Finite Group Databasebeta

← #368993

Subgroups — #368993

Order 512.

1284 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21325C211yesmaximal, characteristic
4C2×#1986C211yesmaximal, characteristic
5#14160C211yesmaximal, characteristic
6#11217C212yesmaximal
7#11217C212yesmaximal
8#10899C212yesmaximal
9#10899C212yesmaximal
10#10851C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#125C2211yescharacteristic
14#3166—24no—
15#3166C418yescomplemented
16C2×#501—21no—
17C2×#501—21no—
18C2×#499—22no—
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×C4×M16C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2370—22no—
24#2369—21no—
25#2365—21no—
26#2355—21no—
27#2355—22no—
28Q8×C4⋊C4—21no—
29#2329—22no—
30#2317C2212yes—
31C4×C22⋊Q8—22no—
32#2285C2211yescharacteristic
33#2274C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1992—21no—
37#1992—21no—
38#1991—22no—
39#1986—24no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48#1441—22no—
49#1436—22no—
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2211yescharacteristic
52C4⋊#95C2211yescharacteristic
53C4⋊#95C2211yescharacteristic
54C22×M16C2312yes—
55C22×M16C2312yes—
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533D414yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125—21no—
63C2×#125—21no—
64C2×#125—22no—
65C2×#125—21no—
66C2×#125—21no—
67C2×#125—22no—
68C2×#125—22no—
69C2×#125C2×C414yes—
70C2×#125D412yes—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×#117C2312yes—
77C2×#117C2312yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C42D412yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#501—28no—
90#501—28no—
91#499—216no—
92#428—24no—
93#428—24no—
94#428C2×C418yes—
95#428C2×C418yes—
96C2×#107—22no—
97C2×#107—22no—
98C2×#106—22no—
99C2×#106—22no—
100C4×M16—22no—
101C4×M16—22no—
102C4×M16—22no—
103C4×M16C2×C414yes—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#391—24no—
109#391—22no—
110#389—24no—
111#389—22no—
112#389—22no—
113#389—22no—
114#387—24no—
115#387—24no—
116#387—24no—
117#385—24no—
118#384—22no—
119#384D411yescharacteristic
120#384D411yescharacteristic
121#382—22no—
122#382D412yes—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C22⋊C4—24no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95D412yes—
146#343—24no—
147#343Q818yescomplemented
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#339—24no—
153#339—24no—
154#339—24no—
155#339—24no—
156C22×Q8—21no—
157C22×Q8—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16C22×C414yes—
176C2×M16C2×Q814yes—
177C2×M16C2×Q814yes—
178C2×M16#4114yes—
179C22×C8—22no—
180C22×C8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8C22⋊C414yes—
184C4⋊Q8C22⋊C414yes—
185#125—24no—
186#125—24no—
187#125—24no—
188#125—24no—
189#125—24no—
190#125—42no—
191#125—42no—
192#125—42no—
193#125—42no—
194#125C22⋊C418yes—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—84no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117C22×C414yes—
211#117C2×D414yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×Q814yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—82no—
274C2×C42C22×C412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42#4112yes—
281C2×C42#4112yes—
282#107—28no—
283#107—28no—
284#106—28no—
285#106—28no—
286#97—24no—
287#97—42no—
288C4×C8—24no—
289C4×C8—42no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—44no—
299#95—44no—
300#95—42no—
301#95—82no—
302#95—82no—
303C24C22×D411yescharacteristic
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—81no—
309C2×Q8—81no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12011yescharacteristic
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4C4⋊Q814yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q812yes—
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4C2×#4111yescharacteristic
368C22×C42−1+412yes—
369C22×C42−1+412yes—
370M16—24no—
371M16—24no—
372M16—24no—
373M16—24no—
374M16—24no—
375M16—42no—
376M16—42no—
377M16C4×Q818yescomplemented
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—48no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4C2×C22⋊C414yes—
418C4⋊C4C2×C22⋊C414yes—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C42—24no—
424C42—24no—
425C42—44no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—82no—
433C42—84no—
434C42—84no—
435C42C2×C22⋊C412yes—
436C42C2×C22⋊C412yes—
437C42C4×D412yes—
438C42C4×D412yes—
439C42#12112yes—
440C42#12112yes—
441C42#12312yes—
442C42#12312yes—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23C2×C4×D411yescharacteristic
448C23C2×#12011yescharacteristic
449C23C2×#12111yescharacteristic
450C23C2×C4⋊Q811yescharacteristic
451C23#53211yescharacteristic
452C23#53611yescharacteristic
453C23#53611yescharacteristic
454Q8—48no—
455Q8—84no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4#38612yes—
515C2×C4#38612yes—
516C2×C4#38612yes—
517C2×C4#38612yes—
518C2×C4C22×C22⋊C411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C4×Q812yes—
521C2×C4#51912yes—
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12311yescharacteristic
524C2×C4#53312yes—
525C2×C4#54112yes—
526C2×C4#54712yes—
527C2×C4#54812yes—
528C2×C4#54812yes—
529C2×C4D4×Q812yes—
530C2×C4D4×Q812yes—
531C2×C4D4×Q812yes—
532C2×C4#55612yes—
533C2×C4#55712yes—
534C8—28no—
535C8—44no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22C2×#38611yescharacteristic
549C22C4×C4⋊Q811yescharacteristic
550C22#232711yescharacteristic
551C22#243211yescharacteristic
552C22#243411yescharacteristic
553C22#254511yescharacteristic
554C22#254611yescharacteristic
555C22#301811yescharacteristic
556C22#301811yescharacteristic
557C22#301911yescharacteristic
558C22#302011yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4Q8×C22⋊C412yes—
574C4#236512yes—
575C4#242912yes—
576C4#244612yes—
577C2—21no—
578C2—21no—
579C2—21no—
580C2—21no—
581C2#1402411yescharacteristic
582C2#2094211yescharacteristic
583C2#2094311yescharacteristic
584C2#2569711yescharacteristic
585C2#2569811yescharacteristic
586C2#2582711yescharacteristic
587C2#2582811yescharacteristic
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
15. #3166C4semidirect
147. #343Q8semidirect
377. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
#342132
#213252
C2×#19862
#141602
#112172
#108992
#108512
C4×#3432