Finite Group Databasebeta

← #204713

Subgroups — #204713

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic, complemented
3C4×#432C211yesmaximal, characteristic, complemented
4C2×#1534C212yesmaximal, complemented
5C2×#1533C212yesmaximal
6#9688C214yesmaximal, complemented
7#9687C214yesmaximal, complemented
8C22×#125C2211yescharacteristic
9C22×#124C2211yescharacteristic
10C2×#432C2212yes—
11C2×#432C2212yes—
12#2861—21no—
13#2861—21no—
14#2861—21no—
15#2861—21no—
16#2861C412yes—
17#2861C412yes—
18#2861C2212yes—
19#2861C2212yes—
20C2×#391C411yescharacteristic
21C2×#391C411yescharacteristic
22C2×#390C411yescharacteristic
23C2×#390C411yescharacteristic
24C2×#388—21no—
25C2×#388—21no—
26C2×#388—21no—
27C2×#388—21no—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831—21no—
31#1797—21no—
32#1797—21no—
33#1767—21no—
34#1767C2212yescomplemented
35#1767C2212yescomplemented
36C4×C4⋊C8—21no—
37#1749—21no—
38C4×C22⋊C8—21no—
39#1745C2212yescomplemented
40C4×#97C2212yescomplemented
41#1534—28no—
42#1533—28no—
43C2×#330C2218yescomplemented
44C2×#329C2218yes—
45C23×Q8C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C2×#125—21no—
48C2×#125—21no—
49C2×#125—24no—
50C2×#125—24no—
51C2×#125C2×C412yes—
52C2×#125C2312yes—
53C2×#124—21no—
54C2×#124—21no—
55C2×#124—24no—
56C2×#124—24no—
57C2×#124C2×C412yes—
58C2×#124C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76#432—24no—
77#432—24no—
78#432—24no—
79#432D418yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—22no—
87C2×#97C2314yes—
88C2×#97C2314yes—
89C2×C4×C8—22no—
90#391—21no—
91#391—21no—
92#391—21no—
93#391—21no—
94#391C2×C412yes—
95#391C2×C412yes—
96#391C2×C412yes—
97#391C2×C412yes—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390C2×C412yes—
103#390C2×C412yes—
104#390C2×C412yes—
105#390C2×C412yes—
106#388—22no—
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—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C43C2×C411yescharacteristic
127C43C2×C411yescharacteristic
128C43D411yescharacteristic
129C43D411yescharacteristic
130C43D411yescharacteristic
131C43D411yescharacteristic
132C43C2311yescharacteristic
133C43C2311yescharacteristic
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#330—216no—
139#329—216no—
140C25—21no—
141C22×Q8—21no—
142C22×Q8—24no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8C22×C412yes—
146C22×D4—21no—
147C22×D4—24no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4C22×C412yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C22×C8—24no—
162C22×C8—24no—
163#125—28no—
164#125—24no—
165#125—24no—
166#125—44no—
167#125—42no—
168#125—42no—
169#124—28no—
170#124—24no—
171#124—24no—
172#124—44no—
173#124—42no—
174#124—42no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
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—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D414yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4114yes—
242C2×C42C2411yescharacteristic
243C4⋊C8—28no—
244C4⋊C8—44no—
245C4⋊C8—44no—
246C4⋊C8—44no—
247C22⋊C8—28no—
248C22⋊C8—44no—
249#97—24no—
250#97—24no—
251#97—24no—
252#97—44no—
253#97C2×D418yes—
254C4×C8—44no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24C2×C22⋊C411yescharacteristic
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—88no—
279C2×D4—24no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—88no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
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—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
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×C4C4×D414yes—
328C22×C4#12114yes—
329C22×C4#12412yes—
330C22×C4#12712yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C2×C8—28no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—44no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42C4×D414yes—
387C42#12114yes—
388C42#12412yes—
389C42#12712yes—
390C42C22×D412yes—
391C42C2×#4112yes—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—84no—
407C23—84no—
408C23—84no—
409C23—84no—
410C23#39012yes—
411C23#41312yes—
412C23#41312yes—
413C23C22×C22⋊C411yescharacteristic
414Q8—84no—
415Q8—84no—
416D4—84no—
417D4—84no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
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—28no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
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—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4#39012yes—
471C2×C4#39012yes—
472C2×C4#39012yes—
473C2×C4#42112yes—
474C2×C4#42112yes—
475C2×C4#42112yes—
476C2×C4#42112yes—
477C2×C4C2×C4×D412yes—
478C2×C4C2×#12112yes—
479C2×C4C2×#12411yescharacteristic
480C2×C4C2×#12711yescharacteristic
481C8—48no—
482C8—84no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—168no—
497C22#196714yes—
498C22C2×#39011yescharacteristic
499C22C2×#41311yescharacteristic
500C22C2×#41311yescharacteristic
501C22C2×#42111yescharacteristic
502C22C2×#42111yescharacteristic
503C22C2×#42111yescharacteristic
504C22C2×#42111yescharacteristic
505C4—21no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—82no—
521C4—82no—
522C4—82no—
523C2—24no—
524C2—162no—
525C2#1765911yescharacteristic
526C2#1765911yescharacteristic
527C2#1766911yescharacteristic
528C2#1766911yescharacteristic
529C2#2120911yescharacteristic
530C2C2×#196711yescharacteristic
531C2#2125211yescharacteristic
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. #20810C2semidirect
3. C4×#432C2semidirect
4. C2×#1534C2semidirect
6. #9688C2semidirect
7. #9687C2semidirect
34. #1767C22semidirect
35. #1767C22semidirect
39. #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
#208102
C4×#4322
C2×#15342
C2×#15332
#96882
#96872