Finite Group Databasebeta

← #410019

Subgroups — #410019

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic
4C2×#1873C211yesmaximal, characteristic
5C2×#1824C211yesmaximal, characteristic
6C2×#1818C211yesmaximal, characteristic
7#11327C214yesmaximal, complemented
8#11067C214yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#483—24no—
15C2×#483—24no—
16C2×#483—24no—
17C22×#106C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2625—24no—
23#2625—24no—
24C2×#400—21no—
25C2×#398—21no—
26C2×#397—22no—
27#2317—24no—
28#2317—24no—
29C2×#391C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#385—22no—
33C2×#384—21no—
34C2×#382—21no—
35C2×#380C2211yescharacteristic
36#1873—24no—
37#1824—24no—
38#1818—24no—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#336C2214yes—
43C23×Q8C2311yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8D414yes—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—28no—
62C2×#117D414yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D414yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42—21no—
84C22×C42C2312yes—
85#483—232no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—28no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8D414yes—
100#400—28no—
101#398—28no—
102#397—216no—
103#391—24no—
104#391—24no—
105#389—28no—
106#389—28no—
107#389—24no—
108#389—24no—
109#385—216no—
110#384—28no—
111#382—28no—
112#380—22no—
113#380D414yes—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—24no—
119#340—28no—
120#340—24no—
121#340—28no—
122#336—28no—
123C25C2×D411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8C2×D412yes—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—84no—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—24no—
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8C2×D414yes—
163C22⋊Q8—28no—
164C22⋊Q8—28no—
165C22⋊Q8—44no—
166C22⋊Q8—44no—
167#117—216no—
168#117—216no—
169#117—44no—
170#117—44no—
171#117—44no—
172#117—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—216no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—88no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4#4118yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—82no—
223C2×C42—28no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231#106—216no—
232#103—216no—
233#103—48no—
234C22⋊C8—28no—
235C22⋊C8—44no—
236#95—216no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—24no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—24no—
245C24—22no—
246C24#12012yes—
247C24#12012yes—
248C24#12112yes—
249C24C22×D411yescharacteristic
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—28no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—44no—
261C2×Q8—44no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—48no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—48no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4—88no—
309C22×C4#12014yes—
310C22×C4#12012yes—
311C22×C4#12012yes—
312C22×C4#12114yes—
313C22×C4#12114yes—
314C22×C4#12112yes—
315C22×C4#12714yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4114yes—
323C22×C42−1+414yes—
324C2×C8—24no—
325C2×C8—28no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—44no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—44no—
333C2×C8—44no—
334C4⋊C4—24no—
335C4⋊C4—28no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—28no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—816no—
351C4⋊C4—88no—
352C4⋊C4#12318yes—
353C22⋊C4—216no—
354C22⋊C4—48no—
355C22⋊C4—48no—
356C22⋊C4—84no—
357C42—216no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—22no—
367C23—24no—
368C23—24no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23#39214yes—
378C23C2×#12011yescharacteristic
379C23C2×#12011yescharacteristic
380C23C2×#12111yescharacteristic
381C23#52512yes—
382C23#53612yes—
383C23#57714yes—
384C23#57712yes—
385C23#57812yes—
386Q8—44no—
387Q8—44no—
388Q8—82no—
389Q8—82no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—816no—
423C2×C4—81no—
424C2×C4—84no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—84no—
431C2×C4—82no—
432C2×C4#39214yes—
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4C2×#12312yes—
437C2×C4#52512yes—
438C2×C4C2×#12711yescharacteristic
439C2×C4#53612yes—
440C2×C4#53612yes—
441C2×C4#54714yes—
442C2×C4#54714yes—
443C2×C4#54714yes—
444C2×C4#55214yes—
445C8—44no—
446C8—82no—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—44no—
451C22—44no—
452C22—48no—
453C22—44no—
454C22—44no—
455C22C2×#39211yescharacteristic
456C22#240612yes—
457C22#241312yes—
458C22#261411yescharacteristic
459C22#284911yescharacteristic
460C22#300911yescharacteristic
461C22#300912yes—
462C22#301012yes—
463C22#301111yescharacteristic
464C22#301312yes—
465C22#305712yes—
466C22#306312yes—
467C22#312014yes—
468C22#312514yes—
469C22#324214yes—
470C22#324414yes—
471C4—24no—
472C4—48no—
473C4—84no—
474C4—82no—
475C4—84no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4#240612yes—
480C4#241312yes—
481C4#248612yes—
482C4#250112yes—
483C2—44no—
484C2#1506911yescharacteristic
485C2#2338811yescharacteristic
486C2#2339411yescharacteristic
487C2#2407112yes—
488C2#2407412yes—
489C2#2461614yes—
490C2#2964812yes—
491C2#3301612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11327C2semidirect
8. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#26252
C2×#23172
C2×#18732
C2×#18242
C2×#18182
#113272
#110672
#107312