Finite Group Databasebeta

← #417161

Subgroups — #417161

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3141C211yesmaximal, characteristic
2C2×#2868C211yesmaximal, characteristic, complemented
3#35289C211yesmaximal, characteristic
4C2×#1997C211yesmaximal, characteristic, complemented
5#21313C211yesmaximal, characteristic, complemented
6C2×#1962C211yesmaximal, characteristic
7#11078C218yesmaximal, complemented
8C4×#339C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8—22no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13#3141—24no—
14#3141—24no—
15C2×#489—22no—
16C2×#487—21no—
17C2×#486—21no—
18C2×#423C2211yescharacteristic
19C22×#104C2214yes—
20C2×#417C2214yes—
21C2×C4×M16C2211yescharacteristic
22#2868—24no—
23#2868—24no—
24#2862—21no—
25#2858—21no—
26#2640—22no—
27#2563—22no—
28#2547—22no—
29#2504—22no—
30#2436—22no—
31#2311C2214yes—
32C2×#384—22no—
33C2×#384C2214yes—
34C4×#117C2211yescharacteristic
35C2×C43C2211yescharacteristic
36#1997—24no—
37#1988—22no—
38#1987—22no—
39#1962—24no—
40C2×#343C2214yes—
41#1736C2214yes—
42C2×#339C2214yes—
43C22×#41C2312yes—
44C23×Q8—21no—
45C23×Q8D412yes—
46C22×M16C2312yes—
47C22×M16C2312yes—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8—216no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—416no—
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8D414yes—
71C2×#127D411yescharacteristic
72C2×#124—24no—
73C2×#124D412yes—
74C2×#121—21no—
75C2×#121—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117C2312yes—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D412yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#489—216no—
89#487—28no—
90#486—28no—
91#423—22no—
92#423—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×#104—28no—
96C2×#104—216no—
97#417—28no—
98#417—28no—
99C2×#103—24no—
100C2×#102—24no—
101C4×M16—22no—
102C4×M16D414yes—
103C2×#97—21no—
104C2×C4×C8—21no—
105#397—28no—
106#393—28no—
107#391—24no—
108#391—24no—
109#390—24no—
110#390—24no—
111#386—28no—
112#384—216no—
113#384—28no—
114#384—28no—
115#382—24no—
116#380—28no—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D412yes—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121#376—21no—
122C43—21no—
123C43—22no—
124C43D414yes—
125#343—28no—
126#340—28no—
127#339—28no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41—28no—
133C2×#41—28no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—24no—
139C22×Q8—48no—
140C22×Q8—48no—
141C22×Q8—41no—
142C22×Q8—41no—
143C22×Q8—42no—
144C22×Q8C2×D412yes—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×D412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D414yes—
155C23×C4C2411yescharacteristic
156C2×M16—22no—
157C2×M16—24no—
158C2×M16—24no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—28no—
162C2×M16C2×D414yes—
163C22×C8—22no—
164C22×C8—22no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—22no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—22no—
171C4⋊Q8—464no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184#127—22no—
185#127—22no—
186#124—24no—
187#124—24no—
188#124—88no—
189#121—42no—
190#121—42no—
191#121—42no—
192#121—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197#117—24no—
198#117—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—232no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—432no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—216no—
228C2×C42—28no—
229C2×C42—28no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42C2×D414yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42#4118yes—
246C2×C42#4112yes—
247C4⋊C8—42no—
248C4⋊C8—42no—
249#104—416no—
250#103—216no—
251#102—216no—
252#97—24no—
253C4×C8—24no—
254#95—216no—
255#95—22no—
256#95—22no—
257C24—22no—
258C24C22×D411yescharacteristic
259#41—44no—
260#41—44no—
261#41—44no—
262#41—44no—
263#41—48no—
264#41—48no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—28no—
268C2×Q8—416no—
269C2×Q8—416no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—48no—
275C2×Q8—84no—
276C2×Q8—88no—
277C2×Q8—84no—
278C2×Q8—88no—
279C2×Q8—84no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—28no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—48no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—44no—
289C2×D4—44no—
290C22×C4—28no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4#12012yes—
311C22×C4#12012yes—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12112yes—
317C22×C4#12112yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4114yes—
325C22×C42−1+414yes—
326M16—24no—
327M16—24no—
328M16—44no—
329M16—44no—
330M16—44no—
331C2×C8—24no—
332C2×C8—44no—
333C2×C8—42no—
334C2×C8—44no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—464no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—816no—
356C42—216no—
357C42—216no—
358C42—24no—
359C42—28no—
360C42—44no—
361C42—44no—
362C42—44no—
363C42#12012yes—
364C42#12012yes—
365C42#12112yes—
366C42#12112yes—
367C42#12112yes—
368C42#12112yes—
369C42#12418yescomplemented
370C42#12712yes—
371C42#12712yes—
372C23—22no—
373C23—22no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23C2×#12011yescharacteristic
379C23C2×#12011yescharacteristic
380C23C2×#12111yescharacteristic
381C23#54814yes—
382Q8—44no—
383Q8—44no—
384Q8—88no—
385Q8—88no—
386Q8—88no—
387Q8—88no—
388D4—44no—
389D4—44no—
390D4—84no—
391D4—84no—
392D4—84no—
393D4—84no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—232no—
397C2×C4—28no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—44no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4#39212yes—
421C2×C4#39212yes—
422C2×C4#39212yes—
423C2×C4#39212yes—
424C2×C4#45314yes—
425C2×C4#45614yes—
426C2×C4C2×#12011yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×#12111yescharacteristic
429C2×C4C2×#12412yes—
430C2×C4C2×#12711yescharacteristic
431C2×C4#53212yes—
432C2×C4#53212yes—
433C2×C4#53612yes—
434C2×C4#53612yes—
435C2×C4#53612yes—
436C2×C4#54714yes—
437C2×C4#54814yes—
438C2×C4#55614yes—
439C8—44no—
440C8—82no—
441C22—28no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—88no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22C2×#39211yescharacteristic
451C22#243212yes—
452C22#243612yes—
453C22#254512yes—
454C22C2×#45312yes—
455C22C2×#45612yes—
456C4—216no—
457C4—22no—
458C4—22no—
459C4—84no—
460C4—84no—
461C4—84no—
462C4—82no—
463C4#200214yes—
464C4#200514yes—
465C4#243212yes—
466C4#254512yes—
467C4#254712yes—
468C4#265811yescharacteristic
469C4#285811yescharacteristic
470C2—22no—
471C2—22no—
472C2—162no—
473C2#1524311yescharacteristic
474C2C2×#200211yescharacteristic
475C2C2×#200511yescharacteristic
476C2#2566614yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2868C2semidirect
4. C2×#1997C2semidirect
5. #21313C2semidirect
7. #11078C2semidirect
8. C4×#339C2semidirect
369. C42#124semidirect

Maximal subgroups

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

subgroupindex
C2×#31412
C2×#28682
#352892
C2×#19972
#213132
C2×#19622
#110782
C4×#3392