Finite Group Databasebeta

← #390440

Subgroups — #390440

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2C2×#1894C212yesmaximal
3C2×#1878C212yesmaximal, complemented
4#13969C212yesmaximal
5#11315C212yesmaximal, complemented
6#11124C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#10789C212yesmaximal, complemented
9#10715C211yesmaximal, characteristic
10C23×Q16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#2001C418yescomplemented
23#1894—28no—
24#1878—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1441—22no—
33#1439—22no—
34#1437—22no—
35#1436—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C23×Q8C2312yes—
39C24×C4C2311yescharacteristic
40C22×Q16—24no—
41C22×Q16—24no—
42C22×Q16—41no—
43C22×Q16—41no—
44C22×Q16C2×C414yes—
45C22×M16—22no—
46C22×M16—24no—
47C22×M16—42no—
48C22×M16—41no—
49C22×M16—41no—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8C2×C418yes—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8—42no—
64C2×C4×Q8—42no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C42—22no—
74C22×C42C2312yes—
75C2×#107—42no—
76C2×#107—41no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103C2×C418yes—
85C2×#103D418yes—
86C2×C22⋊C8—41no—
87C2×C22⋊C8—41no—
88C2×C22⋊C8C2×C414yes—
89#391—24no—
90#391—42no—
91#380—24no—
92#380—42no—
93C4×C22⋊C4—41no—
94C4×C22⋊C4—41no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104#340—24no—
105#340—24no—
106#340—24no—
107#340—41no—
108#340C2×C414yes—
109#336—24no—
110#336—42no—
111C25C2×D411yescharacteristic
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8C22×C414yes—
124C22×Q8C2×D414yes—
125C22×Q8#4114yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4C22×C412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4#4112yes—
143C23×C4C2411yescharacteristic
144C2×Q16—216no—
145C2×Q16—44no—
146C2×Q16—44no—
147C2×Q16—48no—
148C2×Q16—44no—
149C2×M16—44no—
150C2×M16—48no—
151C2×M16—48no—
152C2×M16—44no—
153C2×M16—44no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8C22×C412yes—
169C22×C8C22×C412yes—
170C22×C8C2×D414yes—
171C22⋊Q8—216no—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174C22⋊Q8—48no—
175C4×Q8—48no—
176C4×Q8—48no—
177C4×Q8—48no—
178C4×Q8—48no—
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—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C2×D414yes—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—82no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42—44no—
221#107—44no—
222#103—48no—
223#103—48no—
224#103—48no—
225C22⋊C8—44no—
226#95—41no—
227#95—41no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—82no—
234#95—82no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24C4×D412yes—
246C24C22×D411yescharacteristic
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8C4×D418yes—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
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—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4C4×D412yes—
339C22×C4#13714yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C2×#4112yes—
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C42−1+411yescharacteristic
350Q16—416no—
351M16—416no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—416no—
374C4⋊C4—416no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C22⋊C4—416no—
384C22⋊C4—48no—
385C22⋊C4—88no—
386C42—416no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C23—22no—
392C23—24no—
393C23—24no—
394C23—22no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23C2×C4×D411yescharacteristic
415C23#53411yescharacteristic
416C23#53711yescharacteristic
417C23D4212yes—
418C23#54612yes—
419C23C2×#13712yes—
420C23#57512yes—
421C23#57611yescharacteristic
422C23#57811yescharacteristic
423C23#57811yescharacteristic
424C23#57811yescharacteristic
425Q8—28no—
426Q8—28no—
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—44no—
432Q8—44no—
433C2×C4—22no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4C2×C4×D412yes—
487C2×C4#51812yes—
488C2×C4#51811yescharacteristic
489C2×C4#52011yescharacteristic
490C2×C4#53412yes—
491C2×C4#53712yes—
492C2×C4#53812yes—
493C2×C4#53811yescharacteristic
494C2×C4#53912yes—
495C2×C4#53911yescharacteristic
496C2×C4D4211yescharacteristic
497C2×C4#54612yes—
498C2×C4#54612yes—
499C2×C4#55111yescharacteristic
500C2×C4#55112yes—
501C2×C4#56011yescharacteristic
502C2×C4#56012yes—
503C2×C4#56011yescharacteristic
504C8—42no—
505C8—42no—
506C8—44no—
507C22—44no—
508C22—44no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—82no—
517C22—82no—
518C22#232511yescharacteristic
519C22#235812yes—
520C22#266812yes—
521C22#267511yescharacteristic
522C22#269311yescharacteristic
523C22C4×#13712yes—
524C22#295011yescharacteristic
525C22#295111yescharacteristic
526C22#319612yes—
527C22#320211yescharacteristic
528C22#320311yescharacteristic
529C22#325511yescharacteristic
530C22#325611yescharacteristic
531C22#326111yescharacteristic
532C22#326211yescharacteristic
533C22#326812yes—
534C22#327012yes—
535C22#327612yes—
536C22#327712yes—
537C22#331212yes—
538C22#331312yes—
539C22#332712yes—
540C22#333312yes—
541C4—44no—
542C4—42no—
543C4—48no—
544C4—84no—
545C4—82no—
546C4—84no—
547C4—162no—
548C4#232512yes—
549C4#235811yescharacteristic
550C4#238011yescharacteristic
551C4#266811yescharacteristic
552C4#269312yes—
553C4#276011yescharacteristic
554C2—82no—
555C2#1457211yescharacteristic
556C2#2218911yescharacteristic
557C2#2219711yescharacteristic
558C2#2286912yes—
559C2#2287912yes—
560C2#3042612yes—
561C2#3042712yes—
562C2#3086311yescharacteristic
563C2#3086411yescharacteristic
564C2#3156511yescharacteristic
565C2#3156811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1878C2semidirect
5. #11315C2semidirect
6. #11124C2semidirect
8. #10789C2semidirect
22. #2001C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#18942
C2×#18782
#139692
#113152
#111242
#110962
#107892
#107152