Finite Group Databasebeta

← #233966

Subgroups — #233966

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2C22×#477C211yesmaximal, characteristic, complemented
3#35178C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic, complemented
7#10236C218yesmaximal, complemented
8#8039C211yesmaximal, characteristic, complemented
9C2×#536—21no—
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#3227—24no—
14#3227—24no—
15C2×#498—21no—
16C2×#486—21no—
17C2×#481—22no—
18C2×#477—24no—
19C2×#477—24no—
20C2×#477—22no—
21C2×#477C2214yes—
22C2×#476C2214yes—
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C22×#107—21no—
26C22×C4⋊C8C2211yescharacteristic
27C22×#103—21no—
28#2854—22no—
29#2618—22no—
30#2548—21no—
31#2546—21no—
32#2545—22no—
33#2538C2214yes—
34#2536—22no—
35#2311—21no—
36#2285C2211yescharacteristic
37#1981—22no—
38#1956C2214yes—
39#1787—22no—
40#1480—24no—
41#1479—24no—
42C2×Q8⋊C8C2214yes—
43C2×D4⋊C8C2214yes—
44#1296C2214yes—
45C22×#41—21no—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48C23×C8—21no—
49#536—44no—
50#536—44no—
51#532—22no—
52#532—22no—
53#532—22no—
54#532—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D414yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×#127D411yescharacteristic
66C2×#124—22no—
67C2×#124—21no—
68C2×#123—24no—
69C2×C22⋊Q8—24no—
70C2×#121—21no—
71C2×#121D412yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C2×C4×D4—21no—
81C2×C4×D4C2312yes—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#498—28no—
90#486—28no—
91#481—216no—
92#477—216no—
93#477—28no—
94#477—28no—
95#476—28no—
96#476—28no—
97#431—22no—
98#431—22no—
99#431—22no—
100#431D414yes—
101C2×#107—28no—
102C2×#107—24no—
103C2×#106—24no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8D414yes—
108C2×C4⋊C8C2312yes—
109C2×C4⋊C8C2312yes—
110C2×#103—28no—
111C2×#103—24no—
112C2×#103—28no—
113C2×#102—24no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8C2312yes—
118#398—24no—
119#396—28no—
120#394—24no—
121#391—22no—
122#390—22no—
123#390—24no—
124#389—24no—
125#388—28no—
126#386—24no—
127#384C2312yes—
128#382—21no—
129#380—21no—
130#380—24no—
131#380D412yes—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D412yes—
134Q8⋊C8—28no—
135D4⋊C8—28no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—41no—
152C22×Q8—42no—
153C22×Q8—84no—
154C22×Q8SD1614yes—
155C22×Q8C2×D412yes—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C2×D412yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—41no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C22×C8—24no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—42no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—24no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—48no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185C4⋊Q8C2×D414yes—
186#127—22no—
187#127—22no—
188#124—48no—
189#124—42no—
190#124—42no—
191#123—416no—
192C22⋊Q8—48no—
193C22⋊Q8—48no—
194#121—24no—
195#121—24no—
196#121—42no—
197#121—42no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—42no—
213C4×D4—42no—
214#117—42no—
215#117—42no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D414yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—28no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42#4112yes—
255C2×C42C2411yescharacteristic
256#107—216no—
257#106—216no—
258C4⋊C8—24no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—42no—
262C4⋊C8C2×D414yes—
263C4⋊C8C2×D414yes—
264#103—216no—
265#103—88no—
266#102—216no—
267C22⋊C8—24no—
268C22⋊C8—42no—
269C4×C8—24no—
270C4×C8—22no—
271C4×C8—22no—
272#95—28no—
273#95—22no—
274#95—22no—
275C24—22no—
276C24#12011yescharacteristic
277#41—84no—
278#41—84no—
279#41—84no—
280#41—84no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—28no—
289C2×Q8—28no—
290C2×Q8—48no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—44no—
300C2×Q8—84no—
301C2×Q8—88no—
302C2×Q8—88no—
303C2×Q8C2×SD1614yes—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—28no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—82no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—82no—
334C22×C4—84no—
335C22×C4#12012yes—
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4#12012yes—
339C22×C4#12011yescharacteristic
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4#12712yes—
344C22×C4C2×SD1612yes—
345C22×C4#13512yes—
346C22×C4AGL(1,ℤ/8)12yes—
347C22×C4#13712yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—44no—
356C2×C8—48no—
357C2×C8—44no—
358C2×C8—42no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—44no—
366C4⋊C4—416no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C22⋊C4—416no—
380C22⋊C4—44no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C22⋊C4—88no—
385C42—42no—
386C42—42no—
387C42—44no—
388C42—44no—
389C42#12012yes—
390C42#12012yes—
391C42#12112yes—
392C42#12112yes—
393C42#12112yes—
394C42#12712yes—
395C42C22×D411yescharacteristic
396C42C22×D411yescharacteristic
397C42C22×D411yescharacteristic
398C42C2×#4111yescharacteristic
399C23—22no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23#39212yes—
407C23#45012yes—
408C23#45212yes—
409C23C2×#12011yescharacteristic
410Q8—24no—
411Q8—24no—
412Q8—48no—
413Q8—42no—
414Q8—48no—
415Q8—42no—
416Q8—84no—
417Q8—88no—
418Q8—84no—
419D4—44no—
420D4—44no—
421D4—84no—
422D4—88no—
423D4—84no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—48no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—84no—
445C2×C4—88no—
446C2×C4—84no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#39212yes—
452C2×C4#39212yes—
453C2×C4#39212yes—
454C2×C4#44814yes—
455C2×C4#44914yes—
456C2×C4#45012yes—
457C2×C4#45212yes—
458C2×C4#46514yes—
459C2×C4#46814yes—
460C2×C4C2×#12011yescharacteristic
461C2×C4C2×#12011yescharacteristic
462C2×C4C2×#12111yescharacteristic
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12711yescharacteristic
466C2×C4C22×SD1611yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
469C2×C4C2×#13711yescharacteristic
470C2×C4#57814yes—
471C8—44no—
472C8—44no—
473C8—82no—
474C22—22no—
475C22—22no—
476C22—24no—
477C22—24no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—84no—
482C22—84no—
483C22#200014yes—
484C22C2×#39211yescharacteristic
485C22C2×#45011yescharacteristic
486C22C2×#45211yescharacteristic
487C22#301112yes—
488C22#302112yes—
489C4—22no—
490C4—21no—
491C4—21no—
492C4—22no—
493C4—44no—
494C4—82no—
495C4—82no—
496C4—84no—
497C4—84no—
498C4#200014yes—
499C4C2×#44811yescharacteristic
500C4C2×#44911yescharacteristic
501C4#301112yes—
502C4C2×#46511yescharacteristic
503C4C2×#46811yescharacteristic
504C4#305912yes—
505C2—24no—
506C2—162no—
507C2#1889912yes—
508C2#1891312yes—
509C2C2×#200011yescharacteristic
510C2#2338911yescharacteristic
511C2#2340711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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. C22×#477C2semidirect
4. #21804C2semidirect
6. C2×#1479C2semidirect
7. #10236C2semidirect
8. #8039C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
C22×#4772
#351782
#218042
C2×#14802
C2×#14792
#102362
#80392