Finite Group Databasebeta

← #391304

Subgroups — #391304

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2023C211yesmaximal, characteristic
2#14714C211yesmaximal, characteristic, complemented
3#14700C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic, complemented
5#11324C211yesmaximal, characteristic, complemented
6#11292C212yesmaximal, complemented
7#11256C212yesmaximal, complemented
8#11189C212yesmaximal
9#11123C212yesmaximal
10#11049C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C2×#391—21no—
17C2×#391—21no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388—21no—
23C2×#388—21no—
24C2×#384C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#2023C418yescomplemented
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1437—21no—
37#1437—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C23×Q8—21no—
45C23×D4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—42no—
50C2×C4⋊Q8—42no—
51C2×C4⋊Q8C2×C414yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#107—42no—
75C2×#107—42no—
76C2×#107—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102C2×C418yes—
83#391—44no—
84#391—44no—
85#390—22no—
86#390—42no—
87#390—42no—
88#390C2×C414yes—
89#389—42no—
90#388—44no—
91#388—44no—
92#384—42no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C43—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340C2×C418yes—
108#336—22no—
109#336—42no—
110#336—42no—
111#336C2×C414yes—
112C25—21no—
113C22×Q8—24no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4C22×C412yes—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8C22×C414yes—
156C4⋊Q8—48no—
157C4⋊Q8—48no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C22×C414yes—
193C2×C4⋊C4C22×C412yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42D814yes—
234C2×C42C22×C412yes—
235C2×C42#4112yes—
236#107—48no—
237#102—48no—
238#102—48no—
239#95—42no—
240#95—42no—
241#95—44no—
242#95—42no—
243#95—42no—
244#95—44no—
245#95—44no—
246#95—42no—
247#95—44no—
248C24—22no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24C22×D411yescharacteristic
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4C4×D412yes—
355C22×C4C4×D412yes—
356C22×C4C2×D812yes—
357C22×C4C2×D812yes—
358C22×C4C2×D812yes—
359C22×C4#13512yes—
360C22×C4AGL(1,ℤ/8)12yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4#13712yes—
363C22×C4#13712yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4H(5,𝔽2)12yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)12yes—
370C22×C42−1+411yescharacteristic
371C22×C42−1+411yescharacteristic
372C22×C42−1+411yescharacteristic
373C22×C42−1+412yes—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—48no—
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—84no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—24no—
413C42—24no—
414C42—44no—
415C42—44no—
416C42—44no—
417C42—44no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—81no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23C2×C4×D411yescharacteristic
458C23#53411yescharacteristic
459C23#53511yescharacteristic
460C23#53511yescharacteristic
461C23#53611yescharacteristic
462C23#53611yescharacteristic
463C23#53711yescharacteristic
464C23C22×D811yescharacteristic
465C23C2×#13511yescharacteristic
466C23C2×AGL(1,ℤ/8)11yescharacteristic
467C23C2×AGL(1,ℤ/8)11yescharacteristic
468C23C2×#13711yescharacteristic
469C23C2×#13711yescharacteristic
470C23#57511yescharacteristic
471C23#57511yescharacteristic
472Q8—816no—
473D4—44no—
474D4—44no—
475D4—82no—
476D4—82no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4C4×D814yes—
553C2×C4#44014yes—
554C2×C4#51812yes—
555C2×C4#51911yescharacteristic
556C2×C4#52012yes—
557C2×C4#52011yescharacteristic
558C2×C4#53812yes—
559C2×C4#54111yescharacteristic
560C2×C4#54212yes—
561C2×C4#54211yescharacteristic
562C2×C4H(3,𝔽4)12yes—
563C2×C4#56212yes—
564C2×C4#56212yes—
565C2×C4#56212yes—
566C2×C4#56212yes—
567C2×C4#56312yes—
568C2×C4#56312yes—
569C2×C4#56312yes—
570C8—44no—
571C8—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—161no—
580C22—161no—
581C22—161no—
582C22—161no—
583C22—161no—
584C22—161no—
585C22—161no—
586C22—161no—
587C22#232111yescharacteristic
588C22#232311yescharacteristic
589C22#232611yescharacteristic
590C22#281411yescharacteristic
591C22#281611yescharacteristic
592C22#281911yescharacteristic
593C22#282111yescharacteristic
594C22C2×C4×D811yescharacteristic
595C22C2×#44011yescharacteristic
596C22C4×AGL(1,ℤ/8)11yescharacteristic
597C22C4×#13711yescharacteristic
598C22#318611yescharacteristic
599C22#318811yescharacteristic
600C22#319311yescharacteristic
601C22#319611yescharacteristic
602C22#320511yescharacteristic
603C22#320511yescharacteristic
604C22#321011yescharacteristic
605C22#321111yescharacteristic
606C22#321311yescharacteristic
607C22#321311yescharacteristic
608C22#321611yescharacteristic
609C22#321811yescharacteristic
610C22#322611yescharacteristic
611C22#322611yescharacteristic
612C22#322811yescharacteristic
613C22#323311yescharacteristic
614C22#323511yescharacteristic
615C22#323811yescharacteristic
616C22#323911yescharacteristic
617C22#324011yescharacteristic
618C22#324511yescharacteristic
619C22#324811yescharacteristic
620C22#325211yescharacteristic
621C22#325311yescharacteristic
622C4—24no—
623C4—24no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—84no—
629C4—82no—
630C4—161no—
631C4—161no—
632C4—162no—
633C4#237812yes—
634C4#237912yes—
635C4#283112yes—
636C4#283212yes—
637C2—162no—
638C2#1462511yescharacteristic
639C2#2204011yescharacteristic
640C2#2204711yescharacteristic
641C2#2211011yescharacteristic
642C2#2211711yescharacteristic
643C2#2222011yescharacteristic
644C2#2222311yescharacteristic
645C2#3273811yescharacteristic
646C2#3273911yescharacteristic
647C2#3277011yescharacteristic
648C2#3277211yescharacteristic
649C2#3280611yescharacteristic
650C2#3281111yescharacteristic
651C2#3283411yescharacteristic
652C2#3284011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14714C2semidirect
4. #11384C2semidirect
5. #11324C2semidirect
6. #11292C2semidirect
7. #11256C2semidirect
11. #10795C2semidirect
26. #2023C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20232
#147142
#147002
#113842
#113242
#112922
#112562
#111892
#111232
#110492
#107952