Finite Group Databasebeta

← #431000

Subgroups — #431000

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic
3C2×#2059C212yesmaximal, complemented
4C2×#1864C212yesmaximal
5C2×#1844C211yesmaximal, characteristic, complemented
6#15151C211yesmaximal, characteristic, complemented
7#11371C211yesmaximal, characteristic, complemented
8#11325C212yesmaximal, complemented
9#11121C212yesmaximal
10#10925C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2566—24no—
21#2566—24no—
22C2×#398—22no—
23C2×#396—22no—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#385—22no—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2093—24no—
33#2059—28no—
34#1864—28no—
35#1844—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1445—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—24no—
52C2×#125—42no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—24no—
56C2×C4×D4—42no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—42no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×#106—42no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102Q818yes—
89C2×C4×C8—24no—
90C2×C4×C8—22no—
91#398—48no—
92#396—216no—
93#390—22no—
94#390—22no—
95#390—22no—
96#390D414yes—
97#389—24no—
98#389—24no—
99#385—216no—
100#384—22no—
101#384—22no—
102#384—22no—
103#384D414yes—
104#382—22no—
105#382—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C2×#95—22no—
109C2×#95—24no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—24no—
116#336—24no—
117#336—22no—
118#336D418yes—
119C25—21no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×D412yes—
127C22×D4C2×Q814yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—84no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8C2×D414yes—
153C22×C8C2×D414yes—
154C22×C8C2×Q814yes—
155C22×C8#4114yes—
156#125—48no—
157#125—48no—
158C4×D4—48no—
159C4×D4—48no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—84no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×Q814yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—84no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D414yes—
220#107—48no—
221#106—48no—
222#106—48no—
223#102—216no—
224#102—48no—
225C4×C8—28no—
226#95—28no—
227#95—24no—
228#95—48no—
229#95—44no—
230#95—44no—
231C24—22no—
232C24—24no—
233C24—41no—
234C24—41no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—44no—
239C24C22×D411yescharacteristic
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4C4⋊Q814yes—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—48no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—88no—
295C22×C4—88no—
296C22×C4—88no—
297C22×C4#12014yes—
298C22×C4#12712yes—
299C22×C4#12712yes—
300C22×C4#13514yes—
301C22×C4AGL(1,ℤ/8)14yes—
302C22×C4C22×D412yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q812yes—
306C22×C4C2×#4112yes—
307C22×C4H(5,𝔽2)12yes—
308C22×C42−1+412yes—
309C22×C42−1+412yes—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8C22⋊Q818yes—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—48no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—88no—
342C4⋊C4—88no—
343C4⋊C4—88no—
344C22⋊C4—48no—
345C22⋊C4—416no—
346C22⋊C4—88no—
347C42—22no—
348C42—22no—
349C42—24no—
350C42—24no—
351C42—24no—
352C42—41no—
353C42—41no—
354C42—42no—
355C23—24no—
356C23—48no—
357C23—48no—
358C23—42no—
359C23—42no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—84no—
365C23—84no—
366C23—81no—
367C23—81no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23C2×#12011yescharacteristic
372C23C2×#12711yescharacteristic
373C23#53511yescharacteristic
374C23#54712yes—
375C23D4×Q812yes—
376C23C2×#13512yes—
377C23C2×AGL(1,ℤ/8)12yes—
378C23#57714yes—
379D4—28no—
380D4—44no—
381D4—44no—
382D4—44no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—816no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4#44918yes—
426C2×C4C2×C22⋊Q812yes—
427C2×C4#52511yescharacteristic
428C2×C4C2×C4⋊Q811yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#53612yes—
431C2×C4#53611yescharacteristic
432C2×C4#53912yes—
433C2×C4#54412yes—
434C2×C4D4212yes—
435C2×C4D4212yes—
436C2×C4#54712yes—
437C2×C4#54712yes—
438C2×C4D4×Q812yes—
439C2×C4D4×Q812yes—
440C2×C4#55412yes—
441C2×C4#56312yes—
442C8—28no—
443C8—44no—
444C22—44no—
445C22—48no—
446C22—84no—
447C22—88no—
448C22—84no—
449C22—82no—
450C22—82no—
451C22—84no—
452C22#240611yescharacteristic
453C22#243311yescharacteristic
454C22#243411yescharacteristic
455C22#254212yes—
456C22#272012yes—
457C22C2×#44912yes—
458C22#300912yes—
459C22#314711yescharacteristic
460C22#315111yescharacteristic
461C22#315412yes—
462C22#320712yes—
463C22#322212yes—
464C22#333714yes—
465C22#334514yes—
466C22#338114yes—
467C22#338414yes—
468C4—28no—
469C4—42no—
470C4—42no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4—84no—
476C4—162no—
477C4#244012yes—
478C4#244912yes—
479C4#267712yes—
480C4#268012yes—
481C2—84no—
482C2#1543011yescharacteristic
483C2#2409112yes—
484C2#2572011yescharacteristic
485C2#2573211yescharacteristic
486C2#2583512yes—
487C2#2853312yes—
488C2#2854812yes—
489C2#3234914yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2059C2semidirect
5. C2×#1844C2semidirect
6. #15151C2semidirect
7. #11371C2semidirect
8. #11325C2semidirect
11. #10874C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#20932
C2×#20592
C2×#18642
C2×#18442
#151512
#113712
#113252
#111212
#109252
#108742