Finite Group Databasebeta

← #238728

Subgroups — #238728

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic, complemented
3#20817C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic, complemented
6C22×D4⋊C8C211yesmaximal, characteristic
7#10436C214yesmaximal, complemented
8#10434C214yesmaximal, complemented
9#8045C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×#418—21no—
15C22×C4×C8C2211yescharacteristic
16#2848—22no—
17#2626—22no—
18#2580—22no—
19#2566—22no—
20#2563—21no—
21#2542—21no—
22#2538C2214yes—
23#2317—21no—
24#2309—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#1987—21no—
28#1981—21no—
29#1892—22no—
30#1845C2212yescomplemented
31#1834C2212yescomplemented
32#1808—22no—
33#1804—22no—
34#1803C2214yescomplemented
35#1802C2214yes—
36C4×#107—22no—
37#1736—21no—
38#1493—24no—
39C2×Q8⋊C8C2212yescomplemented
40C2×Q8⋊C8C2212yescomplemented
41C2×D4⋊C8—24no—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1296C2214yescomplemented
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C22×M16—21no—
48C23×C8—21no—
49C23×C8—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54C2×C4⋊Q8D411yescharacteristic
55C2×C4⋊Q8D411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8D412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2312yes—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4D412yes—
69C2×C4×D4C2312yes—
70C2×#117—21no—
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85#418—44no—
86C2×#103—24no—
87C2×#102—24no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8D414yes—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#399—24no—
94#396—28no—
95#393—24no—
96#390—22no—
97#390—22no—
98#390—24no—
99#389—24no—
100#385—28no—
101#385—24no—
102#384—22no—
103#384C2312yes—
104#384C2312yes—
105#382—22no—
106#382—21no—
107#382—21no—
108#380—21no—
109#380—24no—
110#380—21no—
111#380—21no—
112#380D412yes—
113#379—24no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D412yes—
118#340—24no—
119#340—24no—
120#340—24no—
121Q8⋊C8—24no—
122Q8⋊C8—24no—
123D4⋊C8—24no—
124D4⋊C8—24no—
125C25—21no—
126C22×Q8—21no—
127C22×Q8C2×D412yes—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4Q1614yes—
135C22×D4C2×D412yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—42no—
147C2×M16—42no—
148C2×M16—42no—
149C22×C8—21no—
150C22×C8—24no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—44no—
155C22×C8—42no—
156C22×C8—82no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161#125—42no—
162#125—42no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C22⋊Q8—84no—
168C22⋊Q8—84no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185#117—42no—
186#117—42no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D414yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—41no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—41no—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4112yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228#107—48no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235#103—84no—
236#102—48no—
237C4×C8—24no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8C2×D412yes—
241C4×C8C2×D412yes—
242#95—28no—
243#95—22no—
244#95—22no—
245#95—41no—
246#95—41no—
247C24—22no—
248C24—24no—
249C24—42no—
250C24—42no—
251C24—82no—
252C24—82no—
253C24#12011yescharacteristic
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—42no—
259C2×Q8—82no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—28no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4C2×Q1614yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—84no—
288C22×C4—88no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—82no—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12312yes—
299C22×C4#12312yes—
300C22×C4#12412yes—
301C22×C4C2×Q1612yes—
302C22×C4#13512yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4#13712yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308M16—84no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—48no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—44no—
319C2×C8—84no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—88no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—88no—
348C22⋊C4—88no—
349C22⋊C4—82no—
350C42—42no—
351C42—42no—
352C42—44no—
353C42—82no—
354C42#12112yes—
355C42#12112yes—
356C42#12112yes—
357C42#12312yes—
358C42#12312yes—
359C42#12412yes—
360C42C22×D411yescharacteristic
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C23—22no—
365C23—22no—
366C23—24no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—48no—
372C23—88no—
373C23—84no—
374C23—84no—
375C23—88no—
376C23#39412yes—
377C23#44912yes—
378C23#45112yes—
379C23C2×#12011yescharacteristic
380Q8—44no—
381Q8—42no—
382Q8—42no—
383Q8—84no—
384D4—24no—
385D4—24no—
386D4—42no—
387D4—42no—
388D4—48no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—88no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4#39412yes—
415C2×C4#39412yes—
416C2×C4#39412yes—
417C2×C4#44912yes—
418C2×C4#45112yes—
419C2×C4#46714yes—
420C2×C4#46814yes—
421C2×C4#48414yes—
422C2×C4#48514yes—
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×#12111yescharacteristic
425C2×C4C2×#12111yescharacteristic
426C2×C4C2×#12311yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C2×C4C2×#12411yescharacteristic
429C2×C4C22×Q1611yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4#57612yes—
434C2×C4#57712yes—
435C8—24no—
436C8—42no—
437C8—82no—
438C8—82no—
439C22—22no—
440C22—22no—
441C22—24no—
442C22—24no—
443C22—44no—
444C22—48no—
445C22—816no—
446C22—82no—
447C22—82no—
448C22—84no—
449C22#203614yes—
450C22C2×#39411yescharacteristic
451C22C2×#45111yescharacteristic
452C22C2×#44911yescharacteristic
453C22#301212yes—
454C22#301812yes—
455C4—22no—
456C4—21no—
457C4—21no—
458C4—22no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—84no—
463C4—162no—
464C4#203614yes—
465C4C2×#46711yescharacteristic
466C4C2×#46811yescharacteristic
467C4#305711yescharacteristic
468C4#305811yescharacteristic
469C4C2×#48511yescharacteristic
470C4C2×#48411yescharacteristic
471C4#309612yes—
472C2—24no—
473C2—84no—
474C2#1866512yes—
475C2#1867812yes—
476C2C2×#203611yescharacteristic
477C2#2348611yescharacteristic
478C2#2349811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21885C2semidirect
3. #20817C2semidirect
5. C2×#1493C2semidirect
7. #10436C2semidirect
8. #10434C2semidirect
9. #8045C2semidirect
30. #1845C22semidirect
31. #1834C22semidirect
34. #1803C22semidirect
39. C2×Q8⋊C8C22semidirect
40. C2×Q8⋊C8C22semidirect
44. #1296C22semidirect

Maximal subgroups

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

subgroupindex
#351722
#218852
#208172
#206462
C2×#14932
C22×D4⋊C82
#104362
#104342
#80452