Finite Group Databasebeta

← #395851

Subgroups — #395851

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#106C211yesmaximal, characteristic
2C2×#2085C212yesmaximal, complemented
3C2×#1852C211yesmaximal, characteristic
4#13907C211yesmaximal, characteristic
5#13881C211yesmaximal, characteristic
6#11325C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106—28no—
18C22×#106—24no—
19C22×#106C2212yes—
20C22×#106C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C2×#389C2212yes—
23C2×#382C2212yes—
24C2×#380—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#2085—28no—
29#1852—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1442—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C24×C4—21no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8Q814yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4D414yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42C2312yes—
78C2×#107—24no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—42no—
82C2×#106—232no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106D418yes—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8Q814yes—
92#389—24no—
93#389—42no—
94#382—24no—
95#382—42no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—44no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—22no—
117#340—22no—
118#340—42no—
119#336—24no—
120#336—24no—
121C25C2×D411yescharacteristic
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×Q812yes—
147C23×C4C2×Q812yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—216no—
153C22×C8—28no—
154C22×C8—28no—
155C22×C8—28no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—216no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C2×D414yes—
170C22×C8C2×Q812yes—
171C22×C8C2×Q812yes—
172C22×C8C2×Q814yes—
173C22×C8#4114yes—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—28no—
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—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233#107—44no—
234#107—44no—
235#106—216no—
236#106—216no—
237C22⋊C8—28no—
238#95—24no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—42no—
244#95—42no—
245#95—82no—
246#95—82no—
247#95—82no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24#12112yes—
253C24#12312yes—
254C24C4⋊Q812yes—
255C24#13512yes—
256C24#13512yes—
257C24AGL(1,ℤ/8)12yes—
258C24#13712yes—
259C24C22×D411yescharacteristic
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—48no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—48no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—84no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4#12112yes—
335C22×C4C22⋊Q814yes—
336C22×C4C22⋊Q814yes—
337C22×C4#12312yes—
338C22×C4#12514yes—
339C22×C4#12514yes—
340C22×C4C4⋊Q812yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×Q811yescharacteristic
344C22×C4C22×Q811yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4H(5,𝔽2)12yes—
350C22×C42−1+412yes—
351C2×C8—232no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8C22⋊Q818yes—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—88no—
391C4⋊C4#12014yes—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23#39814yes—
417C23#46314yes—
418C23#46414yes—
419C23#48214yes—
420C23#48514yes—
421C23#49914yes—
422C23#50114yes—
423C23C2×#12111yescharacteristic
424C23C2×#12311yescharacteristic
425C23C2×C4⋊Q811yescharacteristic
426C23D4212yes—
427C23#54712yes—
428C23C2×#13511yescharacteristic
429C23C2×#13511yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×#13711yescharacteristic
432C23#57712yes—
433C23#57712yes—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—816no—
470C2×C4—816no—
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—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4#39814yes—
495C2×C4C2×#12011yescharacteristic
496C2×C4C2×C22⋊Q811yescharacteristic
497C2×C4C2×C22⋊Q811yescharacteristic
498C2×C4C2×C22⋊Q812yes—
499C2×C4#52511yescharacteristic
500C2×C4#52512yes—
501C2×C4C2×#12511yescharacteristic
502C2×C4C2×#12511yescharacteristic
503C2×C4#53411yescharacteristic
504C2×C4#53611yescharacteristic
505C2×C4#53912yes—
506C2×C4#54412yes—
507C2×C4#54612yes—
508C2×C4#54812yes—
509C2×C4D4×Q812yes—
510C2×C4#55212yes—
511C2×C4#55312yes—
512C2×C4#55412yes—
513C8—28no—
514C8—42no—
515C8—42no—
516C22—24no—
517C22—24no—
518C22—28no—
519C22—28no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—28no—
525C22—28no—
526C22#208518yescomplemented
527C22C2×#39811yescharacteristic
528C22#241011yescharacteristic
529C22#241311yescharacteristic
530C22#244012yes—
531C22#244312yes—
532C22C2×#46311yescharacteristic
533C22C2×#46411yescharacteristic
534C22#304211yescharacteristic
535C22#304311yescharacteristic
536C22C2×#48211yescharacteristic
537C22C2×#48511yescharacteristic
538C22#309912yes—
539C22C2×#49911yescharacteristic
540C22C2×#50111yescharacteristic
541C22#316312yes—
542C22#326512yes—
543C22#326612yes—
544C22#328212yes—
545C22#328212yes—
546C22#333712yes—
547C22#333712yes—
548C22#334412yes—
549C22#334512yes—
550C4—24no—
551C4—44no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—82no—
556C4—161no—
557C4—162no—
558C4—161no—
559C4#240911yescharacteristic
560C4#244212yes—
561C4#247312yes—
562C4#257911yescharacteristic
563C4#262711yescharacteristic
564C4#284811yescharacteristic
565C2—28no—
566C2#1472811yescharacteristic
567C2C2×#208511yescharacteristic
568C2#2366111yescharacteristic
569C2#2435711yescharacteristic
570C2#2435911yescharacteristic
571C2#2457911yescharacteristic
572C2#2458111yescharacteristic
573C2#2606212yes—
574C2#2606812yes—
575C2#2613612yes—
576C2#2613812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2085C2semidirect
6. #11325C2semidirect
7. #11257C2semidirect
8. #11220C2semidirect
526. C22#2085semidirect

Maximal subgroups

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

subgroupindex
C23×#1062
C2×#20852
C2×#18522
#139072
#138812
#113252
#112572
#112202
#107302
#107212
#107142