Finite Group Databasebeta

← #204681

Subgroups — #204681

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic, complemented
3C4×#423C211yesmaximal, characteristic, complemented
4C2×#1528C212yesmaximal, complemented
5C2×#1527C212yesmaximal
6#9688C214yesmaximal, complemented
7#9687C214yesmaximal, complemented
8C22×#125C2211yescharacteristic
9C22×#124C2211yescharacteristic
10C2×#432C2212yes—
11C2×#423—21no—
12C2×C4×M16C2212yes—
13#2861—21no—
14#2861—21no—
15#2861—21no—
16#2861—21no—
17#2861C412yes—
18#2861C412yes—
19#2861C2212yes—
20#2861C2212yes—
21C2×#391C411yescharacteristic
22C2×#391C411yescharacteristic
23C2×#390C411yescharacteristic
24C2×#390C411yescharacteristic
25C2×#388—21no—
26C2×#388—21no—
27C2×#388—21no—
28C2×#388—21no—
29C2×C43C2211yescharacteristic
30#1767—21no—
31#1767C2212yescomplemented
32#1767C2212yescomplemented
33C4×C4⋊C8—21no—
34C4×C4⋊C8—21no—
35C4×C4⋊C8—21no—
36#1749—21no—
37C4×C22⋊C8—21no—
38#1745C2212yescomplemented
39#1728—21no—
40C4×#97C2212yescomplemented
41#1528—28no—
42#1527—28no—
43C2×#330C2218yescomplemented
44C2×#329C2218yes—
45C23×Q8C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C22×M16—22no—
48C2×#125—21no—
49C2×#125—21no—
50C2×#125—24no—
51C2×#125—24no—
52C2×#125C2×C412yes—
53C2×#125C2312yes—
54C2×#124—21no—
55C2×#124—21no—
56C2×#124—24no—
57C2×#124—24no—
58C2×#124C2×C412yes—
59C2×#124C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77#432—24no—
78#432D418yes—
79#423—28no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—22no—
84C4×M16—24no—
85C4×M16—24no—
86C2×#97—22no—
87C2×#97C2314yes—
88C2×#97C2314yes—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91#391—21no—
92#391—21no—
93#391—21no—
94#391—21no—
95#391C2×C412yes—
96#391C2×C412yes—
97#391C2×C412yes—
98#391C2×C412yes—
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#390C2×C412yes—
104#390C2×C412yes—
105#390C2×C412yes—
106#390C2×C412yes—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C43C2×C411yescharacteristic
128C43C2×C411yescharacteristic
129C43D411yescharacteristic
130C43D411yescharacteristic
131C43D411yescharacteristic
132C43D411yescharacteristic
133C43C2311yescharacteristic
134C43C2311yescharacteristic
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#330—216no—
140#329—216no—
141C25—21no—
142C22×Q8—21no—
143C22×Q8—24no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8C22×C412yes—
147C22×D4—21no—
148C22×D4—24no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4C22×C412yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×M16—28no—
163C2×M16—44no—
164C2×M16—44no—
165C22×C8—24no—
166C22×C8—24no—
167#125—28no—
168#125—24no—
169#125—24no—
170#125—44no—
171#125—42no—
172#125—42no—
173#124—28no—
174#124—24no—
175#124—24no—
176#124—44no—
177#124—42no—
178#124—42no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C42—24no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D414yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4114yes—
246C2×C42C2411yescharacteristic
247C4⋊C8—28no—
248C4⋊C8—28no—
249C4⋊C8—28no—
250C22⋊C8—28no—
251#97—24no—
252#97—24no—
253#97—24no—
254#97—44no—
255#97C2×D418yes—
256C4×C8—44no—
257C4×C8—44no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266C24—22no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24C2×C22⋊C411yescharacteristic
275C2×Q8—24no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—88no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—88no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4#10414yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C4×D414yes—
331C22×C4#12114yes—
332C22×C4#12412yes—
333C22×C4#12712yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337M16—416no—
338C2×C8—28no—
339C2×C8—28no—
340C2×C8—44no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—44no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42C4×D414yes—
391C42#12114yes—
392C42#12412yes—
393C42#12712yes—
394C42C22×D412yes—
395C42C2×#4112yes—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—84no—
408C23—84no—
409C23—84no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23—84no—
414C23#39012yes—
415C23C2×#10412yes—
416C23#42112yes—
417C23C22×C22⋊C411yescharacteristic
418Q8—84no—
419Q8—84no—
420D4—84no—
421D4—84no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4#39012yes—
473C2×C4#39012yes—
474C2×C4#39012yes—
475C2×C4#41312yes—
476C2×C4#41312yes—
477C2×C4C2×#10412yes—
478C2×C4C2×#10412yes—
479C2×C4#42112yes—
480C2×C4#42112yes—
481C2×C4C2×C4×D412yes—
482C2×C4C2×#12112yes—
483C2×C4C2×#12411yescharacteristic
484C2×C4C2×#12711yescharacteristic
485C8—48no—
486C8—48no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—168no—
501C22#198014yes—
502C22C2×#39011yescharacteristic
503C22C2×#41311yescharacteristic
504C22C2×#41311yescharacteristic
505C22C22×#10411yescharacteristic
506C22C2×#42111yescharacteristic
507C22C2×#42111yescharacteristic
508C22C2×#42111yescharacteristic
509C4—21no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4#196714yes—
527C2—24no—
528C2—162no—
529C2#1764211yescharacteristic
530C2#1765011yescharacteristic
531C2#1766811yescharacteristic
532C2#1767011yescharacteristic
533C2#2120911yescharacteristic
534C2C2×#196711yescharacteristic
535C2C2×#198011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#432C2semidirect
3. C4×#423C2semidirect
4. C2×#1528C2semidirect
6. #9688C2semidirect
7. #9687C2semidirect
31. #1767C22semidirect
32. #1767C22semidirect
38. #1745C22semidirect
40. C4×#97C22semidirect
43. C2×#330C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
C4×#4322
C4×#4232
C2×#15282
C2×#15272
#96882
#96872