Finite Group Databasebeta

← #402393

Subgroups — #402393

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1864C211yesmaximal, characteristic
2C2×#1802C211yesmaximal, characteristic
3#15153C211yesmaximal, characteristic, complemented
4#13820C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11255C211yesmaximal, characteristic, complemented
7#11228C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11189C212yesmaximal
10#10908C212yesmaximal, complemented
11#10807C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C2×#400—22no—
19C2×#390C2212yes—
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4—21no—
24#1864—24no—
25#1802—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1445—21no—
36#1445—21no—
37#1437—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C4×#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×D4—24no—
49C2×C4×D4—22no—
50C2×C4×D4—42no—
51C2×C4×D4D414yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—22no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#106—22no—
67C2×#106—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—42no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—24no—
74C2×#102—22no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78#400—48no—
79#390—24no—
80#390—24no—
81#389—22no—
82#389—22no—
83#384—22no—
84#384—22no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—42no—
87C4×C22⋊C4—44no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95D412yes—
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99#340—24no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#336—24no—
106#336—22no—
107#336—42no—
108C25—21no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4C2×D412yes—
118C22×D4#4114yes—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—44no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—82no—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8#4112yes—
151C4×D4—28no—
152C4×D4—44no—
153C4×D4—48no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4#4114yes—
185C2×C4⋊C4#4112yes—
186C2×C4⋊C4#4114yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—44no—
204C2×C42—44no—
205C2×C42—82no—
206C2×C42C2×D412yes—
207C2×C42#4112yes—
208#106—44no—
209C4⋊C8—44no—
210#102—28no—
211#102—44no—
212#102—48no—
213#95—22no—
214#95—22no—
215#95—24no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—42no—
220#95—42no—
221#95—44no—
222#95—44no—
223#95—81no—
224C24—22no—
225C24—24no—
226C24—42no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—42no—
232C24—41no—
233C24—42no—
234C24—41no—
235C24C22×D411yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4#12414yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—48no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
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—84no—
305C22×C4—88no—
306C22×C4—88no—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#13512yes—
310C22×C4AGL(1,ℤ/8)14yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4111yescharacteristic
319C22×C4H(5,𝔽2)12yes—
320C22×C42−1+411yescharacteristic
321C22×C42−1+411yescharacteristic
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4#12314yes—
360C22⋊C4—48no—
361C22⋊C4—416no—
362C22⋊C4—88no—
363C42—22no—
364C42—22no—
365C42—24no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—88no—
371C23—24no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—44no—
379C23—44no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—82no—
385C23—81no—
386C23—82no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23C2×#12111yescharacteristic
395C23#52512yes—
396C23#54612yes—
397C23#54711yescharacteristic
398C23#54811yescharacteristic
399C23C2×#13511yescharacteristic
400C23C2×AGL(1,ℤ/8)12yes—
401C23C2×AGL(1,ℤ/8)11yescharacteristic
402C23#57711yescharacteristic
403C23#57712yes—
404C23#57811yescharacteristic
405D4—24no—
406D4—24no—
407D4—44no—
408D4—42no—
409D4—42no—
410D4—44no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—816no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4#46314yes—
465C2×C4C2×#12311yescharacteristic
466C2×C4#52511yescharacteristic
467C2×C4C2×#12411yescharacteristic
468C2×C4#52912yes—
469C2×C4#52912yes—
470C2×C4#52912yes—
471C2×C4#53211yescharacteristic
472C2×C4#53411yescharacteristic
473C2×C4#53611yescharacteristic
474C2×C4#53912yes—
475C2×C4#54212yes—
476C2×C4#55112yes—
477C2×C4#55212yes—
478C2×C4#55212yes—
479C2×C4#55211yescharacteristic
480C2×C4#55312yes—
481C2×C4#55312yes—
482C2×C4#55611yescharacteristic
483C8—42no—
484C8—82no—
485C8—81no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—82no—
490C22—82no—
491C22—84no—
492C22—84no—
493C22—82no—
494C22—82no—
495C22—84no—
496C22—84no—
497C22#241211yescharacteristic
498C22#241511yescharacteristic
499C22#246012yes—
500C22#246812yes—
501C22#285011yescharacteristic
502C22C2×#46311yescharacteristic
503C22#303811yescharacteristic
504C22#304211yescharacteristic
505C22#304611yescharacteristic
506C22#310612yes—
507C22#310912yes—
508C22#312012yes—
509C22#312312yes—
510C22#329612yes—
511C22#330812yes—
512C22#331012yes—
513C22#333212yes—
514C22#334011yescharacteristic
515C22#334512yes—
516C22#335311yescharacteristic
517C22#336411yescharacteristic
518C22#336811yescharacteristic
519C22#337312yes—
520C4—24no—
521C4—44no—
522C4—42no—
523C4—44no—
524C4—82no—
525C4—84no—
526C4—82no—
527C4—162no—
528C4—161no—
529C4#239511yescharacteristic
530C4#250311yescharacteristic
531C4#253012yes—
532C4#257712yes—
533C4#259111yescharacteristic
534C4#266311yescharacteristic
535C2—84no—
536C2#1484111yescharacteristic
537C2#2452711yescharacteristic
538C2#2454211yescharacteristic
539C2#2468111yescharacteristic
540C2#2469211yescharacteristic
541C2#2676512yes—
542C2#2676912yes—
543C2#2710712yes—
544C2#2711112yes—
545C2#3303011yescharacteristic
546C2#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. #11255C2semidirect
7. #11228C2semidirect
10. #10908C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18642
C2×#18022
#151532
#138202
#113572
#112552
#112282
#111892
#109082
#108072