Finite Group Databasebeta

← #238215

Subgroups — #238215

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4#21311C211yesmaximal, characteristic
5#10357C212yesmaximal, complemented
6#10355C212yesmaximal, complemented
7#10350C212yesmaximal, complemented
8#10349C212yesmaximal, complemented
9C2×#1296C211yesmaximal, characteristic
10#7893C211yesmaximal, characteristic
11#7891C211yesmaximal, characteristic
12C2×#544—21no—
13C2×#533C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#2868—21no—
18#2859—21no—
19#2660—22no—
20#2552—21no—
21#2552C2212yes—
22#2551—21no—
23#2551C2212yes—
24#2440—22no—
25#2440—22no—
26#2317—21no—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1990C2212yes—
31#1989—21no—
32#1989—21no—
33#1988C2211yescharacteristic
34#1988C2211yescharacteristic
35#1987C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2212yes—
38#1981C2212yes—
39#1956C2212yes—
40#1892—22no—
41#1872—22no—
42#1803—22no—
43#1803—21no—
44#1803—21no—
45#1735—21no—
46#1735—21no—
47#1296C2212yes—
48#1296C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C23×C8—21no—
54C23×C8—21no—
55#544—44no—
56#544—44no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533Q814yescomplemented
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×#125—22no—
68C2×#125—22no—
69C2×#125—21no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×#117—21no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D412yes—
82C22×C42C2311yescharacteristic
83#431—22no—
84#431—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—24no—
96C2×#103—22no—
97C2×#103—22no—
98C2×C22⋊C8—21no—
99C2×C4×C8C2312yes—
100#398—24no—
101#398—24no—
102#395—24no—
103#395—24no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#385—24no—
111#384—22no—
112#384—22no—
113#384—22no—
114#384—21no—
115#384—21no—
116#384D411yescharacteristic
117#384D411yescharacteristic
118#384D412yes—
119#384C2312yes—
120#382—22no—
121#382—21no—
122#380—21no—
123#380D412yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132#376—21no—
133C43—21no—
134#340—24no—
135#340—22no—
136#340—22no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145C22×Q8—24no—
146C22×Q8—21no—
147C22×Q8—21no—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8—81no—
161C22×C8—81no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—48no—
169C4⋊Q8—44no—
170C4⋊Q8—44no—
171C4⋊Q8—82no—
172C4⋊Q8—82no—
173C4⋊Q8—84no—
174C4⋊Q8C2×Q812yes—
175C4⋊Q8C2×Q812yes—
176#125—24no—
177#125—48no—
178#125—44no—
179#125—44no—
180#125—42no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—416no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—42no—
189C4×Q8—42no—
190#117—22no—
191#117—22no—
192#117—24no—
193#117—42no—
194#117—42no—
195#117—42no—
196#117SD1614yes—
197#117Q1614yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—81no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D414yes—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—41no—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×Q812yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42C2411yescharacteristic
250C4⋊C8—22no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259#103—82no—
260#103—82no—
261#103—84no—
262C22⋊C8—42no—
263C4×C8—22no—
264C4×C8—22no—
265#95—24no—
266#95—24no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—22no—
273C24#12011yescharacteristic
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—48no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—84no—
286C2×Q8—84no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—44no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4C22⋊Q812yes—
309C22×C4C22⋊Q812yes—
310C22×C4C22⋊Q812yes—
311C22×C4#12412yes—
312C22×C4#12412yes—
313C22×C4#12412yes—
314C22×C4C2×SD1612yes—
315C22×C4C2×Q1612yes—
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—84no—
334C2×C8—84no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—82no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C4⋊C4C2×SD1612yes—
371C4⋊C4C2×SD1612yes—
372C4⋊C4C2×Q1612yes—
373C4⋊C4C2×Q1612yes—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C42—24no—
379C42—24no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—44no—
389C42C22⋊Q812yes—
390C42C22⋊Q812yes—
391C42C22⋊Q812yes—
392C42#12412yes—
393C42#12412yes—
394C42#12412yes—
395C42C22×Q811yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23#39312yes—
404C23#45012yes—
405C23#45112yes—
406C23C2×#12011yescharacteristic
407Q8—44no—
408Q8—42no—
409Q8—42no—
410Q8—82no—
411Q8—81no—
412Q8—81no—
413Q8—168no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—164no—
454C2×C4#39312yes—
455C2×C4#39312yes—
456C2×C4#39312yes—
457C2×C4#45012yes—
458C2×C4#45112yes—
459C2×C4#47514yes—
460C2×C4#47714yes—
461C2×C4#48714yes—
462C2×C4#48914yes—
463C2×C4C2×C22⋊Q811yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4C2×#12411yescharacteristic
467C2×C4C2×#12411yescharacteristic
468C2×C4C2×#12411yescharacteristic
469C2×C4C22×SD1611yescharacteristic
470C2×C4C22×Q1611yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57711yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57811yescharacteristic
477C8—82no—
478C8—81no—
479C8—81no—
480C8—82no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22#202414yes—
488C22C2×#39311yescharacteristic
489C22C2×#45011yescharacteristic
490C22C2×#45111yescharacteristic
491C22#301211yescharacteristic
492C22#301311yescharacteristic
493C22#301911yescharacteristic
494C22#302011yescharacteristic
495C4—22no—
496C4—21no—
497C4—21no—
498C4—22no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—82no—
505C4—84no—
506C4—82no—
507C4—82no—
508C4#202414yes—
509C4C2×#47511yescharacteristic
510C4C2×#47711yescharacteristic
511C4#308311yescharacteristic
512C4#308411yescharacteristic
513C4C2×#48711yescharacteristic
514C4C2×#48911yescharacteristic
515C4#313611yescharacteristic
516C4#313711yescharacteristic
517C2—24no—
518C2#1887111yescharacteristic
519C2#1887311yescharacteristic
520C2#1888311yescharacteristic
521C2#1888511yescharacteristic
522C2C2×#202411yescharacteristic
523C2#2345811yescharacteristic
524C2#2346211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10357C2semidirect
6. #10355C2semidirect
7. #10350C2semidirect
8. #10349C2semidirect
60. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#218852
#218602
#213112
#103572
#103552
#103502
#103492
C2×#12962
#78932
#78912