Finite Group Databasebeta

← #405832

Subgroups — #405832

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1825C211yesmaximal, characteristic
2C2×#1736C211yesmaximal, characteristic
3#13939C211yesmaximal, characteristic
4#13881C211yesmaximal, characteristic
5#11358C212yesmaximal, complemented
6#11316C212yesmaximal, complemented
7#11315C212yesmaximal, complemented
8#11268C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C2×#382C2212yes—
17C2×#380C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1825—24no—
21#1736—24no—
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1442—21no—
35#1442—21no—
36#1440—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—21no—
42C22⋊#95—22no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—24no—
49C22×M16—42no—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117D414yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2312yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79#382—24no—
80#382—42no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99#340—24no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#336—24no—
110#336—42no—
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—44no—
123C23×C4—42no—
124C23×C4—44no—
125C23×C4—81no—
126C23×C4—81no—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4C2411yescharacteristic
137C2×M16—28no—
138C2×M16—48no—
139C2×M16—48no—
140C2×M16—44no—
141C2×M16—44no—
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—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8#4114yes—
159#117—28no—
160#117—28no—
161#117—44no—
162#117—44no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
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—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
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—42no—
219C2×C42—42no—
220C22⋊C8—44no—
221#95—22no—
222#95—22no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
229#95—82no—
230#95—82no—
231#95—82no—
232#95—82no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24#12112yes—
242C24#12312yes—
243C24#12412yes—
244C24C22×D411yescharacteristic
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
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—48no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
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—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—88no—
313C22×C4—88no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4#12112yes—
317C22×C4#12314yes—
318C22×C4#12314yes—
319C22×C4#12312yes—
320C22×C4#12412yes—
321C22×C4#12614yes—
322C22×C4#12614yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4112yes—
329C22×C4C2×#4112yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+412yes—
333M16—416no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—48no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—84no—
377C22⋊C4—84no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—48no—
383C23—24no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—22no—
390C23—22no—
391C23—24no—
392C23—24no—
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#39714yes—
406C23C2×#12111yescharacteristic
407C23C2×#12311yescharacteristic
408C23C2×#12411yescharacteristic
409C23#54612yes—
410C23#54712yes—
411C23#57512yes—
412C23#57512yes—
413C23#57611yescharacteristic
414C23#57711yescharacteristic
415C23#57711yescharacteristic
416C23#57811yescharacteristic
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—42no—
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—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—816no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
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—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#39714yes—
479C2×C4C2×#12311yescharacteristic
480C2×C4C2×#12311yescharacteristic
481C2×C4#52511yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4C2×#12611yescharacteristic
484C2×C4C2×#12611yescharacteristic
485C2×C4#52912yes—
486C2×C4#52912yes—
487C2×C4#53511yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#53912yes—
490C2×C4#53912yes—
491C2×C4#55112yes—
492C2×C4#55112yes—
493C2×C4#55112yes—
494C2×C4#55212yes—
495C2×C4#55212yes—
496C2×C4#55212yes—
497C8—44no—
498C8—81no—
499C8—81no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—42no—
510C22—42no—
511C22C2×#39711yescharacteristic
512C22#241111yescharacteristic
513C22#241311yescharacteristic
514C22#241912yes—
515C22#248812yes—
516C22#304011yescharacteristic
517C22#304011yescharacteristic
518C22#304511yescharacteristic
519C22#304611yescharacteristic
520C22#309312yes—
521C22#309811yescharacteristic
522C22#309911yescharacteristic
523C22#313611yescharacteristic
524C22#313711yescharacteristic
525C22#314212yes—
526C22#331112yes—
527C22#331312yes—
528C22#332212yes—
529C22#332612yes—
530C22#334712yes—
531C22#334812yes—
532C22#335112yes—
533C22#335212yes—
534C4—24no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—84no—
540C4—161no—
541C4—162no—
542C4—161no—
543C4#248211yescharacteristic
544C4#248411yescharacteristic
545C4#249312yes—
546C4#249312yes—
547C4#262311yescharacteristic
548C4#285311yescharacteristic
549C2—44no—
550C2#1491911yescharacteristic
551C2#2363111yescharacteristic
552C2#2363411yescharacteristic
553C2#2449311yescharacteristic
554C2#2449711yescharacteristic
555C2#2459511yescharacteristic
556C2#2459911yescharacteristic
557C2#2495912yes—
558C2#2496312yes—
559C2#2771012yes—
560C2#2772412yes—
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
5. #11358C2semidirect
6. #11316C2semidirect
7. #11315C2semidirect
8. #11268C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18252
C2×#17362
#139392
#138812
#113582
#113162
#113152
#112682
#107302
#107222
#107152