Finite Group Databasebeta

← #188308

Subgroups — #188308

Order 512.

1938 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2#20626C212yesmaximal, complemented
3C2×#1462C212yesmaximal
4#8561C218yesmaximal, complemented
5#7909C212yesmaximal, complemented
6C23×C4⋊C4C411yescharacteristic
7C23×C22⋊C4C411yescharacteristic
8C23×C22⋊C4C411yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423—22no—
12C2×#407C2214yes—
13C2×#388—22no—
14C2×#388—22no—
15C2×#388—22no—
16C2×#388—22no—
17C2×#388C414yes—
18C2×#388C2214yes—
19C22×#95C411yescharacteristic
20C22×#95C2211yescharacteristic
21#1797—22no—
22#1797—22no—
23#1797C2214yescomplemented
24#1797C2214yescomplemented
25#1728—22no—
26#1462—28no—
27C2×#323C2218yes—
28#1280C2218yescomplemented
29C26C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—24no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—44no—
41C22×C4⋊C4C2×C414yes—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—24no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—44no—
52C22×C22⋊C4—44no—
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C414yes—
55C22×C22⋊C4C2×C414yes—
56C22×C22⋊C4D414yes—
57C22×C22⋊C4Q814yes—
58C22×C22⋊C4C2312yes—
59C22×C42—44no—
60C22×C42C2×C412yes—
61C22×C42D412yes—
62C22×C42D412yes—
63C22×C42D412yes—
64C22×C42D412yes—
65C22×C42C2312yes—
66#423—48no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69#407—28no—
70#407—28no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8C2318yes—
73#388—24no—
74#388—24no—
75#388—24no—
76#388—24no—
77#388—24no—
78#388—24no—
79#388—24no—
80#388—24no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95C2×C414yes—
87C2×#95C2×C412yes—
88C2×#95C2312yes—
89#336—24no—
90#336—24no—
91#323—216no—
92C25—21no—
93C25—21no—
94C25—22no—
95C25—42no—
96C25—42no—
97C25—42no—
98C25—42no—
99C25—42no—
100C25—42no—
101C25C4⋊C414yes—
102C25C22×C411yescharacteristic
103C25C22×C412yes—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C4⋊C414yes—
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C22×C412yes—
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×Q812yes—
144C23×C4#4114yes—
145C23×C4C2411yescharacteristic
146C2×M16—48no—
147C2×M16—48no—
148C2×M16—48no—
149C22×C8—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—48no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—48no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4C4⋊C418yes—
177C2×C22⋊C4C2×D414yes—
178C2×C22⋊C4C2×Q814yes—
179C2×C42—28no—
180C2×C42—44no—
181C2×C42—44no—
182C2×C42—44no—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C4⋊C8—48no—
194C4⋊C8—48no—
195C22⋊C8—28no—
196C22⋊C8—28no—
197C22⋊C8—48no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—24no—
211C24—24no—
212C24—24no—
213C24—24no—
214C24—48no—
215C24—42no—
216C24—42no—
217C24—48no—
218C24—48no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—48no—
226C24—42no—
227C24—42no—
228C24—48no—
229C24—48no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—88no—
237C24—88no—
238C24—88no—
239C24—88no—
240C24C2×C22⋊C411yescharacteristic
241C24C2×C22⋊C411yescharacteristic
242C24C2×C4⋊C412yes—
243C24C2×C4⋊C414yes—
244C24#11712yes—
245C24C23×C411yescharacteristic
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—48no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—416no—
289C22×C4—416no—
290C22×C4—48no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—48no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—88no—
314C22×C4—88no—
315C22×C4#9914yes—
316C22×C4#10014yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C4⋊C414yes—
323C22×C4C2×C4⋊C412yes—
324C22×C4#11712yes—
325C22×C4#12114yes—
326C22×C4#12114yes—
327C22×C4C22⋊Q814yes—
328C22×C4C22⋊Q814yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C2×#4112yes—
333M16—816no—
334C2×C8—48no—
335C2×C8—48no—
336C2×C8—48no—
337C4⋊C4—48no—
338C4⋊C4—48no—
339C22⋊C4—28no—
340C22⋊C4—28no—
341C22⋊C4—216no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—48no—
345C22⋊C4—48no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—24no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C23—28no—
355C23—28no—
356C23—28no—
357C23—28no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—28no—
378C23—48no—
379C23—48no—
380C23—44no—
381C23—48no—
382C23—44no—
383C23—48no—
384C23—44no—
385C23—44no—
386C23—48no—
387C23—48no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—44no—
395C23—84no—
396C23—84no—
397C23—84no—
398C23—84no—
399C23—832no—
400C23—832no—
401C23—84no—
402C23—84no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—84no—
407C23—84no—
408C23—84no—
409C23#38012yes—
410C23#38012yes—
411C23C2×#9912yes—
412C23#41012yes—
413C23C2×#10012yes—
414C23#41312yes—
415C23C22×C22⋊C411yescharacteristic
416C23C22×C4⋊C411yescharacteristic
417C23C2×#11711yescharacteristic
418C23#56812yes—
419C23#56812yes—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4#38014yes—
471C2×C4C2×#9914yes—
472C2×C4C2×#10014yes—
473C2×C4C2×#12112yes—
474C2×C4C2×C22⋊Q812yes—
475C8—88no—
476C22—216no—
477C22—216no—
478C22—22no—
479C22—22no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—88no—
502C22—816no—
503C22—816no—
504C22—88no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22#179914yes—
510C22#180014yes—
511C22C2×#38011yescharacteristic
512C22#287911yescharacteristic
513C22#287911yescharacteristic
514C22C22×#9911yescharacteristic
515C22C2×#41011yescharacteristic
516C22C22×#10011yescharacteristic
517C22C2×#41311yescharacteristic
518C22#290512yes—
519C22#292112yes—
520C4—22no—
521C4—22no—
522C4—24no—
523C4—24no—
524C4—24no—
525C4—44no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4—82no—
530C2—24no—
531C2—24no—
532C2—22no—
533C2—22no—
534C2—84no—
535C2#1739211yescharacteristic
536C2#1739311yescharacteristic
537C2#1739611yescharacteristic
538C2#1739811yescharacteristic
539C2#2062711yescharacteristic
540C2C2×#179911yescharacteristic
541C2C2×#180011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
4. #8561C2semidirect
5. #7909C2semidirect
23. #1797C22semidirect
24. #1797C22semidirect
28. #1280C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
#206262
C2×#14622
#85612
#79092