Finite Group Databasebeta

← #365189

Subgroups — #365189

Order 512.

1388 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#381—22no—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1854—24no—
24#1824C418yescomplemented
25#1736—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1445—21no—
37#1445—21no—
38#1437—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1434—21no—
43C4×#95—21no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—42no—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—42no—
57C2×#117D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C414yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42—22no—
77C22×C42C2312yes—
78C2×#107—22no—
79C2×#107—24no—
80C2×#106—22no—
81C2×#106—42no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8C2×C414yes—
85#381—48no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380C2×C414yes—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—42no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105#340—24no—
106#340—22no—
107#340—22no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340C2×C418yes—
112#340C2×C418yes—
113#336—24no—
114#336—42no—
115C25C2×D411yescharacteristic
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—24no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—44no—
128C23×C4—44no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4C22×C412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4#4112yes—
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C2×M16—28no—
142C2×M16—48no—
143C2×M16—48no—
144C2×M16—44no—
145C2×M16—44no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C22×C412yes—
161C22×C8C22×C412yes—
162C22×C8#4114yes—
163#117—28no—
164#117—44no—
165#117—48no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4C22×C414yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4#4114yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—81no—
210C2×C22⋊C4—81no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—82no—
222#107—44no—
223#106—44no—
224C22⋊C8—28no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—44no—
234#95—44no—
235#95—82no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24C4×D412yes—
245C24#12112yes—
246C24#12312yes—
247C24C22×D411yescharacteristic
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—416no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—48no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—88no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4#11714yes—
317C22×C4C4×D414yes—
318C22×C4C4×D412yes—
319C22×C4#12112yes—
320C22×C4#12312yes—
321C22×C4#12414yes—
322C22×C4#12614yes—
323C22×C4AGL(1,ℤ/8)14yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4112yes—
331C22×C4C2×#4111yescharacteristic
332C22×C4H(5,𝔽2)12yes—
333C22×C42−1+412yes—
334M16—416no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—48no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4#12314yes—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—84no—
376C22⋊C4—84no—
377C42—48no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—88no—
383C23—24no—
384C23—22no—
385C23—22no—
386C23—24no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23#38814yes—
406C23C2×C4×D411yescharacteristic
407C23C2×#12111yescharacteristic
408C23C2×#12311yescharacteristic
409C23#54612yes—
410C23#54712yes—
411C23C2×AGL(1,ℤ/8)12yes—
412C23#57512yes—
413C23#57711yescharacteristic
414C23#57711yescharacteristic
415C23#57811yescharacteristic
416C23#57811yescharacteristic
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—816no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
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—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4#38814yes—
477C2×C4C2×#11711yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C4×#4112yes—
480C2×C4#52012yes—
481C2×C4C2×#12311yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4C2×#12411yescharacteristic
484C2×C4C2×#12611yescharacteristic
485C2×C4#52912yes—
486C2×C4#53411yescharacteristic
487C2×C4#53611yescharacteristic
488C2×C4#53912yes—
489C2×C4#55112yes—
490C2×C4#55112yes—
491C2×C4#55112yes—
492C2×C4#55212yes—
493C2×C4#55212yes—
494C2×C4#55212yes—
495C8—44no—
496C8—42no—
497C8—42no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22C2×#38811yescharacteristic
510C22#235912yes—
511C22#241211yescharacteristic
512C22#241311yescharacteristic
513C22#241912yes—
514C22C4×AGL(1,ℤ/8)12yes—
515C22#294911yescharacteristic
516C22#295011yescharacteristic
517C22#303811yescharacteristic
518C22#304011yescharacteristic
519C22#304611yescharacteristic
520C22#304611yescharacteristic
521C22#309412yes—
522C22#309711yescharacteristic
523C22#309711yescharacteristic
524C22#330612yes—
525C22#331712yes—
526C22#332712yes—
527C22#333012yes—
528C22#334312yes—
529C22#334912yes—
530C22#335112yes—
531C22#335412yes—
532C4—24no—
533C4—44no—
534C4—44no—
535C4—84no—
536C4—82no—
537C4—84no—
538C4—82no—
539C4—161no—
540C4—161no—
541C4C4×#12311yescharacteristic
542C4#232211yescharacteristic
543C4#234312yes—
544C4#249312yes—
545C4#250311yescharacteristic
546C4#256111yescharacteristic
547C2—44no—
548C2#1397211yescharacteristic
549C2#2106011yescharacteristic
550C2#2106611yescharacteristic
551C2#2292912yes—
552C2#2294712yes—
553C2#2454311yescharacteristic
554C2#2454411yescharacteristic
555C2#2459311yescharacteristic
556C2#2459411yescharacteristic
557C2#2495412yes—
558C2#2495812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
6. #11316C2semidirect
7. #11257C2semidirect
8. #10907C2semidirect
9. #10806C2semidirect
24. #1824C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#18242
C2×#17362
#139042
#138012
#113162
#112572
#109072
#108062
#107222
#107152