Finite Group Databasebeta

← #194287

Subgroups — #194287

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#8983C214yesmaximal, complemented
7#8982C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3229—24no—
13#3229C418yescomplemented
14C2×#498—21no—
15C2×#487—21no—
16C2×#483—22no—
17C2×#475C2214yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103—21no—
22C22×#103—21no—
23#2370—22no—
24#2369—21no—
25#2365—21no—
26#2355—22no—
27#2355C2212yes—
28Q8×C4⋊C4C2212yes—
29#2329—22no—
30#2317—21no—
31C4×C22⋊Q8—22no—
32#2285C2211yescharacteristic
33#2274—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1990C2212yes—
36#1990C2212yes—
37#1988—22no—
38#1981—22no—
39#1981C2214yes—
40#1823—22no—
41#1481—24no—
42C2×Q8⋊C8C2214yes—
43#1296C2212yes—
44#1296C2212yes—
45#1279C2214yes—
46C23×Q8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533D414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#498—28no—
77#487—28no—
78#483—216no—
79#475—28no—
80#475C2×C4116yescomplemented
81#431—22no—
82#431—22no—
83#431—22no—
84#431C2×C414yes—
85C2×#106—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—44no—
96C2×#103—44no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2312yes—
101#391—24no—
102#391—22no—
103#389—22no—
104#389—22no—
105#389—24no—
106#389—22no—
107#387—24no—
108#387—24no—
109#387—24no—
110#385—24no—
111#384—22no—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#382—22no—
115#382—21no—
116#380—21no—
117#380D412yes—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2312yes—
130C4×C22⋊C4—24no—
131C2×#95—21no—
132#340—24no—
133#340—24no—
134Q8⋊C8—28no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—41no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8C2×D412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—41no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—42no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8C22⋊C414yes—
169C4⋊Q8C22⋊C414yes—
170C4⋊Q8C22×C414yes—
171#125—24no—
172#125—24no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8C22⋊C418yes—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—84no—
182C4×Q8C22×C414yes—
183C4×Q8C2×D414yes—
184#117—24no—
185#117—24no—
186#117—42no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4SD1614yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—82no—
227C2×C42C22×C412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235#106—216no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8C22×C414yes—
242#103—216no—
243#103—48no—
244#103—48no—
245C22⋊C8—24no—
246C22⋊C8—42no—
247C4×C8—24no—
248C4×C8—22no—
249C4×C8—22no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258C24#12011yescharacteristic
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—42no—
265C2×Q8—44no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—81no—
273C2×Q8—81no—
274C2×Q8—82no—
275C2×Q8—82no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8C2×C22⋊C414yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12312yes—
308C22×C4C2×SD1612yes—
309C22×C4#13512yes—
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4#13712yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—84no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4C2×C22⋊C414yes—
354C4⋊C4C2×SD1614yes—
355C22⋊C4—28no—
356C22⋊C4—44no—
357C42—24no—
358C42—24no—
359C42—44no—
360C42—44no—
361C42—82no—
362C42—84no—
363C42—84no—
364C42C2×C22⋊C412yes—
365C42C4×D412yes—
366C42C4×D412yes—
367C42#12112yes—
368C42#12112yes—
369C42#12312yes—
370C42C23×C411yescharacteristic
371C42C22×D411yescharacteristic
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23#38612yes—
379C23#44812yes—
380C23#44912yes—
381C23C2×#12011yescharacteristic
382Q8—28no—
383Q8—44no—
384Q8—44no—
385Q8—44no—
386Q8—84no—
387Q8—88no—
388Q8—82no—
389Q8—82no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—88no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#38612yes—
425C2×C4#38612yes—
426C2×C4#38612yes—
427C2×C4C4×SD1614yes—
428C2×C4#44014yes—
429C2×C4#44812yes—
430C2×C4#44912yes—
431C2×C4#46014yes—
432C2×C4#46414yes—
433C2×C4C22×C22⋊C411yescharacteristic
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12311yescharacteristic
439C2×C4C22×SD1611yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57712yes—
444C2×C4#57812yes—
445C8—44no—
446C8—82no—
447C8—82no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22#188014yes—
455C22C2×#38611yescharacteristic
456C22C2×#44811yescharacteristic
457C22C2×#44911yescharacteristic
458C22#300911yescharacteristic
459C22#301111yescharacteristic
460C22#302211yescharacteristic
461C22#302211yescharacteristic
462C4—22no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—44no—
467C4—44no—
468C4—44no—
469C4—84no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4#188014yes—
474C4#289812yes—
475C4C2×C4×SD1611yescharacteristic
476C4C2×#44011yescharacteristic
477C4C2×#46011yescharacteristic
478C4C2×#46411yescharacteristic
479C4#309712yes—
480C2—24no—
481C2#1890411yescharacteristic
482C2#1890611yescharacteristic
483C2#1892411yescharacteristic
484C2#1892611yescharacteristic
485C2#2091311yescharacteristic
486C2C2×#188011yescharacteristic
487C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #8983C2semidirect
7. #8982C2semidirect
13. #3229C4semidirect
80. #475C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#342132
#218332
C2×#19812
C2×#14812
#89832
#89822
#80392
#78922