Finite Group Databasebeta

← #406799

Subgroups — #406799

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic
5#14682C211yesmaximal, characteristic
6C2×C4⋊#95C211yesmaximal, characteristic
7#11379C212yesmaximal, complemented
8#11378C212yesmaximal, complemented
9#11372C212yesmaximal, complemented
10#11325C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C22×#95—22no—
24#1853—24no—
25#1852—24no—
26#1824—24no—
27#1824—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C4⋊#95—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41#1434—22no—
42C22⋊#95—22no—
43C24×C4—21no—
44C24×C4—21no—
45C24×C4D412yes—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4Q814yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—24no—
72C22×C42—42no—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—42no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8D414yes—
87C2×C22⋊C8D414yes—
88#382—24no—
89#382—42no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—24no—
101C2×#95—24no—
102C2×#95—48no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336—24no—
110#336—42no—
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—44no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×Q812yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
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×C8C2×D414yes—
162C22×C8C2×D414yes—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
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—48no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C2×Q814yes—
202C2×C4⋊C4#4114yes—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—48no—
217C2×C42—48no—
218C2×C42—42no—
219C2×C42#4114yes—
220#107—48no—
221#106—48no—
222C22⋊C8—28no—
223C22⋊C8—28no—
224#95—44no—
225#95—48no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—42no—
230#95—44no—
231#95—48no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24C22⋊Q812yes—
239C24#12312yes—
240C24#12712yes—
241C24#13514yes—
242C24#13714yes—
243C24C22×D411yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—216no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
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—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—416no—
283C22×C4—416no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—48no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—88no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—88no—
318C22×C4—84no—
319C22×C4C22⋊Q814yes—
320C22×C4C22⋊Q814yes—
321C22×C4C22⋊Q812yes—
322C22×C4#12314yes—
323C22×C4#12314yes—
324C22×C4#12314yes—
325C22×C4#12312yes—
326C22×C4#12712yes—
327C22×C4AGL(1,ℤ/8)14yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C42−1+412yes—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—48no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C22⋊C4—84no—
371C22⋊C4—84no—
372C42—24no—
373C42—24no—
374C42—48no—
375C42—42no—
376C42—42no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23#39914yes—
396C23#47918yes—
397C23#48518yes—
398C23#49514yes—
399C23#49714yes—
400C23C2×C22⋊Q811yescharacteristic
401C23C2×#12311yescharacteristic
402C23C2×#12311yescharacteristic
403C23#52511yescharacteristic
404C23C2×#12711yescharacteristic
405C23#53411yescharacteristic
406C23#53611yescharacteristic
407C23C2×#13512yes—
408C23C2×AGL(1,ℤ/8)12yes—
409C23C2×#13712yes—
410C23#57512yes—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—432no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—816no—
441C2×C4—816no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#39914yes—
469C2×C4C2×C22⋊Q811yescharacteristic
470C2×C4C2×C22⋊Q811yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4#52912yes—
474C2×C4#53312yes—
475C2×C4#53912yes—
476C2×C4#54412yes—
477C2×C4#54612yes—
478C2×C4#54612yes—
479C2×C4#54712yes—
480C2×C4#54712yes—
481C2×C4#55112yes—
482C2×C4#55112yes—
483C2×C4#55212yes—
484C2×C4#55212yes—
485C8—44no—
486C8—44no—
487C22—24no—
488C22—24no—
489C22—28no—
490C22—24no—
491C22—28no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—28no—
496C22—28no—
497C22#209718yes—
498C22C2×#39911yescharacteristic
499C22#248311yescharacteristic
500C22#248411yescharacteristic
501C22#250111yescharacteristic
502C22#253911yescharacteristic
503C22#262711yescharacteristic
504C22#285511yescharacteristic
505C22C2×#47912yes—
506C22#307712yes—
507C22C2×#48512yes—
508C22#309112yes—
509C22#309212yes—
510C22#309412yes—
511C22#310512yes—
512C22#311712yes—
513C22C2×#49511yescharacteristic
514C22C2×#49711yescharacteristic
515C22#315111yescharacteristic
516C22#315311yescharacteristic
517C22#318812yes—
518C22#319612yes—
519C22#323212yes—
520C22#323912yes—
521C4—24no—
522C4—44no—
523C4—44no—
524C4—84no—
525C4—84no—
526C4—161no—
527C4—161no—
528C4—161no—
529C4—161no—
530C4#241912yes—
531C4#241912yes—
532C4#243812yes—
533C4#243812yes—
534C2—28no—
535C2#1493611yescharacteristic
536C2C2×#209711yescharacteristic
537C2#2373111yescharacteristic
538C2#2760412yes—
539C2#2761712yes—
540C2#2791211yescharacteristic
541C2#2791311yescharacteristic
542C2#2830311yescharacteristic
543C2#2831511yescharacteristic
544C2#2997312yes—
545C2#3315012yes—
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
7. #11379C2semidirect
8. #11378C2semidirect
9. #11372C2semidirect
10. #11325C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C2×#18522
C2×#18242
#146822
C2×C4⋊#952
#113792
#113782
#113722
#113252
#107122