Finite Group Databasebeta

← #380113

Subgroups — #380113

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1895C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1729C211yesmaximal, characteristic, complemented
4#14264C211yesmaximal, characteristic, complemented
5#14257C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#11189C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#1895C418yescomplemented
25#1893—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33#1729—24no—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1444—21no—
37#1442—21no—
38#1439—21no—
39#1439—21no—
40#1439—22no—
41#1438—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×D4—21no—
45C23×D4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—42no—
50C22×M16—42no—
51C22×M16C2×C414yes—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—42no—
55C23×C8—42no—
56C23×C8D414yes—
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121C2×C414yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—42no—
69C2×#117—42no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42C2312yes—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102C2×C418yes—
93#390—42no—
94#390—42no—
95#389—42no—
96#389—42no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380—22no—
102#380C2×C414yes—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111#340—22no—
112#340—22no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340C2×C418yes—
118#336—24no—
119#336—24no—
120C25—21no—
121C25C2×D411yescharacteristic
122C22×D4—24no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—82no—
135C22×D4—82no—
136C22×D4C22×C412yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C2×M16—28no—
157C2×M16—48no—
158C2×M16—44no—
159C2×M16—44no—
160C2×M16—48no—
161C22×C8—28no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—44no—
177C22×C8—44no—
178C22×C8—48no—
179C22×C8—48no—
180C22×C8C22×C414yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183#121—28no—
184#121—28no—
185#121—44no—
186#121—44no—
187#117—44no—
188#117—44no—
189#117—48no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
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—21no—
201C2×C4⋊C4—22no—
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—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C414yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—81no—
239C2×C22⋊C4—81no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—81no—
242C2×C22⋊C4—81no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249#102—48no—
250#102—44no—
251#102—44no—
252#95—24no—
253#95—44no—
254#95—42no—
255#95—42no—
256#95—82no—
257#95—82no—
258#95—82no—
259#95—82no—
260C24—22no—
261C24—22no—
262C24—21no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—21no—
268C24—22no—
269C24—41no—
270C24—41no—
271C24—42no—
272C24—42no—
273C24—41no—
274C24—42no—
275C24—41no—
276C24—42no—
277C24—41no—
278C24—41no—
279C24C4×D412yes—
280C24#12112yes—
281C24#12412yes—
282C24C22×D411yescharacteristic
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—48no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—88no—
303C2×D4—88no—
304C2×D4—84no—
305C2×D4—84no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4#11714yes—
366C22×C4C4×D414yes—
367C22×C4C4×D412yes—
368C22×C4#12112yes—
369C22×C4#12314yes—
370C22×C4#12412yes—
371C22×C4#12614yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)12yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382M16—416no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—416no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—88no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—84no—
425C22⋊C4—88no—
426C22⋊C4—88no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—24no—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—81no—
461C23—81no—
462C23—82no—
463C23—81no—
464C23—81no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23#38814yes—
479C23C2×C4×D411yescharacteristic
480C23C2×#12111yescharacteristic
481C23C2×#12411yescharacteristic
482C23#53412yes—
483C23#53712yes—
484C23#57611yescharacteristic
485C23#57612yes—
486C23#57711yescharacteristic
487C23#57712yes—
488C23#57711yescharacteristic
489C23#57811yescharacteristic
490D4—44no—
491D4—42no—
492D4—42no—
493D4—816no—
494D4—81no—
495D4—81no—
496D4—82no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4#38814yes—
551C2×C4C2×#11711yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4#51812yes—
554C2×C4#52012yes—
555C2×C4C2×#12311yescharacteristic
556C2×C4C2×#12611yescharacteristic
557C2×C4#53411yescharacteristic
558C2×C4#53411yescharacteristic
559C2×C4#53511yescharacteristic
560C2×C4#53611yescharacteristic
561C2×C4#53812yes—
562C2×C4#53812yes—
563C2×C4#53812yes—
564C2×C4#53812yes—
565C2×C4#53912yes—
566C2×C4#53912yes—
567C2×C4#53912yes—
568C2×C4#53912yes—
569C8—44no—
570C8—42no—
571C8—42no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—84no—
587C22—84no—
588C22—161no—
589C22—161no—
590C22—161no—
591C22—161no—
592C22—162no—
593C22—162no—
594C22C2×#38811yescharacteristic
595C22#232512yes—
596C22#261711yescharacteristic
597C22#261811yescharacteristic
598C22#263912yes—
599C22#294612yes—
600C22#294611yescharacteristic
601C22#294711yescharacteristic
602C22#305711yescharacteristic
603C22#305811yescharacteristic
604C22#306011yescharacteristic
605C22#306111yescharacteristic
606C22#313612yes—
607C22#313611yescharacteristic
608C22#313711yescharacteristic
609C22#319912yes—
610C22#320012yes—
611C22#320312yes—
612C22#320412yes—
613C22#325712yes—
614C22#325712yes—
615C22#325712yes—
616C22#325812yes—
617C4—24no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—84no—
622C4—161no—
623C4—161no—
624C4—162no—
625C4#231812yes—
626C4#232111yescharacteristic
627C4#232211yescharacteristic
628C4#262211yescharacteristic
629C4#262311yescharacteristic
630C4#264312yes—
631C2—44no—
632C2—162no—
633C2#1425911yescharacteristic
634C2#2109111yescharacteristic
635C2#2109211yescharacteristic
636C2#2220112yes—
637C2#2220212yes—
638C2#2973811yescharacteristic
639C2#2974111yescharacteristic
640C2#2977611yescharacteristic
641C2#2978011yescharacteristic
642C2#3019112yes—
643C2#3019212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×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
3. C2×#1729C2semidirect
4. #14264C2semidirect
6. #11316C2semidirect
7. #11292C2semidirect
8. #11189C2semidirect
9. #11066C2semidirect
10. #10722C2semidirect
11. #10721C2semidirect
24. #1895C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18952
C2×#18932
C2×#17292
#142642
#142572
#113162
#112922
#111892
#110662
#107222
#107212