Finite Group Databasebeta

← #204153

Subgroups — #204153

Order 512.

1283 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34965C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic, complemented
3#20366C211yesmaximal, characteristic, complemented
4C2×#1494C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic, complemented
6C2×#1490C211yesmaximal, characteristic, complemented
7C22×D4⋊C8C211yesmaximal, characteristic, complemented
8#9645C212yesmaximal, complemented
9#9644C212yesmaximal, complemented
10#9642C212yesmaximal, complemented
11#9640C212yesmaximal, complemented
12C2×#529C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2603—22no—
19#2566C412yes—
20#2544C412yes—
21#2504C411yescharacteristic
22#2502C411yescharacteristic
23#2435C411yescharacteristic
24#2433C411yescharacteristic
25#2411—22no—
26#2392—22no—
27C4×#127C2212yes—
28C4×#124C2212yes—
29C4×#120—22no—
30C4×#117C2211yescharacteristic
31#1940—22no—
32#1850—22no—
33#1842C2212yescomplemented
34#1831C2212yescomplemented
35#1746—22no—
36#1744C2212yescomplemented
37#1744C2212yescomplemented
38#1729—22no—
39#1494—24no—
40#1490—24no—
41#1483—24no—
42C2×Q8⋊C8C2212yescomplemented
43C2×Q8⋊C8C2211yescharacteristic, complemented
44C2×Q8⋊C8C2211yescharacteristic, complemented
45C2×D4⋊C8—24no—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50C2×D4⋊C8C2212yescomplemented
51C2×D4⋊C8C2211yescharacteristic, complemented
52C2×D4⋊C8C2211yescharacteristic, complemented
53C23×D4C2×C411yescharacteristic
54C24×C4—21no—
55C22×M16—21no—
56C23×C8—21no—
57C23×C8—21no—
58C2×#127C2×C412yes—
59#529—22no—
60#529—22no—
61#529—22no—
62#529—22no—
63C2×#126—21no—
64C2×#125C2×C411yescharacteristic
65C2×#124C2×C411yescharacteristic
66C2×#124C2×C412yes—
67C2×#123—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×#121C2×C411yescharacteristic
70C2×#120—22no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—24no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4C2×C412yes—
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2312yes—
83C2×#117—21no—
84C2×#117D412yes—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C22⋊C4—21no—
87C22×C42C2311yescharacteristic
88#432—22no—
89#432—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2311yescharacteristic
97C2×C22⋊C8—21no—
98C4×M16—22no—
99C4×M16D414yescomplemented
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—22no—
104C2×C4×C8D414yescomplemented
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2312yes—
108#398—22no—
109#397—22no—
110#396—24no—
111#394—24no—
112#394—22no—
113#392—22no—
114#391C2×C412yes—
115#390—24no—
116#390C2×C412yes—
117#390C2×C412yes—
118#390C2×C412yes—
119#388—24no—
120#386—22no—
121#386—24no—
122#385—22no—
123#385—24no—
124#384C2×C412yes—
125#382C2×C412yes—
126#379—22no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4D412yes—
129C4×C22⋊C4—24no—
130C4×C22⋊C4C2×C411yescharacteristic
131C4×C22⋊C4C2×C411yescharacteristic
132C4×C22⋊C4D412yes—
133#376D412yes—
134C43C2312yes—
135#336—24no—
136#336—24no—
137#336—24no—
138Q8⋊C8—24no—
139Q8⋊C8—22no—
140Q8⋊C8—22no—
141D4⋊C8—24no—
142D4⋊C8—24no—
143D4⋊C8—24no—
144D4⋊C8—24no—
145D4⋊C8—24no—
146D4⋊C8—22no—
147D4⋊C8—22no—
148C25—21no—
149C22×Q8C22×C411yescharacteristic
150C22×D4—24no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—42no—
157C22×D4C22×C411yescharacteristic
158C22×D4C22×C412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—24no—
171C2×M16—44no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—44no—
180C22×C8—42no—
181C22×C8—82no—
182#127—24no—
183#127—42no—
184#126—42no—
185#126—42no—
186#125—22no—
187#125—22no—
188#124—24no—
189#124—22no—
190#124—22no—
191#124—42no—
192#123—42no—
193#123—42no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196#121—22no—
197#121—22no—
198#120—48no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—84no—
220C4×D4—84no—
221#117—44no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4C22⋊C414yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4C22⋊C412yes—
250C2×C22⋊C4C22⋊C412yes—
251C2×C42—21no—
252C2×C42C22×C412yes—
253C2×C42C22×C412yes—
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D414yes—
256C2×C42C2×D414yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—22no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—42no—
271C4⋊C8—84no—
272C22⋊C8—42no—
273#97—24no—
274#97—42no—
275C4×C8—24no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—22no—
279C4×C8C2×D412yes—
280C4×C8C2×D412yes—
281C4×C8C2×D414yes—
282#95—24no—
283#95—24no—
284#95C22⋊C412yes—
285#95C22⋊C412yes—
286#95C22⋊C414yes—
287C24—22no—
288C24—21no—
289C24—44no—
290C24—42no—
291C24—44no—
292C24—82no—
293C24C2×C22⋊C411yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×D4—28no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—48no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—42no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—84no—
317C2×D4#13114yes—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—84no—
338C22×C4—88no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4—82no—
342C22×C4C2×C22⋊C414yes—
343C22×C4C2×C22⋊C414yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12412yes—
352C22×C4#12712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356M16—48no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—48no—
363C2×C8—44no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—44no—
369C2×C8—42no—
370C2×C8—84no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—44no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—84no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—48no—
388C22⋊C4—48no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—44no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—24no—
405C42—24no—
406C42—44no—
407C42—41no—
408C42—41no—
409C42C4×D412yes—
410C42C4×D412yes—
411C42#12112yes—
412C42#12112yes—
413C42#12412yes—
414C42#12712yes—
415C42C22×D411yescharacteristic
416C42C22×D411yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C23—22no—
420C23—22no—
421C23—24no—
422C23—48no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—816no—
432C23—84no—
433C23—84no—
434C23#39012yes—
435C23#42114yes—
436C23C22×C22⋊C411yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—42no—
440D4—24no—
441D4—24no—
442D4—48no—
443D4—41no—
444D4—42no—
445D4—41no—
446D4—42no—
447D4—42no—
448D4—816no—
449D4—84no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—88no—
477C2×C4—84no—
478C2×C4—82no—
479C2×C4—84no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4#39012yes—
485C2×C4#39012yes—
486C2×C4#39012yes—
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12411yescharacteristic
492C2×C4C2×#12711yescharacteristic
493C2×C4C2×#13112yes—
494C2×C4#56812yes—
495C2×C4#57612yes—
496C2×C4#57712yes—
497C8—42no—
498C8—42no—
499C8—44no—
500C8—81no—
501C8—81no—
502C8—82no—
503C22—28no—
504C22—22no—
505C22—22no—
506C22—48no—
507C22—44no—
508C22—816no—
509C22—84no—
510C22—82no—
511C22—82no—
512C22—84no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#196314yes—
517C22C2×#39011yescharacteristic
518C22#288012yes—
519C22#289712yes—
520C22C2×#42112yes—
521C4—22no—
522C4—22no—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—44no—
527C4—44no—
528C4—81no—
529C4—81no—
530C4—82no—
531C4—81no—
532C4#292911yescharacteristic
533C4#293011yescharacteristic
534C4#293312yes—
535C4#313612yes—
536C4#315411yescharacteristic
537C4#315511yescharacteristic
538C2—24no—
539C2—84no—
540C2—161no—
541C2#1752114yes—
542C2#2119111yescharacteristic
543C2#2120611yescharacteristic
544C2C2×#196311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2433C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21976C2semidirect
3. #20366C2semidirect
4. C2×#1494C2semidirect
5. C2×#1483C2semidirect
6. C2×#1490C2semidirect
7. C22×D4⋊C8C2semidirect
8. #9645C2semidirect
9. #9644C2semidirect
10. #9642C2semidirect
11. #9640C2semidirect
33. #1842C22semidirect
34. #1831C22semidirect
36. #1744C22semidirect
37. #1744C22semidirect
42. C2×Q8⋊C8C22semidirect
43. C2×Q8⋊C8C22semidirect
44. C2×Q8⋊C8C22semidirect
50. C2×D4⋊C8C22semidirect
51. C2×D4⋊C8C22semidirect
52. C2×D4⋊C8C22semidirect
99. C4×M16D4semidirect
104. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#349652
#219762
#203662
C2×#14942
C2×#14832
C2×#14902
C22×D4⋊C82
#96452
#96442
#96422
#96402