Finite Group Databasebeta

← #147067

Subgroups — #147067

Order 512.

1180 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
1C2×#1800C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#13872C211yesmaximal, characteristic
4C2×#1390C214yesmaximal
5C22×C4×D4C2211yescharacteristic
6C23×C4⋊C4C411yescharacteristic, complemented
7C2×#423—21no—
8C22×#100C2212yes—
9C22×#99C2212yes—
10C2×#386C2212yes—
11C2×#380—21no—
12C2×#380C411yescharacteristic, complemented
13C2×C4×C22⋊C4—21no—
14#1800—22no—
15#1800C414yescomplemented
16C4×#99—22no—
17C4×#99C414yescomplemented
18C2×#343—21no—
19C2×#342—21no—
20#1441—21no—
21#1439—21no—
22C4⋊#95—21no—
23C22⋊#95C412yescomplemented
24#1390—28no—
25C23×D4C2311yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C412yes—
28C24×C4D411yescharacteristic
29C22×M16—22no—
30C2×C4×D4—41no—
31C2×C4×D4—41no—
32C2×C4×D4—42no—
33C2×C4×D4C2×C412yes—
34C2×C4×D4C2×C412yes—
35C2×C4×D4D412yes—
36C2×C4×D4Q812yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—44no—
41C22×C4⋊C4—41no—
42C22×C4⋊C4—41no—
43C22×C4⋊C4D411yescharacteristic
44C22×C22⋊C4—22no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42—41no—
50C22×C42D411yescharacteristic
51#423—24no—
52#423—42no—
53C2×C4⋊C8—41no—
54C2×C4⋊C8—41no—
55C2×#100—24no—
56C2×#100—24no—
57C2×#100—22no—
58C2×#100C2×C414yes—
59C2×#99—22no—
60C2×#99—22no—
61C2×#99—22no—
62C2×#99C2×C414yes—
63C2×#99D414yes—
64C2×#99Q814yes—
65#386—24no—
66#386—24no—
67#380—44no—
68#380—42no—
69C4×C22⋊C4—24no—
70C4×C22⋊C4—42no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—41no—
76#343—24no—
77#342—24no—
78C25C22×C411yescharacteristic
79C25C2×D411yescharacteristic
80C22×D4—41no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4C22×C411yescharacteristic
85C22×D4C22×C411yescharacteristic
86C22×D4C2×D411yescharacteristic
87C22×D4C2×D412yes—
88C22×D4C2×Q811yescharacteristic
89C22×D4C2×Q812yes—
90C22×D4#4112yes—
91C22×D4#4112yes—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—24no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—44no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—82no—
110C23×C4C4212yes—
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C411yescharacteristic
116C23×C4C4⋊C412yes—
117C23×C4C4⋊C412yes—
118C23×C4C2×D411yescharacteristic
119C2×M16—24no—
120C2×M16—42no—
121C2×M16—42no—
122C2×M16—42no—
123C2×M16—42no—
124C2×M16—42no—
125C22×C8—42no—
126C4×D4—42no—
127C4×D4—42no—
128C4×D4—42no—
129C4×D4—42no—
130C4×D4—42no—
131C4×D4—42no—
132C4×D4—42no—
133C4×D4—42no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C4⋊C412yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—82no—
166C2×C22⋊C4C4212yes—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C4⋊C412yes—
169C2×C22⋊C4C4⋊C412yes—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—44no—
174C2×C42—82no—
175C2×C42—82no—
176C2×C42C22⋊C412yes—
177C2×C42C4⋊C412yes—
178C4⋊C8—44no—
179C4⋊C8—44no—
180#100—28no—
181#100—44no—
182#99—24no—
183#99—24no—
184#99—24no—
185#99C4⋊C418yescomplemented
186#95—42no—
187#95—44no—
188#95—44no—
189#95—42no—
190#95—42no—
191#95—82no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—41no—
201C24—41no—
202C24—41no—
203C24—41no—
204C24—41no—
205C24—41no—
206C24—41no—
207C24—41no—
208C24C2×C4211yescharacteristic
209C24C2×C22⋊C411yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C24C2×C4⋊C411yescharacteristic
212C24C2×C4⋊C411yescharacteristic
213C24C4×D412yes—
214C24C4×Q812yes—
215C24#12112yes—
216C24C22⋊Q812yes—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4C2×C4⋊C412yes—
234C2×D4#11712yes—
235C2×D4C4×D412yes—
236C2×D4C4×Q812yes—
237C2×D4#12112yes—
238C2×D4C22⋊Q812yes—
239C2×D4#12512yes—
240C2×D4C4⋊Q812yes—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—84no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—88no—
301C22×C4—88no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4#9512yes—
309C22×C4#9512yes—
310C22×C4#9512yes—
311C22×C4#9512yes—
312C22×C4#9912yes—
313C22×C4#9911yescharacteristic
314C22×C4#9911yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C4⋊C411yescharacteristic
317C22×C4#12411yescharacteristic
318C22×C4#12411yescharacteristic
319C22×C4#12711yescharacteristic
320C22×C4C4⋊Q811yescharacteristic
321M16—44no—
322M16—44no—
323M16—82no—
324M16—82no—
325C2×C8—44no—
326C2×C8—82no—
327C2×C8—82no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C4⋊C4—44no—
333C4⋊C4—84no—
334C22⋊C4—28no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—88no—
343C22⋊C4—84no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—88no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23C2×#9511yescharacteristic
406C23C4×C4⋊C412yes—
407C23#38012yes—
408C23#38912yes—
409C23#38912yes—
410C23#39011yescharacteristic
411C23#39111yescharacteristic
412C23C2×#9911yescharacteristic
413C23C2×#9911yescharacteristic
414C23#41011yescharacteristic
415C23#41011yescharacteristic
416D4—42no—
417D4—42no—
418D4—42no—
419D4—42no—
420D4—42no—
421D4—42no—
422D4—42no—
423D4—42no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—88no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—88no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#34212yes—
487C2×C4#34212yes—
488C2×C4AΣL(1,𝔽16)12yes—
489C2×C4#35212yes—
490C2×C4#38211yescharacteristic
491C2×C4#38411yescharacteristic
492C2×C4#39011yescharacteristic
493C2×C4#39111yescharacteristic
494C8—84no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22—82no—
539C22C4⋊#9511yescharacteristic
540C22C2×#34211yescharacteristic
541C22#174011yescharacteristic
542C22#196411yescharacteristic
543C22#196411yescharacteristic
544C22#199611yescharacteristic
545C22#199611yescharacteristic
546C22C2×AΣL(1,𝔽16)11yescharacteristic
547C22C2×#35211yescharacteristic
548C22#212211yescharacteristic
549C22#212311yescharacteristic
550C4—22no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—88no—
559C4—84no—
560C4—162no—
561C4#139014yescomplemented
562C2—21no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—42no—
567C2—42no—
568C2—42no—
569C2—42no—
570C2#1085411yescharacteristic
571C2C2×#139011yescharacteristic
572C2#1257511yescharacteristic
573C2#1322011yescharacteristic
574C2#1322111yescharacteristic
575C2#1329511yescharacteristic
576C2#1329611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C23×C4⋊C4C4semidirect
12. C2×#380C4semidirect
15. #1800C4semidirect
17. C4×#99C4semidirect
23. C22⋊#95C4semidirect
185. #99C4⋊C4semidirect
561. C4#1390semidirect

Maximal subgroups

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

subgroupindex
C2×#18002
C2×C4×#992
#138722
C2×#13902