Finite Group Databasebeta

← #192061

Subgroups — #192061

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34057C211yesmaximal, characteristic
2#20843C212yesmaximal, complemented
3C2×#1459C211yesmaximal, characteristic
4#13839C211yesmaximal, characteristic
5#8855C214yesmaximal, complemented
6#8854C212yesmaximal, complemented
7#8853C212yesmaximal, complemented
8#8012C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C2×#427—22no—
12C22×C22⋊C8C2212yes—
13#2318—24no—
14#2315C2212yes—
15#2313C2212yes—
16#2312—21no—
17#2308—21no—
18#2290—22no—
19#2290—22no—
20#2283—21no—
21#2280C2212yes—
22C2×C4×C22⋊C4—21no—
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1854—24no—
26#1853C2214yes—
27#1852C2214yes—
28#1735—22no—
29#1459C418yescomplemented
30C2×#324C2212yes—
31C2×#323C2211yescharacteristic
32C2×#323C2211yescharacteristic
33#1441C2212yes—
34#1440—21no—
35#1439C2212yes—
36#1438—21no—
37C4×#95—21no—
38#1294C2214yes—
39#1294C2214yes—
40C23×D4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—24no—
47C2×C4×D4C2×C414yes—
48C2×C4×D4Q814yes—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—44no—
52C2×#117D412yes—
53C2×#117D412yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42D411yescharacteristic
67#427—48no—
68C2×#107—24no—
69C2×#106—24no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8C2×C418yescomplemented
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75#388—24no—
76#388—24no—
77#387—24no—
78#387—24no—
79#386—24no—
80#386—24no—
81#385—24no—
82#385—24no—
83#383—22no—
84#382—22no—
85#381—22no—
86#381—22no—
87#380—22no—
88#380D414yes—
89#380C2314yes—
90#379—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—44no—
100#376—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95C2312yes—
106#340—24no—
107#340—24no—
108#324C2×C418yescomplemented
109#323C2×C414yes—
110#323C2×C414yes—
111C25C2×D411yescharacteristic
112C25C2×D411yescharacteristic
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—24no—
116C22×D4—42no—
117C22×D4C4⋊C414yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C4⋊C414yes—
130C23×C4C22×C412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×Q812yes—
138C23×C4#4114yes—
139C23×C4C2411yescharacteristic
140C22×C8—28no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—44no—
144C22×C8—44no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—28no—
148C4×D4—28no—
149#117—28no—
150#117—48no—
151#117—44no—
152#117—44no—
153#117—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×Q812yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4C4⋊C414yes—
183C2×C22⋊C4C22×C412yes—
184C2×C22⋊C4C2×Q812yes—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—42no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—82no—
200C2×C42—82no—
201C2×C42C4⋊C414yes—
202#107—48no—
203#106—48no—
204C22⋊C8—28no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8C22×C414yes—
208C22⋊C8C22×C414yes—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—22no—
213#95—22no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—44no—
218#95—44no—
219#95—44no—
220C24—21no—
221C24—21no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—41no—
226C24—42no—
227C24—41no—
228C24—41no—
229C24—42no—
230C24—41no—
231C24C2×C4⋊C412yes—
232C24C2×C4⋊C412yes—
233C24#12012yes—
234C24#12312yes—
235C24#13514yes—
236C24C22×D411yescharacteristic
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—44no—
240C2×D4—48no—
241C2×D4—48no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—48no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—48no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—84no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4C2×C4⋊C412yes—
295C22×C4C2×C4⋊C412yes—
296C22×C4C2×C4⋊C412yes—
297C22×C4C2×C4⋊C412yes—
298C22×C4C4×D414yes—
299C22×C4C4×D414yes—
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4C22⋊Q814yes—
303C22×C4C22⋊Q814yes—
304C22×C4#12312yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D412yes—
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4112yes—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—416no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—28no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C22⋊C4—28no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C22⋊C4—416no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—84no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—24no—
337C42—24no—
338C42—44no—
339C42—44no—
340C42—44no—
341C42—44no—
342C42—84no—
343C42—82no—
344C42—82no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—42no—
365C23—42no—
366C23—44no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23—81no—
371C23—81no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23#38514yes—
379C23#42714yescomplemented
380C23C22×C4⋊C411yescharacteristic
381C23C2×#12011yescharacteristic
382C23C2×#12311yescharacteristic
383C23C2×#13512yes—
384C23#57512yes—
385D4—416no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—28no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—48no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—84no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—88no—
433C2×C4—88no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4#38512yes—
438C2×C4#38512yes—
439C2×C4B(4,𝔽2)12yes—
440C2×C4#45812yes—
441C2×C4C2×C4×D412yes—
442C2×C4C2×C22⋊Q812yes—
443C8—44no—
444C8—44no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—84no—
465C22—84no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22C2×#38511yescharacteristic
470C22C2×#42711yescharacteristic
471C22#291111yescharacteristic
472C22#300212yes—
473C22#301812yes—
474C22C2×B(4,𝔽2)11yescharacteristic
475C22C2×#45811yescharacteristic
476C22#309112yes—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—84no—
484C4—84no—
485C4—84no—
486C4—84no—
487C4#186714yes—
488C2—22no—
489C2—21no—
490C2—21no—
491C2—44no—
492C2—44no—
493C2#1810012yes—
494C2#1810712yes—
495C2#2084311yescharacteristic
496C2#2085511yescharacteristic
497C2C2×#186711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20843C2semidirect
5. #8855C2semidirect
6. #8854C2semidirect
7. #8853C2semidirect
8. #8012C2semidirect
29. #1459C4semidirect
72. C2×C22⋊C8C2×C4semidirect
108. #324C2×C4semidirect
379. C23#427semidirect

Maximal subgroups

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

subgroupindex
#340572
#208432
C2×#14592
#138392
#88552
#88542
#88532
#80122