Finite Group Databasebeta

← #359724

Subgroups — #359724

Order 512.

2238 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3C2×#1821C211yesmaximal, characteristic, complemented
4#20600C212yesmaximal
5#11055C212yesmaximal, complemented
6#11054C212yesmaximal, complemented
7#10958C212yesmaximal, complemented
8#10957C212yesmaximal, complemented
9#10720C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#419C2214yes—
14C2×#418C2214yes—
15C2×#407C2212yes—
16#2345—24no—
17#2344—24no—
18#2316C2212yes—
19#2313—22no—
20#2311—24no—
21#2311C418yescomplemented
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#386—22no—
27#2280—22no—
28#2280C2212yes—
29C2×#380—22no—
30#2274C2212yes—
31#1821—24no—
32#1788—24no—
33#1788—24no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C22×#41—24no—
39C22×#41—24no—
40C22×#41—22no—
41C22×#41—22no—
42C22×#41—22no—
43C22×#41—24no—
44C22×#41—44no—
45C22×#41C2×C414yes—
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4D412yes—
51C24×C4C2311yescharacteristic
52C22×M16C2314yes—
53C2×#117—22no—
54C2×#117—24no—
55C2×#117—44no—
56C2×#117C2×C414yes—
57C2×#117C2314yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C414yes—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4D414yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#419—28no—
73#419—44no—
74#418—28no—
75#418—44no—
76C2×#103—28no—
77C2×#102—28no—
78#407—24no—
79#407—24no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82#391—22no—
83#391—22no—
84#391C2×C414yes—
85#391C2×C414yes—
86#390—22no—
87#390—22no—
88#390C2×C414yes—
89#390C2×C414yes—
90#389—28no—
91#387—216no—
92#386—216no—
93#381—24no—
94#381—22no—
95#380—24no—
96#380—24no—
97#380—216no—
98#380D414yes—
99#380D414yes—
100#379—22no—
101C4×C4⋊C4—28no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C2×#95—22no—
107C2×#95—22no—
108#343—24no—
109#343—24no—
110#339—44no—
111#339—44no—
112C25—21no—
113C25C2×D411yescharacteristic
114C2×#41—216no—
115C2×#41—28no—
116C2×#41—28no—
117C2×#41—28no—
118C2×#41—28no—
119C2×#41—48no—
120C2×#41—48no—
121C2×#41—48no—
122C2×#41—416no—
123C2×#41—48no—
124C2×#41—48no—
125C2×#41—416no—
126C2×#41—48no—
127C2×#41—44no—
128C2×#41—44no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8C22×C412yes—
135C22×Q8#4114yes—
136C22×D4—28no—
137C22×D4—24no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—44no—
144C22×D4—41no—
145C22×D4—41no—
146C22×D4—42no—
147C22×D4C22×C412yes—
148C22×D4#4114yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—84no—
164C23×C4C22⋊C418yes—
165C23×C4C22×C412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4#4114yes—
173C23×C4C2411yescharacteristic
174C2×M16—24no—
175C2×M16—24no—
176C2×M16—24no—
177C2×M16—24no—
178C2×M16—24no—
179C2×M16—24no—
180C22×C8—24no—
181#117—24no—
182#117—24no—
183#117—24no—
184#117—24no—
185#117—48no—
186#117—48no—
187#117—88no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C22⋊C418yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C2×D418yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—84no—
215C2×C22⋊C4—84no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—24no—
221C2×C42—28no—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224#103—416no—
225#102—416no—
226C22⋊C8—44no—
227C22⋊C8—44no—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—28no—
232#95—24no—
233#95—24no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—24no—
238C24—42no—
239C24—42no—
240C24—44no—
241C24—44no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24C2×C22⋊C412yes—
246C24#12011yescharacteristic
247C24#12011yescharacteristic
248C24#12312yes—
249C24C22×D411yescharacteristic
250#41—416no—
251#41—416no—
252#41—416no—
253#41—416no—
254#41—432no—
255#41—432no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—28no—
261C2×Q8—28no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8#11714yescomplemented
267C2×D4—216no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—28no—
273C2×D4—28no—
274C2×D4—48no—
275C2×D4—48no—
276C2×D4—48no—
277C2×D4—416no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4#11714yescomplemented
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—216no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—28no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—28no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—28no—
310C22×C4—48no—
311C22×C4—416no—
312C22×C4—416no—
313C22×C4—48no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—48no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—88no—
327C22×C4—84no—
328C22×C4—88no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C414yes—
331C22×C4C2×C22⋊C414yes—
332C22×C4C4×D414yes—
333C22×C4C4×D414yes—
334C22×C4#12012yes—
335C22×C4#12114yes—
336C22×C4#12114yes—
337C22×C4#12312yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D412yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4C2×#4112yes—
344C22×C4H(5,𝔽2)14yes—
345M16—44no—
346M16—44no—
347M16—44no—
348M16—44no—
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—44no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—216no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C4⋊C4—48no—
360C4⋊C4—88no—
361C22⋊C4—48no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—88no—
367C22⋊C4—88no—
368C42—24no—
369C42—24no—
370C42—42no—
371C42—42no—
372C42—44no—
373C42—44no—
374C42—88no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—24no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—48no—
392C23—48no—
393C23—44no—
394C23—416no—
395C23—44no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—88no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23#38612yes—
406C23#38612yes—
407C23C22×C22⋊C411yescharacteristic
408C23C2×#12011yescharacteristic
409C23C2×#12311yescharacteristic
410C23#52512yes—
411C23#53512yes—
412Q8—28no—
413Q8—44no—
414Q8—48no—
415D4—28no—
416D4—416no—
417D4—416no—
418D4—44no—
419D4—48no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—432no—
434C2×C4—416no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—88no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4#38614yes—
458C2×C4C2×#11711yescharacteristic
459C2×C4C2×#11711yescharacteristic
460C2×C4C2×C4×D412yes—
461C2×C4#51812yes—
462C2×C4C2×#12112yes—
463C2×C4#52512yes—
464C2×C4#53512yes—
465C2×C4#54614yes—
466C2×C4#54614yes—
467C2×C4#55114yes—
468C2×C4#55114yes—
469C8—84no—
470C22—24no—
471C22—24no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—44no—
483C22—44no—
484C22—48no—
485C22—48no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—84no—
490C22—88no—
491C22—88no—
492C22—82no—
493C22—82no—
494C22#228011yescharacteristic
495C22C2×#38611yescharacteristic
496C22#230911yescharacteristic
497C22#240412yes—
498C22#248212yes—
499C22#301912yes—
500C22#302012yes—
501C4—24no—
502C4—44no—
503C4—44no—
504C4—44no—
505C4—82no—
506C4—82no—
507C4—88no—
508C4#234212yes—
509C4#234212yes—
510C4#241114yes—
511C2—21no—
512C2—21no—
513C2—22no—
514C2—42no—
515C2—42no—
516C2—84no—
517C2#1386811yescharacteristic
518C2#2095311yescharacteristic
519C2#2095411yescharacteristic
520C2#2395114yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34178C2semidirect
3. C2×#1821C2semidirect
5. #11055C2semidirect
6. #11054C2semidirect
7. #10958C2semidirect
8. #10957C2semidirect
9. #10720C2semidirect
21. #2311C4semidirect
266. C2×Q8#117semidirect
284. C2×D4#117semidirect

Maximal subgroups

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

subgroupindex
#341782
C2×#23112
C2×#18212
#206002
#110552
#110542
#109582
#109572
#107202