Finite Group Databasebeta

← #204214

Subgroups — #204214

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic
4C2×#1472C214yesmaximal
5#9654C214yesmaximal, complemented
6#9650C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic
8C22×C22⋊Q8C2212yes—
9C23×C42C2211yescharacteristic
10C22×C4⋊C8—21no—
11C2×#407C2212yes—
12C22×C22⋊C8C2212yes—
13C2×C4×M16—21no—
14#2857—21no—
15#2857—21no—
16#2857—21no—
17#2857—21no—
18#2857—21no—
19#2857—21no—
20#2857—21no—
21#2857—21no—
22C2×#391C412yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#384C411yescharacteristic
26C2×#380C412yes—
27C2×#380C412yes—
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—
37C2×#336—21no—
38#1472—216no—
39C2×#324C22116yes—
40C23×Q8C2×C412yes—
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C23×C8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—44no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—28no—
56C2×C22⋊Q8—28no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2314yes—
63C22×C22⋊C4C2×C414yes—
64C22×C42—44no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—44no—
80#407—24no—
81#407D418yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—28no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8C2314yes—
86C2×C22⋊C8C2314yes—
87C4×M16—44no—
88C2×#97—22no—
89C2×C4×C8—22no—
90#391—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#384—21no—
103#384—21no—
104#384—21no—
105#384—21no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114C2×#95C2×C414yes—
115C2×#95C2×C414yes—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#324—232no—
121C25C22×C411yescharacteristic
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—28no—
125C22×Q8—44no—
126C22×Q8—44no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C414yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C414yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D414yes—
155C23×C4#4114yes—
156C23×C4C2411yescharacteristic
157C2×M16—28no—
158C2×M16—44no—
159C2×M16—44no—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8—44no—
163C22×C8—44no—
164C22×C8—44no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C22⋊Q8—48no—
170C22⋊Q8—48no—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—216no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C4⋊C8—44no—
226C4⋊C8—44no—
227C22⋊C8—28no—
228C22⋊C8—24no—
229C22⋊C8—28no—
230C22⋊C8—44no—
231C22⋊C8C2×D418yes—
232#97—44no—
233C4×C8—44no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24C2×C22⋊C411yescharacteristic
245C24C2×C22⋊C411yescharacteristic
246C24C4×D414yes—
247C24C23×C411yescharacteristic
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—816no—
255C22×C4—22no—
256C22×C4—22no—
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—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—41no—
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—416no—
309C22×C4—416no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4#10114yes—
313C22×C4#10114yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
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×C4C4×D414yes—
325C22×C4#12114yes—
326C22×C4#12114yes—
327C22×C4#12412yes—
328C22×C4#12412yes—
329C22×C4#12712yes—
330C22×C4#12712yes—
331C22×C4#13114yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C2×#4112yes—
336M16—416no—
337C2×C8—28no—
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—44no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—88no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42#9914yes—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—28no—
390C23—28no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23#39012yes—
397C23#39012yes—
398C23C2×#10112yes—
399C23C2×#10112yes—
400C23C22×C22⋊C411yescharacteristic
401C23C2×C4×D412yes—
402C23C2×#13112yes—
403C23#56812yes—
404Q8—88no—
405Q8—88no—
406C2×C4—28no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—416no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—88no—
457C2×C4#39012yes—
458C2×C4#39012yes—
459C2×C4#40812yes—
460C2×C4#40812yes—
461C2×C4C2×#9912yes—
462C2×C4C2×#9912yes—
463C2×C4C2×#9912yes—
464C2×C4#41012yes—
465C2×C4C2×#10112yes—
466C2×C4C2×#10112yes—
467C2×C4C2×#10112yes—
468C2×C4C2×#10112yes—
469C2×C4C2×#12112yes—
470C2×C4C2×#12411yescharacteristic
471C2×C4C2×#12711yescharacteristic
472C8—48no—
473C8—84no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—416no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22#196614yes—
495C22C2×#39011yescharacteristic
496C22C2×#40811yescharacteristic
497C22#287911yescharacteristic
498C22C22×#9911yescharacteristic
499C22C2×#41011yescharacteristic
500C22C22×#10111yescharacteristic
501C22C22×#10111yescharacteristic
502C22#293514yes—
503C4—22no—
504C4—21no—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—41no—
513C4—41no—
514C4—84no—
515C4—84no—
516C4#147214yescomplemented
517C4#196414yes—
518C2—24no—
519C2—24no—
520C2—42no—
521C2C2×#147211yescharacteristic
522C2#1738511yescharacteristic
523C2#1739711yescharacteristic
524C2#1740411yescharacteristic
525C2#2118911yescharacteristic
526C2C2×#196411yescharacteristic
527C2C2×#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
516. C4#1472semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#17972
C4×#4072
C2×#14722
#96542
#96502