Finite Group Databasebeta

← #192125

Subgroups — #192125

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34057C211yesmaximal, characteristic
2#21941C212yesmaximal, complemented
3C2×#1459C211yesmaximal, characteristic
4#14344C211yesmaximal, characteristic
5#8855C214yesmaximal, complemented
6#8854C212yesmaximal, complemented
7#8853C212yesmaximal, complemented
8#8024C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2318—22no—
12#2318—22no—
13#2315C2212yes—
14#2313C2212yes—
15#2312—21no—
16#2308—21no—
17#2290—24no—
18#2283—21no—
19#2280C2212yes—
20C2×#381C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4—21no—
23#1854—24no—
24#1853C2214yes—
25#1852C2214yes—
26#1808—22no—
27#1807—22no—
28#1459C418yescomplemented
29C2×#324C2212yes—
30C2×#323C2211yescharacteristic
31C2×#323C2211yescharacteristic
32#1441C2212yes—
33#1440—21no—
34#1439C2212yes—
35#1438—21no—
36#1434—21no—
37C22⋊#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×#117D414yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4C2311yescharacteristic
58C22×C42—22no—
59C22×C42D411yescharacteristic
60C2×#107—24no—
61C2×#106—24no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8C2×C418yescomplemented
65C2×C22⋊C8C2312yes—
66C2×C22⋊C8C2312yes—
67#388—24no—
68#388—24no—
69#387—24no—
70#387—24no—
71#386—24no—
72#386—24no—
73#385—24no—
74#385—24no—
75#383—22no—
76#382—21no—
77#382—21no—
78#381—22no—
79#381—22no—
80#381—24no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380D412yes—
85#380D412yes—
86#380C2314yes—
87#379—22no—
88C4×C4⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94#376—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95C2312yes—
103#340—24no—
104#340—24no—
105#324C2×C418yescomplemented
106#323C2×C414yes—
107#323C2×C414yes—
108C25C2×D411yescharacteristic
109C25C2×D411yescharacteristic
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—24no—
113C22×D4—42no—
114C22×D4C4⋊C414yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—82no—
123C23×C4—82no—
124C23×C4C4⋊C414yes—
125C23×C4C22×C412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×Q812yes—
133C23×C4#4114yes—
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—28no—
138C22×C8—84no—
139C4×D4—28no—
140C4×D4—28no—
141C4×D4—28no—
142C4×D4—28no—
143#117—84no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4C22×C412yes—
151C2×C4⋊C4C2×D414yes—
152C2×C4⋊C4C2×D414yes—
153C2×C4⋊C4C2×Q812yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—82no—
169C2×C22⋊C4—82no—
170C2×C22⋊C4—82no—
171C2×C22⋊C4—82no—
172C2×C22⋊C4C4⋊C414yes—
173C2×C22⋊C4C22×C412yes—
174C2×C22⋊C4C2×Q812yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—84no—
181C2×C42C4⋊C414yes—
182#107—84no—
183#106—84no—
184C22⋊C8—28no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8C22×C414yes—
188C22⋊C8C22×C414yes—
189#95—22no—
190#95—22no—
191#95—24no—
192#95—22no—
193#95—22no—
194#95—24no—
195#95—22no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—44no—
200#95—41no—
201#95—41no—
202#95—42no—
203#95—82no—
204#95—82no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—41no—
211C24—42no—
212C24—41no—
213C24—41no—
214C24—42no—
215C24—41no—
216C24C2×C4⋊C412yes—
217C24C2×C4⋊C412yes—
218C24#12012yes—
219C24#12312yes—
220C24#13514yes—
221C24C22×D411yescharacteristic
222C2×D4—28no—
223C2×D4—28no—
224C2×D4—44no—
225C2×D4—48no—
226C2×D4—48no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C2×C4⋊C412yes—
274C22×C4C2×C4⋊C412yes—
275C22×C4C2×C4⋊C412yes—
276C22×C4C2×C4⋊C412yes—
277C22×C4C4×D414yes—
278C22×C4C4×D414yes—
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4C22⋊Q814yes—
282C22×C4C22⋊Q814yes—
283C22×C4#12312yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D412yes—
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4112yes—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—88no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—48no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—84no—
310C22⋊C4—84no—
311C22⋊C4—84no—
312C22⋊C4—88no—
313C42—24no—
314C42—24no—
315C42—48no—
316C42—84no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—24no—
321C23—24no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—22no—
326C23—24no—
327C23—24no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23—44no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—81no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—81no—
350C23#38514yes—
351C23#42714yes—
352C23C22×C4⋊C411yescharacteristic
353C23C2×#12011yescharacteristic
354C23C2×#12311yescharacteristic
355C23C2×#13512yes—
356C23#57512yes—
357D4—416no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—84no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—88no—
393C2×C4—88no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4#38512yes—
409C2×C4#38512yes—
410C2×C4C2×C4×D412yes—
411C2×C4C2×C22⋊Q812yes—
412C8—44no—
413C8—44no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—24no—
423C22—24no—
424C22—24no—
425C22—24no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22—42no—
432C22—42no—
433C22—84no—
434C22—84no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22C2×#38511yescharacteristic
439C22C2×#42711yescharacteristic
440C22#291111yescharacteristic
441C22#300212yes—
442C22#302111yescharacteristic
443C22#302211yescharacteristic
444C22#302612yes—
445C22#309112yes—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—84no—
454C4—84no—
455C4—162no—
456C4—162no—
457C2—22no—
458C2—21no—
459C2—21no—
460C2—44no—
461C2—44no—
462C2#1829012yes—
463C2#1829112yes—
464C2#2084311yescharacteristic
465C2#2085811yescharacteristic
466C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21941C2semidirect
5. #8855C2semidirect
6. #8854C2semidirect
7. #8853C2semidirect
8. #8024C2semidirect
28. #1459C4semidirect
64. C2×C22⋊C8C2×C4semidirect
105. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340572
#219412
C2×#14592
#143442
#88552
#88542
#88532
#80242