Finite Group Databasebeta

← #409992

Subgroups — #409992

Order 512.

2010 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C23×#106C211yesmaximal, characteristic, complemented
3C2×#2504C211yesmaximal, characteristic
4C2×#1893C211yesmaximal, characteristic
5C2×#1824C211yesmaximal, characteristic, complemented
6#13904C211yesmaximal, characteristic, complemented
7#11325C214yesmaximal, complemented
8#11121C214yesmaximal, complemented
9#10714C211yesmaximal, characteristic, complemented
10C24×C8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#480—24no—
15C2×#480—24no—
16C2×#480—24no—
17C22×#107C2212yes—
18C22×#106—24no—
19C22×#106—24no—
20C22×#106—24no—
21C22×#106C2214yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25#2504—24no—
26#2504—24no—
27C2×#397—22no—
28C2×#390C2212yes—
29C2×#386—22no—
30C2×#382C2212yes—
31C2×#381—22no—
32C2×#380C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1893—24no—
35#1824—24no—
36C2×#340C2212yes—
37C2×#340C2214yes—
38C2×#336C2214yes—
39#1445—22no—
40C4⋊#95C2214yes—
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4—21no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—28no—
49C23×C8D414yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D414yes—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42C2312yes—
81#480—232no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#106—232no—
86C2×#106—28no—
87C2×#106—28no—
88C2×#106—28no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—28no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8D414yes—
97#397—216no—
98#390—24no—
99#390—24no—
100#386—216no—
101#382—24no—
102#382—24no—
103#381—48no—
104#380—22no—
105#380—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—24no—
109C2×#95—28no—
110C2×#95—21no—
111C2×#95D412yes—
112#340—28no—
113#340—24no—
114#336—28no—
115C25—21no—
116C25C2×D411yescharacteristic
117C22×D4—24no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4—44no—
128C22×D4—44no—
129C22×D4C2×D412yes—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—84no—
145C23×C4C2×D414yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C22×C8—216no—
155C22×C8—216no—
156C22×C8—28no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—24no—
160C22×C8—216no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—24no—
164C22×C8—24no—
165C22×C8C2×D414yes—
166C22×C8C2×D414yes—
167C22×C8#4118yes—
168#121—28no—
169#121—28no—
170#121—44no—
171#121—44no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—216no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—88no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—24no—
222#107—216no—
223#106—232no—
224#102—216no—
225#102—48no—
226C22⋊C8—28no—
227#95—216no—
228#95—24no—
229#95—416no—
230#95—42no—
231C24—24no—
232C24—24no—
233C24—22no—
234C24—41no—
235C24—41no—
236C24—44no—
237C24—42no—
238C24—42no—
239C24—44no—
240C24#12012yes—
241C24#12112yes—
242C24#12712yes—
243C24#13514yes—
244C24AGL(1,ℤ/8)14yes—
245C24C22×D411yescharacteristic
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—28no—
249C2×D4—48no—
250C2×D4—416no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—88no—
262C2×D4—84no—
263C2×D4—84no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—416no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—88no—
301C22×C4—88no—
302C22×C4—88no—
303C22×C4—88no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4#12014yes—
307C22×C4#12014yes—
308C22×C4#12012yes—
309C22×C4#12114yes—
310C22×C4#12114yes—
311C22×C4#12112yes—
312C22×C4#12712yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4114yes—
320C22×C42−1+414yes—
321C2×C8—232no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—28no—
325C2×C8—28no—
326C2×C8—24no—
327C2×C8—28no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—44no—
331C2×C8#12318yes—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—816no—
352C22⋊C4—84no—
353C42—42no—
354C42—42no—
355C42—44no—
356C23—216no—
357C23—216no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—42no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—88no—
369C23—88no—
370C23—81no—
371C23—81no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23#39214yes—
378C23#44918yes—
379C23#46318yes—
380C23#49514yes—
381C23#49614yes—
382C23C2×#12011yescharacteristic
383C23C2×#12111yescharacteristic
384C23C2×#12711yescharacteristic
385C23#54714yes—
386C23C2×#13512yes—
387C23C2×AGL(1,ℤ/8)12yes—
388C23#57714yes—
389D4—44no—
390D4—44no—
391D4—816no—
392D4—81no—
393D4—81no—
394D4—82no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—816no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—816no—
423C2×C4—84no—
424C2×C4—88no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#39214yes—
429C2×C4C2×#12011yescharacteristic
430C2×C4C2×#12011yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12312yes—
434C2×C4#52512yes—
435C2×C4#52512yes—
436C2×C4#53612yes—
437C2×C4#53612yes—
438C2×C4#53612yes—
439C2×C4#54714yes—
440C2×C4#54714yes—
441C2×C4#55214yes—
442C8—28no—
443C8—44no—
444C22—216no—
445C22—28no—
446C22—216no—
447C22—28no—
448C22—24no—
449C22—24no—
450C22—88no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—164no—
455C22—162no—
456C22—162no—
457C22#201218yes—
458C22C2×#39211yescharacteristic
459C22#240612yes—
460C22#241312yes—
461C22#250112yes—
462C22C2×#44912yes—
463C22#300912yes—
464C22C2×#46312yes—
465C22#304312yes—
466C22C2×#49511yescharacteristic
467C22C2×#49611yescharacteristic
468C22#315412yes—
469C22#333718yes—
470C22#334518yes—
471C4—24no—
472C4—84no—
473C4—84no—
474C4—82no—
475C4—84no—
476C4—162no—
477C4#240612yes—
478C4#241312yes—
479C4#248612yes—
480C4#261411yescharacteristic
481C4#284911yescharacteristic
482C2—28no—
483C2—162no—
484C2#1506911yescharacteristic
485C2C2×#201211yescharacteristic
486C2#2342311yescharacteristic
487C2#2409114yes—
488C2#2458114yes—
489C2#2792612yes—
490C2#2793412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C23×#106C2semidirect
5. C2×#1824C2semidirect
6. #13904C2semidirect
7. #11325C2semidirect
8. #11121C2semidirect
9. #10714C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C23×#1062
C2×#25042
C2×#18932
C2×#18242
#139042
#113252
#111212
#107142