Finite Group Databasebeta

← #381851

Subgroups — #381851

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2013C211yesmaximal, characteristic
2C2×#1877C212yesmaximal
3#14274C211yesmaximal, characteristic, complemented
4#13785C211yesmaximal, characteristic
5#11268C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11051C211yesmaximal, characteristic
8#11023C212yesmaximal
9#11010C211yesmaximal, characteristic, complemented
10#10932C212yesmaximal, complemented
11#10796C211yesmaximal, characteristic, complemented
12C23×SD16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#2013C418yescomplemented
27#1877—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1445—22no—
36#1445—22no—
37#1437—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C22⋊#95—22no—
41C23×Q8C2312yes—
42C23×D4C2311yescharacteristic
43C24×C4—21no—
44C22×SD16—24no—
45C22×SD16—24no—
46C22×SD16—42no—
47C22×SD16—42no—
48C22×SD16—42no—
49C22×SD16—42no—
50C22×SD16C2×C418yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2×C414yescomplemented
57C2×C4×D4—22no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C2×C4×D4D414yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103C2×C418yescomplemented
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85#391—24no—
86#391—24no—
87#391—42no—
88#391—42no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390C2×C414yes—
93#382—42no—
94C4×C4⋊C4—42no—
95#376—42no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—24no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104C2×#95D412yes—
105#340—24no—
106#340—42no—
107#340—42no—
108#336—24no—
109#336—22no—
110#336—42no—
111#336C2×C414yescomplemented
112C25—21no—
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×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8C22×C414yes—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22×C412yes—
135C22×D4C2×D412yes—
136C22×D4#4112yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C2×SD16—216no—
153C2×SD16—48no—
154C2×SD16—48no—
155C2×SD16—48no—
156C2×SD16—48no—
157C2×SD16—48no—
158C2×SD16—48no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C22×C414yes—
173C4⋊Q8—48no—
174C4⋊Q8—48no—
175C4×D4—48no—
176C4×D4—48no—
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—28no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4#4112yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—42no—
216C2×C42—82no—
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221#103—48no—
222#103—48no—
223#103—48no—
224#102—48no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95—44no—
229#95—48no—
230#95—44no—
231#95—44no—
232#95—82no—
233#95—82no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24C22×D411yescharacteristic
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
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×D4—22no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4C4×D414yes—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—48no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
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×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4C4×D414yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C4H(5,𝔽2)12yes—
357C22×C4H(5,𝔽2)12yes—
358SD16—432no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—416no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C22⋊C4—48no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—42no—
392C42—42no—
393C42—84no—
394C23—24no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—82no—
416C23—82no—
417C23C2×C4×D411yescharacteristic
418C23C2×#12111yescharacteristic
419C23#52511yescharacteristic
420C23#53411yescharacteristic
421C23#53511yescharacteristic
422C23#53511yescharacteristic
423C23#53511yescharacteristic
424C23#57712yes—
425C23#57712yes—
426C23#57712yes—
427C23#57812yes—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—44no—
432Q8—82no—
433Q8—82no—
434Q8—82no—
435Q8—82no—
436D4—28no—
437D4—44no—
438D4—44no—
439D4—44no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—816no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C4×#4111yescharacteristic
496C2×C4#51812yes—
497C2×C4#51812yes—
498C2×C4#52911yescharacteristic
499C2×C4#53211yescharacteristic
500C2×C4#53412yes—
501C2×C4#53512yes—
502C2×C4#53812yes—
503C2×C4#53812yes—
504C2×C4#53812yes—
505C2×C4#53812yes—
506C2×C4#53812yes—
507C2×C4#54012yes—
508C2×C4#54012yes—
509C2×C4#54012yes—
510C8—44no—
511C8—82no—
512C22—48no—
513C22—44no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—84no—
519C22—82no—
520C22—82no—
521C22—84no—
522C22—84no—
523C22C4×#12111yescharacteristic
524C22#232111yescharacteristic
525C22#232811yescharacteristic
526C22#261711yescharacteristic
527C22#261911yescharacteristic
528C22#263111yescharacteristic
529C22#263511yescharacteristic
530C22#294712yes—
531C22#294912yes—
532C22#304212yes—
533C22#304612yes—
534C22#312012yes—
535C22#312312yes—
536C22#320012yes—
537C22#320112yes—
538C22#322112yes—
539C22#322112yes—
540C22#322211yescharacteristic
541C22#322211yescharacteristic
542C22#322211yescharacteristic
543C22#322311yescharacteristic
544C22#322512yes—
545C22#322512yes—
546C4—42no—
547C4—42no—
548C4—44no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—84no—
554C4—162no—
555C4#232112yes—
556C4#233312yes—
557C4#264812yes—
558C4#266112yes—
559C2—84no—
560C2#1431111yescharacteristic
561C2#2151412yes—
562C2#2205812yes—
563C2#2227511yescharacteristic
564C2#2227811yescharacteristic
565C2#2972311yescharacteristic
566C2#2972611yescharacteristic
567C2#2980412yes—
568C2#3004911yescharacteristic
569C2#3005411yescharacteristic
570C2#3012012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14274C2semidirect
5. #11268C2semidirect
6. #11190C2semidirect
9. #11010C2semidirect
10. #10932C2semidirect
11. #10796C2semidirect
26. #2013C4semidirect
56. C2×C4⋊Q8C2×C4semidirect
81. C2×#103C2×C4semidirect
111. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20132
C2×#18772
#142742
#137852
#112682
#111902
#110512
#110232
#110102
#109322
#107962