Finite Group Databasebeta

← #215930

Subgroups — #215930

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#21906C211yesmaximal, characteristic
3#13947C211yesmaximal, characteristic
4#9911C212yesmaximal, complemented
5#9908C212yesmaximal, complemented
6#9889C212yesmaximal, complemented
7#9888C212yesmaximal, complemented
8#8712C212yesmaximal
9#8712C212yesmaximal
10C2×#543—21no—
11#2857C2212yes—
12#2857C2212yes—
13#2659—21no—
14#2659—21no—
15#2549—24no—
16#2440C414yescomplemented
17#2440C414yescomplemented
18#2315—21no—
19C2×#380C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1924—24no—
22#1797C2212yes—
23#1779C2212yes—
24#1749C2212yes—
25#1443—21no—
26#1440—21no—
27#1437—21no—
28#1434—21no—
29#1434C2211yescharacteristic
30#1434C2211yescharacteristic
31#1434C2211yescharacteristic
32C22⋊#95C2211yescharacteristic
33C22⋊#95C2211yescharacteristic
34C4×#95C2211yescharacteristic
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C24×C4C2311yescharacteristic
40#543—44no—
41#543—44no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C412yes—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8C2314yes—
62C2×#97—22no—
63#398—24no—
64#398—24no—
65#395—24no—
66#395—24no—
67#391—22no—
68#391—22no—
69#389—22no—
70#389—22no—
71#389C2×C414yes—
72#389C2×C414yes—
73#385—24no—
74#385—24no—
75#384C2×C412yes—
76#384C2×C412yes—
77#380—22no—
78#380—22no—
79#380—22no—
80#380—22no—
81#380C2×C414yes—
82#380C2314yes—
83C4×C22⋊C4C2×C414yes—
84C4×C22⋊C4D414yes—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95D411yescharacteristic
96C2×#95D411yescharacteristic
97C2×#95C2311yescharacteristic
98C2×#95C2311yescharacteristic
99C2×#95C2311yescharacteristic
100C2×#95C2311yescharacteristic
101#336—22no—
102#336—22no—
103C25C2×D411yescharacteristic
104C22×Q8—22no—
105C22×Q8—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—81no—
115C23×C4—81no—
116C23×C4C22×C412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C4⋊Q8—82no—
126C4⋊Q8—82no—
127C22⋊Q8—416no—
128C22⋊Q8—416no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4—82no—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C22×C414yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—81no—
160C2×C22⋊C4—81no—
161C2×C22⋊C4C22⋊C414yes—
162C2×C22⋊C4C22⋊C414yes—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—81no—
166C2×C42—81no—
167C2×C42—81no—
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C4⋊C8—82no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175C22⋊C8—82no—
176C22⋊C8—82no—
177#97—82no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—42no—
187#95—42no—
188#95—42no—
189#95—42no—
190#95—81no—
191#95—81no—
192#95—81no—
193#95—81no—
194#95—81no—
195#95C22⋊C414yes—
196#95C22⋊C414yes—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—41no—
202C24—41no—
203C24—41no—
204C24—41no—
205C24C2×C22⋊C412yes—
206C24AGL(1,ℤ/8)12yes—
207C24AGL(1,ℤ/8)12yes—
208C24C22×D411yescharacteristic
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×Q8—82no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4#13712yes—
285C22×C4#13712yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C42−1+412yes—
289C22×C42−1+412yes—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C22⋊C4—28no—
306C22⋊C4—28no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—48no—
310C22⋊C4—84no—
311C22⋊C4—84no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—84no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23#41814yes—
344C23C22×C22⋊C411yescharacteristic
345C23#53611yescharacteristic
346C23#53611yescharacteristic
347C23C2×AGL(1,ℤ/8)11yescharacteristic
348C23C2×AGL(1,ℤ/8)11yescharacteristic
349C23C2×#13711yescharacteristic
350C23C2×#13711yescharacteristic
351Q8—164no—
352Q8—164no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—84no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—164no—
420C2×C4—164no—
421C2×C4#41914yes—
422C2×C4#51912yes—
423C2×C4#53612yes—
424C8—84no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22#231011yescharacteristic
449C22#288311yescharacteristic
450C22#288311yescharacteristic
451C22C2×#41811yescharacteristic
452C22C2×#41911yescharacteristic
453C22#290111yescharacteristic
454C22#290111yescharacteristic
455C22#323811yescharacteristic
456C22#323811yescharacteristic
457C22#323811yescharacteristic
458C22#323811yescharacteristic
459C4—22no—
460C4—22no—
461C4—24no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—82no—
467C4—82no—
468C4—84no—
469C4—84no—
470C4—161no—
471C4—161no—
472C2—21no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2—44no—
477C2#1762011yescharacteristic
478C2#1762011yescharacteristic
479C2#1762111yescharacteristic
480C2#1762711yescharacteristic
481C2#2182711yescharacteristic
482C2#2184011yescharacteristic
483C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9911C2semidirect
5. #9908C2semidirect
6. #9889C2semidirect
7. #9888C2semidirect
16. #2440C4semidirect
17. #2440C4semidirect

Maximal subgroups

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

subgroupindex
#350022
#219062
#139472
#99112
#99082
#98892
#98882
#87122