Finite Group Databasebeta

← #215917

Subgroups — #215917

Order 512.

1052 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
1#35002C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic
3#13948C211yesmaximal, characteristic
4#9911C212yesmaximal, complemented
5#9908C212yesmaximal, complemented
6#9885C212yesmaximal, complemented
7#9884C212yesmaximal, complemented
8#8713C212yesmaximal
9#8713C212yesmaximal
10C2×#543—21no—
11#2857C2212yes—
12#2857C2212yes—
13#2659—21no—
14#2659—21no—
15#2549—24no—
16#2440C414yescomplemented
17#2440C414yescomplemented
18#2315—21no—
19C2×#388—21no—
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#1923—24no—
23#1797C2212yes—
24#1779C2212yes—
25C4×C22⋊C8C2212yes—
26#1442—21no—
27#1439—21no—
28#1436—21no—
29C4⋊#95C2211yescharacteristic
30C4⋊#95C2211yescharacteristic
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⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4D411yescharacteristic
56C22×C22⋊C4D411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8C2314yes—
64C2×C4×C8—22no—
65#398—24no—
66#398—24no—
67#395—24no—
68#395—24no—
69#391—22no—
70#391—22no—
71#389—22no—
72#389—22no—
73#389C2×C414yes—
74#389C2×C414yes—
75#388—44no—
76#385—24no—
77#385—24no—
78#384C2×C412yes—
79#384C2×C412yes—
80#380—22no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380C2×C414yes—
85#380C2314yes—
86C4×C22⋊C4C2×C414yes—
87C4×C22⋊C4D414yes—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98C2×#95C2311yescharacteristic
99C2×#95C2311yescharacteristic
100C2×#95C2311yescharacteristic
101C2×#95C2311yescharacteristic
102#336—22no—
103#336—22no—
104C25C2×D411yescharacteristic
105C22×Q8—22no—
106C22×Q8—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—81no—
116C23×C4—81no—
117C23×C4C22×C412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—24no—
126C4⋊Q8—82no—
127C4⋊Q8—82no—
128C22⋊Q8—416no—
129C22⋊Q8—416no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C22⋊Q8—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—81no—
151C2×C4⋊C4—81no—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C414yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—81no—
165C2×C22⋊C4—81no—
166C2×C22⋊C4C22⋊C414yes—
167C2×C22⋊C4C22⋊C414yes—
168C2×C22⋊C4SD1614yes—
169C2×C22⋊C4Q1614yes—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—81no—
173C2×C42—81no—
174C2×C42—81no—
175C2×C42C22×C412yes—
176C2×C42C22×C412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C4⋊C8—82no—
180C22⋊C8—24no—
181C22⋊C8—24no—
182C22⋊C8—82no—
183C22⋊C8—82no—
184C4×C8—82no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—42no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—81no—
198#95—81no—
199#95—81no—
200#95C22⋊C414yes—
201#95C22⋊C414yes—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—41no—
207C24—41no—
208C24—41no—
209C24—41no—
210C24C2×C22⋊C412yes—
211C24C2×SD1612yes—
212C24C2×Q1612yes—
213C24C22×D411yescharacteristic
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8—82no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
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—41no—
258C22×C4—44no—
259C22×C4—44no—
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—41no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×SD1612yes—
286C22×C4C2×SD1612yes—
287C22×C4C2×Q1612yes—
288C22×C4C2×Q1612yes—
289C22×C4AGL(1,ℤ/8)12yes—
290C22×C4AGL(1,ℤ/8)12yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C42−1+412yes—
294C22×C42−1+412yes—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—48no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C22⋊C4—28no—
313C22⋊C4—48no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—84no—
317C22⋊C4—84no—
318C22⋊C4#103116yes—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—84no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23C2×#10314yes—
351C23C22×C22⋊C411yescharacteristic
352C23#53611yescharacteristic
353C23#53611yescharacteristic
354C23C22×SD1611yescharacteristic
355C23C22×Q1611yescharacteristic
356C23C2×AGL(1,ℤ/8)11yescharacteristic
357C23C2×AGL(1,ℤ/8)11yescharacteristic
358Q8—164no—
359Q8—164no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—84no—
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—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—164no—
425C2×C4—164no—
426C2×C4C2×#10314yes—
427C2×C4C2×#10314yes—
428C2×C4#41814yes—
429C2×C4#51912yes—
430C2×C4#53612yes—
431C8—84no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—24no—
445C22—24no—
446C22—24no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22#231011yescharacteristic
456C22#288311yescharacteristic
457C22#288311yescharacteristic
458C22C22×#10311yescharacteristic
459C22C2×#41811yescharacteristic
460C22#290111yescharacteristic
461C22#290111yescharacteristic
462C22#322711yescharacteristic
463C22#322711yescharacteristic
464C22#323011yescharacteristic
465C22#323011yescharacteristic
466C4—22no—
467C4—22no—
468C4—24no—
469C4—24no—
470C4—24no—
471C4—42no—
472C4—42no—
473C4—82no—
474C4—82no—
475C4—84no—
476C4—84no—
477C4—161no—
478C4—161no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—44no—
484C2#1761411yescharacteristic
485C2#1761411yescharacteristic
486C2#1762011yescharacteristic
487C2#1762611yescharacteristic
488C2#2182711yescharacteristic
489C2#2183411yescharacteristic
490C2#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. #9885C2semidirect
7. #9884C2semidirect
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
#219022
#139482
#99112
#99082
#98852
#98842
#87132