Finite Group Databasebeta

← #204223

Subgroups — #204223

Order 512.

1754 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
1C22×#389C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic
4C2×#1463C214yesmaximal
5#9654C214yesmaximal, complemented
6#9650C214yesmaximal, complemented
7C23×C4⋊C4C411yescharacteristic
8C23×C4⋊C4C411yescharacteristic
9C23×C4⋊C4C2212yes—
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407C2212yes—
13C2×#407C2212yes—
14C2×C4×M16—21no—
15C2×#389—21no—
16C2×#389—21no—
17C2×#389—21no—
18C2×#389—21no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C412yes—
24C2×#389C412yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C22×#95C412yes—
28#1797—21no—
29#1797—21no—
30#1797C2212yes—
31#1797C2212yes—
32#1749—21no—
33#1749C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8C2212yes—
36#1728—21no—
37#1728—21no—
38#1463—216no—
39C2×#324C22116yes—
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C24×C4C2×C412yes—
43C24×C4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—28no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—28no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—28no—
62C22×C4⋊C4—44no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C414yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2314yes—
69C22×C42—44no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#423—44no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85#407—24no—
86#407—28no—
87#407D418yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8C2314yes—
91C2×C22⋊C8C2314yes—
92C4×M16—44no—
93C2×#97—22no—
94C2×C4×C8—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95C2×C414yes—
116C2×#95C2×C414yes—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#324—232no—
122C25C22×C411yescharacteristic
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—28no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—28no—
133C23×C4—28no—
134C23×C4—42no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C414yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C414yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D414yes—
161C23×C4#4114yes—
162C23×C4C2411yescharacteristic
163C2×M16—28no—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—44no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8—24no—
170C22×C8—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—216no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4M1618yescomplemented
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
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×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C4⋊C8—44no—
234C4⋊C8—44no—
235C22⋊C8—28no—
236C22⋊C8—24no—
237C22⋊C8—28no—
238C22⋊C8—44no—
239C22⋊C8C2×D418yes—
240#97—44no—
241C4×C8—44no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24C2×C22⋊C411yescharacteristic
253C24C2×C22⋊C411yescharacteristic
254C24C4×D414yes—
255C24C23×C411yescharacteristic
256C22×C4—232no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—28no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—28no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
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—41no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—41no—
311C22×C4—432no—
312C22×C4—416no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—816no—
326C22×C4#9914yes—
327C22×C4#10114yes—
328C22×C4#10114yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C4×D414yes—
340C22×C4#12114yes—
341C22×C4#12114yes—
342C22×C4#12412yes—
343C22×C4#12412yes—
344C22×C4#12712yes—
345C22×C4#12712yes—
346C22×C4C2×M1618yes—
347C22×C4C2×M1614yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C2×#4112yes—
352M16—416no—
353M16—88no—
354C2×C8—28no—
355C2×C8—44no—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—44no—
359C2×C8—44no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—28no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—28no—
406C23—28no—
407C23—28no—
408C23#39012yes—
409C23#39012yes—
410C23C2×#9912yes—
411C23#41012yes—
412C23C2×#10112yes—
413C23C2×#10112yes—
414C23C22×C22⋊C411yescharacteristic
415C23C2×C4×D412yes—
416C23C22×M1612yes—
417C23#56812yes—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
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—21no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—216no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—416no—
465C2×C4—416no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—88no—
474C2×C4—88no—
475C2×C4—88no—
476C2×C4#39012yes—
477C2×C4#39012yes—
478C2×C4#40712yes—
479C2×C4#40712yes—
480C2×C4C2×#9912yes—
481C2×C4C2×#9912yes—
482C2×C4C2×#10112yes—
483C2×C4C2×#10112yes—
484C2×C4C2×#10112yes—
485C2×C4C2×#10112yes—
486C2×C4C2×#12112yes—
487C2×C4C2×#12411yescharacteristic
488C2×C4C2×#12711yescharacteristic
489C8—48no—
490C8—84no—
491C22—232no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22#196414yes—
511C22#196614yes—
512C22C2×#39011yescharacteristic
513C22C2×#40711yescharacteristic
514C22#287911yescharacteristic
515C22C22×#9911yescharacteristic
516C22C2×#41011yescharacteristic
517C22C22×#10111yescharacteristic
518C22C22×#10111yescharacteristic
519C22D4×M1614yes—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—21no—
525C4—21no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—48no—
531C4—41no—
532C4—41no—
533C4—84no—
534C4#146314yescomplemented
535C2—24no—
536C2—24no—
537C2—24no—
538C2C2×#146311yescharacteristic
539C2#1736511yescharacteristic
540C2#1739411yescharacteristic
541C2#1739711yescharacteristic
542C2#2119011yescharacteristic
543C2C2×#196411yescharacteristic
544C2C2×#196611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×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
5. #9654C2semidirect
6. #9650C2semidirect
204. C2×C4⋊C4M16semidirect
534. C4#1463semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
#206262
C4×#4072
C2×#14632
#96542
#96502