Finite Group Databasebeta

← #195666

Subgroups — #195666

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3141C211yesmaximal, characteristic
2#34236C211yesmaximal, characteristic
3#21841C211yesmaximal, characteristic, complemented
4C2×#1981C211yesmaximal, characteristic, complemented
5C2×#1551C211yesmaximal, characteristic
6#9060C214yesmaximal, complemented
7#9057C214yesmaximal, complemented
8#7904C211yesmaximal, characteristic, complemented
9#7895C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3141—24no—
13#3141C418yescomplemented
14C2×#489—22no—
15C2×#487C2212yes—
16C2×#486C2212yes—
17C2×#432C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×#417—22no—
20C22×#103—22no—
21C2×C4×M16C2211yescharacteristic
22#2369C2212yes—
23#2366C2212yes—
24#2361—22no—
25#2357—21no—
26#2349—21no—
27#2339—22no—
28#2337—22no—
29#2319—21no—
30C4×#121—22no—
31#2283C2211yescharacteristic
32#2275—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1993—21no—
35#1993—21no—
36#1988C2214yes—
37#1981—21no—
38#1981—21no—
39#1981C2214yes—
40#1824—22no—
41#1551—24no—
42C2×#333C2212yes—
43C2×#331C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C22×#41D412yes—
47C23×Q8—21no—
48C22×M16—21no—
49C23×C8—21no—
50#532—22no—
51#532—22no—
52#532C2×C414yes—
53#532D414yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yescomplemented
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×#127C2311yescharacteristic
64C2×#124D412yes—
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4—24no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#489—216no—
78#487—24no—
79#487C2×C418yescomplemented
80#486—24no—
81#486C2×C418yescomplemented
82#432—22no—
83#432—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89#417—216no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—48no—
94C2×#102—24no—
95C2×C22⋊C8—21no—
96C4×M16—22no—
97C4×M16C2×C414yescomplemented
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8C2312yes—
101#391—24no—
102#390—22no—
103#390—22no—
104#389—22no—
105#389—22no—
106#388—24no—
107#388—24no—
108#386—24no—
109#386—24no—
110#385—24no—
111#383—21no—
112#382D411yescharacteristic
113#382D411yescharacteristic
114#381—22no—
115#381—21no—
116#380—22no—
117#380D412yes—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129#376—21no—
130C2×#95—22no—
131C2×#95—21no—
132#340—24no—
133#340—24no—
134#333—24no—
135#331—24no—
136C2×#41—24no—
137C2×#41—48no—
138C2×#41C22⋊C418yes—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—44no—
142C22×Q8—44no—
143C22×Q8—41no—
144C22×Q8—42no—
145C22×Q8C2×D412yes—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C2×D412yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×M16—24no—
158C2×M16—44no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—44no—
163C22×C8—42no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8C22×C414yes—
176C4⋊Q8C22×C412yes—
177C4⋊Q8C2×D412yes—
178#127—21no—
179#127—21no—
180#127C22×C412yes—
181#127C2×D412yes—
182#124—24no—
183#124C22⋊C418yes—
184#121—24no—
185#121—24no—
186#121—24no—
187#121—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4—84no—
195C4×D4—88no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—82no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—42no—
239#103—216no—
240#103—48no—
241#103—48no—
242#102—216no—
243C22⋊C8—42no—
244#97—24no—
245#97—42no—
246C4×C8—22no—
247C4×C8C22×C414yes—
248#95—24no—
249#95—24no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—22no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—22no—
259C24—22no—
260C24#12011yescharacteristic
261#41—416no—
262#41—88no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—44no—
266C2×Q8—416no—
267C2×Q8—42no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—84no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8C2×C22⋊C414yes—
275C2×D4—28no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4C2×C22⋊C414yes—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—28no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—44no—
300C22×C4—48no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—84no—
305C22×C4—82no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4#12312yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321M16—48no—
322C2×C8—24no—
323C2×C8—44no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—44no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—84no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—84no—
347C4⋊C4AGL(1,ℤ/8)14yes—
348C22⋊C4—28no—
349C22⋊C4—28no—
350C22⋊C4—48no—
351C22⋊C4—44no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C42—42no—
355C42—42no—
356C42—44no—
357C42—81no—
358C42—81no—
359C42—84no—
360C42C2×C22⋊C412yes—
361C42C4×D412yes—
362C42C4×D412yes—
363C42#12112yes—
364C42#12112yes—
365C42#12312yes—
366C42C23×C411yescharacteristic
367C42C22×D411yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C23—22no—
371C23—21no—
372C23—21no—
373C23—24no—
374C23—44no—
375C23—42no—
376C23—42no—
377C23—44no—
378C23#38612yes—
379C23#45514yes—
380C23C2×#12011yescharacteristic
381Q8—44no—
382Q8—48no—
383Q8—82no—
384Q8—88no—
385Q8—82no—
386Q8—82no—
387D4—44no—
388D4—416no—
389D4—44no—
390D4—44no—
391D4—42no—
392D4—42no—
393D4—44no—
394D4—82no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4—88no—
425C2×C4—84no—
426C2×C4—168no—
427C2×C4#38612yes—
428C2×C4#38612yes—
429C2×C4#38612yes—
430C2×C4C22×C22⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4C2×AGL(1,ℤ/8)12yes—
437C2×C4#57512yes—
438C2×C4#57712yes—
439C2×C4#57812yes—
440C8—44no—
441C8—82no—
442C8—82no—
443C22—24no—
444C22—24no—
445C22—22no—
446C22—22no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—84no—
451C22—84no—
452C22#188514yes—
453C22C2×#38611yescharacteristic
454C22#300612yes—
455C22#300911yescharacteristic
456C22#301111yescharacteristic
457C22C2×#45512yes—
458C4—22no—
459C4—22no—
460C4—21no—
461C4—22no—
462C4—21no—
463C4—44no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—84no—
469C4C4×AGL(1,ℤ/8)12yes—
470C4#294712yes—
471C4#303811yescharacteristic
472C4#304011yescharacteristic
473C4#304612yes—
474C2—24no—
475C2—84no—
476C2#1857012yes—
477C2#1857212yes—
478C2#2092811yescharacteristic
479C2#2093111yescharacteristic
480C2C2×#188511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ 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
3. #21841C2semidirect
4. C2×#1981C2semidirect
6. #9060C2semidirect
7. #9057C2semidirect
8. #7904C2semidirect
9. #7895C2semidirect
13. #3141C4semidirect
60. C2×C4⋊Q8C2×C4semidirect
79. #487C2×C4semidirect
81. #486C2×C4semidirect
97. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31412
#342362
#218412
C2×#19812
C2×#15512
#90602
#90572
#79042
#78952