Finite Group Databasebeta

← #365687

Subgroups — #365687

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21323C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4#20765C211yesmaximal, characteristic
5#13909C211yesmaximal, characteristic
6#11276C212yesmaximal
7#11276C212yesmaximal
8#10892C212yesmaximal
9#10891C212yesmaximal
10#10851C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2370—22no—
16#2369—21no—
17#2365—21no—
18#2355—21no—
19#2355—22no—
20Q8×C4⋊C4—21no—
21#2329—22no—
22#2317C2212yes—
23C4×C22⋊Q8—22no—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28#2274C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1993—21no—
32#1993—21no—
33#1991—22no—
34#1986C418yescomplemented
35#1836—21no—
36#1836—21no—
37#1835—22no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48#1441—22no—
49#1436—22no—
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2211yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54#533—22no—
55#533—22no—
56#533C2×C414yes—
57#533Q814yes—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D411yescharacteristic
60C2×#125D412yes—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117C2312yes—
67C2×#117C2312yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C4×M16—22no—
79C4×M16—22no—
80C4×M16—22no—
81C4×M16C2×C414yes—
82C2×#97—21no—
83C2×#97—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86#391—24no—
87#391—22no—
88#389—24no—
89#389—22no—
90#389—22no—
91#389—22no—
92#387—24no—
93#387—24no—
94#387—24no—
95#385—24no—
96#384—22no—
97#384—24no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#382—22no—
101#382—24no—
102#382—21no—
103#382—21no—
104#382—21no—
105#382—21no—
106#382D412yes—
107#380—21no—
108#380—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C414yes—
123C4×C22⋊C4—24no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95D412yes—
131#343—24no—
132#343—24no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#339—24no—
142#339—24no—
143#339C2×C418yes—
144#339C2×C418yes—
145C22×Q8—21no—
146C22×Q8—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16C22×C414yes—
167C2×M16#4114yes—
168C22×C8—22no—
169C22×C8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8C4⋊C414yescomplemented
173C4⋊Q8C4⋊C414yescomplemented
174#125—24no—
175#125C4⋊C418yescomplemented
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—84no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117—22no—
190#117—22no—
191#117C22×C414yes—
192#117C2×Q814yes—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4114yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—82no—
244C2×C42C22×C412yes—
245C2×C42C22×C412yes—
246C2×C42C22×C412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×Q812yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254#97—24no—
255#97—42no—
256C4×C8—24no—
257C4×C8—42no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—44no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—44no—
273#95—82no—
274C24C22×D411yescharacteristic
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—81no—
280C2×Q8—81no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
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—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4C22⋊Q812yes—
329C22×C4C22⋊Q812yes—
330C22×C4#12414yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×Q811yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4112yes—
338C22×C4C2×#4111yescharacteristic
339C22×C42−1+412yes—
340C22×C42−1+412yes—
341M16—24no—
342M16—24no—
343M16—24no—
344M16—24no—
345M16—42no—
346M16—42no—
347M16—42no—
348M16—42no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—88no—
381C4⋊C4C2×C4⋊C414yes—
382C4⋊C4C2×C4⋊C414yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—44no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—82no—
397C42—84no—
398C42—84no—
399C42C2×C4⋊C412yes—
400C42C2×C4⋊C412yes—
401C42C4×D412yes—
402C42C4×D412yes—
403C42C22⋊Q812yes—
404C42C22⋊Q812yes—
405C42#12312yes—
406C42#12312yes—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23C2×C4×D411yescharacteristic
412C23C2×#12011yescharacteristic
413C23C2×C22⋊Q811yescharacteristic
414C23C2×#12411yescharacteristic
415C23#53211yescharacteristic
416C23#53611yescharacteristic
417C23#53611yescharacteristic
418Q8—48no—
419Q8—84no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4#38512yes—
479C2×C4#38512yes—
480C2×C4#38512yes—
481C2×C4#38512yes—
482C2×C4C22×C4⋊C411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C4×#4112yes—
485C2×C4#51912yes—
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4#52912yes—
489C2×C4#54112yes—
490C2×C4#54712yes—
491C2×C4#54812yes—
492C2×C4#54812yes—
493C2×C4#54812yes—
494C2×C4#55212yes—
495C2×C4#55612yes—
496C2×C4#55712yes—
497C2×C4#55712yes—
498C8—44no—
499C8—44no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22C2×#38511yescharacteristic
513C22C4×#12411yescharacteristic
514C22#232711yescharacteristic
515C22#243211yescharacteristic
516C22#243611yescharacteristic
517C22#256511yescharacteristic
518C22#256811yescharacteristic
519C22#301811yescharacteristic
520C22#301811yescharacteristic
521C22#301911yescharacteristic
522C22#302011yescharacteristic
523C4—22no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—44no—
528C4—44no—
529C4—44no—
530C4—44no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—161no—
536C4—161no—
537C4#235112yes—
538C4#236212yes—
539C4#242112yes—
540C4#245312yes—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—21no—
545C2#1397811yescharacteristic
546C2#2086311yescharacteristic
547C2#2086411yescharacteristic
548C2#2569711yescharacteristic
549C2#2569811yescharacteristic
550C2#2592111yescharacteristic
551C2#2592211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
34. #1986C4semidirect
172. C4⋊Q8C4⋊C4semidirect
173. C4⋊Q8C4⋊C4semidirect
175. #125C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#213232
C2×#19862
#207652
#139092
#112762
#108922
#108912
#108512
C4×#3432