416no—
356C2×C8—416no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8#12314yes—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—48no—
393C23—24no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23#39714yes—
410C23#46514yes—
411C23#46814yes—
412C23#48114yes—
413C23#48214yes—
414C23#48314yes—
415C23#48514yes—
416C23C2×#12111yescharacteristic
417C23C2×#12311yescharacteristic
418C23C2×#12311yescharacteristic
419C23#53212yes—
420C23#53712yes—
421C23C22×SD1611yescharacteristic
422C23C2×#13511yescharacteristic
423C23C2×#13512yes—
424C23C2×AGL(1,ℤ/8)11yescharacteristic
425C23C2×#13711yescharacteristic
426C23#57512yes—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—416no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—416no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
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—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4#39714yes—
488C2×C4C2×#12311yescharacteristic
489C2×C4C2×#12311yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4C2×#12411yescharacteristic
492C2×C4C2×#12611yescharacteristic
493C2×C4C2×#12611yescharacteristic
494C2×C4#52912yes—
495C2×C4#53411yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53912yes—
498C2×C4#55112yes—
499C2×C4#55112yes—
500C2×C4#55112yes—
501C2×C4#55112yes—
502C2×C4#55212yes—
503C2×C4#55212yes—
504C2×C4#55212yes—
505C2×C4#55212yes—
506C8—24no—
507C8—44no—
508C8—42no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—28no—
516C22—28no—
517C22—28no—
518C22—28no—
519C22#207718yes—
520C22C2×#39711yescharacteristic
521C22#249812yes—
522C22#249812yes—
523C22#261811yescharacteristic
524C22#285411yescharacteristic
525C22C2×#46511yescharacteristic
526C22#305011yescharacteristic
527C22C2×#46811yescharacteristic
528C22#305411yescharacteristic
529C22C2×#48311yescharacteristic
530C22C2×#48211yescharacteristic
531C22C2×#48511yescharacteristic
532C22C2×#48111yescharacteristic
533C22#309112yes—
534C22#309112yes—
535C22#316712yes—
536C22#316812yes—
537C22#318012yes—
538C22#318112yes—
539C22#324612yes—
540C22#324712yes—
541C22#324812yes—
542C22#325012yes—
543C4—24no—
544C4—44no—
545C4—44no—
546C4—161no—
547C4—161no—
548C4—161no—
549C4—161no—
550C4—162no—
551C4—162no—
552C4#248311yescharacteristic
553C4#248411yescharacteristic
554C4#249312yes—
555C4#249312yes—
556C4#250311yescharacteristic
557C4#256111yescharacteristic
558C2—28no—
559C2#1501911yescharacteristic
560C2C2×#207711yescharacteristic
561C2#2363811yescharacteristic
562C2#2785512yes—
563C2#2785712yes—
564C2#2785912yes—
565C2#2786012yes—
566C2#2977411yescharacteristic
567C2#2977511yescharacteristic
568C2#3312811yescharacteristic
569C2#3313411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11378C2semidirect
6. #11316C2semidirect
7. #11315C2semidirect
8. #11257C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18072
C22×#3362
#146852
#113782
#113162
#113152
#112572
#107212