Finite Group Databasebeta

← #120652

Subgroups — #120652

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14347C211yesmaximal, characteristic
2#10730C212yesmaximal
3C2×#1294C212yesmaximal
4C2×#1280C211yesmaximal, characteristic
5#7557C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2212yes—
9C22×#95—22no—
10C22×#95C2211yescharacteristic
11C22×#95C2211yescharacteristic
12C2×#340—22no—
13C2×#340—22no—
14#1442—22no—
15#1442—22no—
16C22⋊#95—22no—
17#1294—24no—
18#1280C414yescomplemented
19C24×C4—21no—
20C24×C4—21no—
21C24×C4D412yes—
22C24×C4C2311yescharacteristic
23C23×C8—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—44no—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4Q812yes—
31C22×C22⋊C4—22no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8C2×C414yes—
38#380—22no—
39#380—22no—
40#380—22no—
41#380—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—21no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—44no—
53C2×#95—44no—
54C2×#95—44no—
55C2×#95C2×C412yes—
56C2×#95D412yes—
57C2×#95Q812yes—
58#340—44no—
59#340—44no—
60C25C2×D411yescharacteristic
61C23×C4—24no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4—41no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4C4⋊C414yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×Q811yescharacteristic
100C23×C4C2×Q811yescharacteristic
101C23×C4#4112yes—
102C23×C4#4112yes—
103C23×C4#4112yes—
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C22×C8—24no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—42no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—416no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—82no—
148#95—82no—
149#95—82no—
150#95—82no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—22no—
156C24—22no—
157C24C2×C4⋊C411yescharacteristic
158C24#12011yescharacteristic
159C24#12111yescharacteristic
160C24C22⋊Q811yescharacteristic
161C24#12311yescharacteristic
162C24#12411yescharacteristic
163C24#12411yescharacteristic
164C24#13512yes—
165C24#13512yes—
166C24AGL(1,ℤ/8)12yes—
167C24#13712yes—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—82no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#11712yes—
268C22×C4C4×D412yes—
269C22×C4C4×D411yescharacteristic
270C22×C4C4×Q811yescharacteristic
271C22×C4#12112yes—
272C22×C4C22⋊Q812yes—
273C22×C4C22⋊Q812yes—
274C22×C4#12312yes—
275C22×C4#12312yes—
276C22×C4#12412yes—
277C22×C4#12512yes—
278C22×C4#12512yes—
279C22×C4#12511yescharacteristic
280C22×C4#12612yes—
281C22×C4#12612yes—
282C22×C4C4⋊Q811yescharacteristic
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—82no—
287C2×C8—82no—
288C2×C8—82no—
289C2×C8—82no—
290C4⋊C4—28no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23#38211yescharacteristic
339C23#38511yescharacteristic
340C23#38911yescharacteristic
341C23#39311yescharacteristic
342C23#39411yescharacteristic
343C23#39711yescharacteristic
344C23#40011yescharacteristic
345C23#42712yes—
346C23#42812yes—
347C23#44912yes—
348C23#45212yes—
349C23#45311yescharacteristic
350C23#45411yescharacteristic
351C23#45511yescharacteristic
352C23#45611yescharacteristic
353C23B(4,𝔽2)12yes—
354C23#45812yes—
355C23#46312yes—
356C23#46412yes—
357C23#47812yes—
358C23#47912yes—
359C23#48212yes—
360C23#48512yes—
361C23#48811yescharacteristic
362C23#48811yescharacteristic
363C23#48812yes—
364C23#48912yes—
365C23#49011yescharacteristic
366C23#49111yescharacteristic
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—44no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—84no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#38712yes—
435C2×C4#38812yes—
436C2×C4#39812yes—
437C2×C4#40012yes—
438C8—84no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—21no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22#144211yescharacteristic
476C22#159712yes—
477C22#160012yes—
478C22#160312yes—
479C22#160612yes—
480C22#160912yes—
481C22#161212yes—
482C22#161512yes—
483C22#161812yes—
484C22#183511yescharacteristic
485C22#183611yescharacteristic
486C22#185412yes—
487C22#185611yescharacteristic
488C22#185711yescharacteristic
489C22#186712yes—
490C22#194912yes—
491C22#202511yescharacteristic
492C22#202611yescharacteristic
493C22#202711yescharacteristic
494C22#202811yescharacteristic
495C22#202911yescharacteristic
496C22#203011yescharacteristic
497C22#203911yescharacteristic
498C22#204011yescharacteristic
499C22#204212yes—
500C22#204412yes—
501C22#207311yescharacteristic
502C22#207411yescharacteristic
503C22#210112yes—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—84no—
509C4—84no—
510C4—84no—
511C4—162no—
512C4—162no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2—22no—
518C2—22no—
519C2#885512yes—
520C2#885812yes—
521C2#1034411yescharacteristic
522C2#1034511yescharacteristic
523C2#1034611yescharacteristic
524C2#1034711yescharacteristic
525C2#1040412yes—
526C2#1040512yes—
527C2#1126811yescharacteristic
528C2#1128711yescharacteristic
529C2#1129611yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1280C4semidirect

Maximal subgroups

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

subgroupindex
#143472
#107302
C2×#12942
C2×#12802
#75572