Finite Group Databasebeta

← #416804

Subgroups — #416804

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1834C211yesmaximal, characteristic, complemented
2C2×#1750C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#14340C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#10843C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23#1834—24no—
24#1750—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1441—21no—
36#1441—21no—
37#1441—21no—
38#1441—21no—
39#1439—21no—
40#1439—21no—
41#1439—21no—
42#1439—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—22no—
79C2×#97—22no—
80C2×#97—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8D414yes—
84#384—22no—
85#384—22no—
86#384—22no—
87#384—22no—
88#382—22no—
89#382—22no—
90#382—22no—
91#382—22no—
92#380—24no—
93#380—24no—
94#380—42no—
95#380—42no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340—42no—
114#336—22no—
115#336—22no—
116C25C2×D411yescharacteristic
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8#4112yes—
168C22×C8#4112yes—
169#117—44no—
170#117—44no—
171#117—44no—
172#117—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
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—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
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—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C2×D412yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235C22⋊C8—28no—
236C22⋊C8—28no—
237#97—44no—
238C4×C8—28no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—44no—
245#95—82no—
246#95—82no—
247#95—82no—
248#95—82no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24#12412yes—
260C24C22×D411yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
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—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
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#12112yes—
338C22×C4#12112yes—
339C22×C4#12312yes—
340C22×C4#12312yes—
341C22×C4#12312yes—
342C22×C4#12312yes—
343C22×C4#12412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+411yescharacteristic
357C22×C42−1+411yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42#12314yes—
417C42#12414yes—
418C42#12614yes—
419C42#12614yes—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C2×#12111yescharacteristic
444C23C2×#12311yescharacteristic
445C23C2×#12311yescharacteristic
446C23C2×#12411yescharacteristic
447C23#53211yescharacteristic
448C23#53511yescharacteristic
449C23#53611yescharacteristic
450C23#57612yes—
451C23#57712yes—
452C23#57712yes—
453C23#57812yes—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
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—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4#39714yes—
516C2×C4#39714yes—
517C2×C4C2×#12311yescharacteristic
518C2×C4#52511yescharacteristic
519C2×C4#52511yescharacteristic
520C2×C4C2×#12411yescharacteristic
521C2×C4C2×#12611yescharacteristic
522C2×C4C2×#12611yescharacteristic
523C2×C4#52911yescharacteristic
524C2×C4#52911yescharacteristic
525C2×C4#53711yescharacteristic
526C2×C4#53711yescharacteristic
527C2×C4#54211yescharacteristic
528C2×C4#54211yescharacteristic
529C2×C4#55211yescharacteristic
530C2×C4#55211yescharacteristic
531C2×C4#55211yescharacteristic
532C2×C4#55211yescharacteristic
533C2×C4#55211yescharacteristic
534C2×C4#55211yescharacteristic
535C2×C4#55311yescharacteristic
536C2×C4#55311yescharacteristic
537C2×C4#55311yescharacteristic
538C2×C4#55311yescharacteristic
539C2×C4#55311yescharacteristic
540C2×C4#55311yescharacteristic
541C2×C4#55511yescharacteristic
542C2×C4#55511yescharacteristic
543C2×C4#55611yescharacteristic
544C2×C4#55611yescharacteristic
545C8—42no—
546C8—42no—
547C8—42no—
548C8—42no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—82no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—82no—
560C22C2×#39711yescharacteristic
561C22#249911yescharacteristic
562C22#250011yescharacteristic
563C22#250311yescharacteristic
564C22#250411yescharacteristic
565C22#262011yescharacteristic
566C22#285011yescharacteristic
567C22#305711yescharacteristic
568C22#305711yescharacteristic
569C22#305811yescharacteristic
570C22#305911yescharacteristic
571C22#309812yes—
572C22#309812yes—
573C22#309912yes—
574C22#309912yes—
575C22#313612yes—
576C22#313712yes—
577C22#317812yes—
578C22#317912yes—
579C22#322412yes—
580C22#322512yes—
581C22#324112yes—
582C22#324212yes—
583C4—24no—
584C4—24no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—162no—
589C4—162no—
590C4—162no—
591C4—162no—
592C4#248511yescharacteristic
593C4#248511yescharacteristic
594C4#249411yescharacteristic
595C4#249411yescharacteristic
596C4#249511yescharacteristic
597C4#249511yescharacteristic
598C4#256211yescharacteristic
599C4#256211yescharacteristic
600C2—82no—
601C2#1523311yescharacteristic
602C2#2363911yescharacteristic
603C2#2364011yescharacteristic
604C2#2788812yes—
605C2#2790412yes—
606C2#2797012yes—
607C2#2798412yes—
608C2#2984711yescharacteristic
609C2#2984811yescharacteristic
610C2#3304511yescharacteristic
611C2#3304611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1834C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect
9. #10843C2semidirect
10. #10843C2semidirect
11. #10842C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18342
C2×#17502
#143402
#113162
#113152
#112002
#111992
#108432
#108422