Finite Group Databasebeta

← #192079

Subgroups — #192079

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34057C211yesmaximal, characteristic
2#21363C212yesmaximal, complemented
3C2×#1466C211yesmaximal, characteristic
4#13941C211yesmaximal, characteristic
5#8858C214yesmaximal, complemented
6#8854C212yesmaximal, complemented
7#8853C212yesmaximal, complemented
8#8017C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11#2318—24no—
12#2315C2211yescharacteristic
13#2315C2211yescharacteristic
14#2313—21no—
15#2312C2212yes—
16#2308—21no—
17C2×#390—21no—
18#2290—24no—
19#2283C2212yes—
20C2×#382C2211yescharacteristic
21#2280—21no—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4—21no—
24#1949—24no—
25#1853C2214yes—
26#1852C2214yes—
27#1776—24no—
28#1466C418yescomplemented
29C2×#323C2212yes—
30C2×#323C2212yes—
31#1439C2212yes—
32#1439C2212yes—
33C4⋊#95—21no—
34C22⋊#95—21no—
35C22⋊#95—21no—
36#1294C2214yes—
37#1294C2214yes—
38C23×D4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—24no—
44C2×C4×D4C2×C414yes—
45C2×C4×D4Q814yes—
46C2×#117—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4C2311yescharacteristic
55C22×C22⋊C4C2311yescharacteristic
56C22×C42—22no—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#432—24no—
60#432C2×C418yescomplemented
61C2×#107—24no—
62C2×#106—24no—
63C2×C4⋊C8—22no—
64C2×C22⋊C8C2314yes—
65C2×#97—22no—
66#390—44no—
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—24no—
77#382D414yes—
78#381—22no—
79#381—22no—
80#380—22no—
81#380—21no—
82#380—21no—
83#380C2312yes—
84#380C2312yes—
85#379—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4D414yes—
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—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2312yes—
101#340—24no—
102#340—24no—
103#323C2×C418yes—
104#323C2×C418yescomplemented
105C25C2×D411yescharacteristic
106C25C2×D411yescharacteristic
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4C4⋊C414yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—82no—
125C23×C4—82no—
126C23×C4C4⋊C414yes—
127C23×C4C4⋊C414yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—24no—
134C4×D4—28no—
135C4×D4—28no—
136C4×D4—28no—
137C4×D4—28no—
138#117—84no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C4⋊C414yes—
149C2×C4⋊C4C2×D414yes—
150C2×C4⋊C4C2×D414yes—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—82no—
170C2×C22⋊C4—82no—
171C2×C22⋊C4C22×C412yes—
172C2×C22⋊C4C22×C412yes—
173C2×C22⋊C4C2×Q812yes—
174C2×C22⋊C4C2×Q812yes—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—84no—
181C2×C42C22×C412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×Q812yes—
185C2×C42#4114yes—
186#107—48no—
187#106—48no—
188C4⋊C8—28no—
189C22⋊C8—24no—
190C22⋊C8C22×C418yes—
191#97—28no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95—22no—
200#95—24no—
201#95—41no—
202#95—41no—
203#95—42no—
204#95—82no—
205#95—82no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24C2×C4⋊C412yes—
221C24C2×C4⋊C412yes—
222C24AGL(1,ℤ/8)12yes—
223C24#13712yes—
224C24C22×D411yescharacteristic
225C2×D4—28no—
226C2×D4—28no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—44no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4C2×C4⋊C412yes—
279C22×C4C2×C4⋊C412yes—
280C22×C4C2×C4⋊C412yes—
281C22×C4C2×C4⋊C412yes—
282C22×C4#12011yescharacteristic
283C22×C4#12012yes—
284C22×C4#12011yescharacteristic
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D412yes—
289C22×C4C22×Q811yescharacteristic
290C22×C4C2×#4112yes—
291C2×C8—28no—
292C2×C8—44no—
293C2×C8—44no—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—84no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—84no—
313C22⋊C4—84no—
314C22⋊C4—88no—
315C22⋊C4—88no—
316C42—48no—
317C42—84no—
318C42—84no—
319C42C4×D414yes—
320C42C4×D414yes—
321C42C22⋊Q814yes—
322C42C22⋊Q814yes—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—44no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23—42no—
343C23—42no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23#42814yes—
359C23C22×C4⋊C411yescharacteristic
360C23C2×#12011yescharacteristic
361C23C2×#12311yescharacteristic
362C23C2×AGL(1,ℤ/8)11yescharacteristic
363C23C2×#13711yescharacteristic
364C23#57511yescharacteristic
365C23#57511yescharacteristic
366D4—416no—
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—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—88no—
405C2×C4—88no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#38514yes—
416C2×C4#38512yes—
417C2×C4#38512yes—
418C2×C4C2×C4×D412yes—
419C2×C4C2×C22⋊Q812yes—
420C8—48no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—24no—
430C22—44no—
431C22—44no—
432C22—42no—
433C22—42no—
434C22—44no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22C2×#38511yescharacteristic
448C22C2×#42811yescharacteristic
449C22#291111yescharacteristic
450C22#300611yescharacteristic
451C22#300711yescharacteristic
452C22#301912yes—
453C22#302512yes—
454C22#309411yescharacteristic
455C22#309511yescharacteristic
456C4—24no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—22no—
461C4—42no—
462C4—42no—
463C4—84no—
464C4—84no—
465C4—84no—
466C4—162no—
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—44no—
472C2—82no—
473C2#1815812yes—
474C2#1815912yes—
475C2#2084611yescharacteristic
476C2#2085611yescharacteristic
477C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21363C2semidirect
5. #8858C2semidirect
6. #8854C2semidirect
7. #8853C2semidirect
8. #8017C2semidirect
28. #1466C4semidirect
60. #432C2×C4semidirect
104. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340572
#213632
C2×#14662
#139412
#88582
#88542
#88532
#80172