Finite Group Databasebeta

← #363387

Subgroups — #363387

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic, complemented
3C2×#1750C211yesmaximal, characteristic, complemented
4#11147C212yesmaximal, complemented
5#10964C212yesmaximal
6#10922C211yesmaximal, characteristic, complemented
7#10921C211yesmaximal, characteristic, complemented
8#10875C211yesmaximal, characteristic, complemented
9#10874C211yesmaximal, characteristic, complemented
10#10795C212yesmaximal, complemented
11#10713C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2293—24no—
22#2293—24no—
23C2×#390C412yes—
24C2×#389C412yes—
25C2×#388C412yes—
26C2×#388C412yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1818—24no—
32#1750—24no—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43C23×D4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—42no—
57C2×#117—42no—
58C2×#117—42no—
59C2×#117—42no—
60C2×#117D414yes—
61C2×#117D414yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C42C2×C412yes—
73C22×C42C2×C412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×#97—22no—
92C2×#97—42no—
93C2×C4×C8—22no—
94C2×C4×C8—42no—
95#390—28no—
96#389—28no—
97#388—28no—
98#388—28no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C43—22no—
109C43—22no—
110#336—24no—
111#336—24no—
112#336—24no—
113#336—22no—
114#336—22no—
115#336—24no—
116#336—24no—
117#336—24no—
118#336—42no—
119#336—42no—
120C25C22×C411yescharacteristic
121C25C22×C411yescharacteristic
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—42no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4#4114yes—
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
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—22no—
177C22×C8—22no—
178C22×C8#4114yes—
179C22×C8#4114yes—
180C4×D4—216no—
181C4×D4—216no—
182#117—44no—
183#117—44no—
184#117—48no—
185#117—44no—
186#117—44no—
187#117—48no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C22⋊C418yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4C22⋊C418yes—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42#4114yes—
244C22⋊C8—216no—
245C22⋊C8—48no—
246C22⋊C8—48no—
247C22⋊C8—48no—
248#97—44no—
249C4×C8—44no—
250#95—24no—
251#95—24no—
252#95—24no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24C2×C22⋊C412yes—
270C24#11712yes—
271C24C23×C411yescharacteristic
272C2×D4—28no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—44no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
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—42no—
349C22×C4C2×C22⋊C414yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C414yes—
352C22×C4#11712yes—
353C22×C4#11712yes—
354C22×C4#11712yes—
355C22×C4#13114yes—
356C22×C4#13114yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C4H(5,𝔽2)12yes—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
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—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—28no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C42—24no—
404C42—24no—
405C42—22no—
406C42—22no—
407C42—24no—
408C42—22no—
409C42—22no—
410C42—24no—
411C42—48no—
412C42—48no—
413C42—42no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—42no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—41no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23C22×C22⋊C411yescharacteristic
462C23C2×#11711yescharacteristic
463C23C2×#11711yescharacteristic
464C23C4×#4112yes—
465C23#51812yes—
466C23C2×#13112yes—
467C23C2×#13112yes—
468C23#56812yes—
469C23#56812yes—
470D4—416no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4#43314yes—
527C2×C4#52512yes—
528C2×C4#52512yes—
529C2×C4#52512yes—
530C2×C4#53211yescharacteristic
531C2×C4#53211yescharacteristic
532C2×C4#53411yescharacteristic
533C2×C4#53411yescharacteristic
534C2×C4#53412yes—
535C2×C4#55112yes—
536C2×C4#55112yes—
537C2×C4#55312yes—
538C2×C4#55312yes—
539C2×C4#55312yes—
540C2×C4#55312yes—
541C2×C4#55912yes—
542C2×C4#55912yes—
543C8—44no—
544C8—44no—
545C8—82no—
546C8—82no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22#228011yescharacteristic
566C22#229711yescharacteristic
567C22#230811yescharacteristic
568C22#235312yes—
569C22#235712yes—
570C22#288012yes—
571C22#288012yes—
572C22C2×#43311yescharacteristic
573C22#292111yescharacteristic
574C22#292211yescharacteristic
575C22#292311yescharacteristic
576C22#292412yes—
577C22#292512yes—
578C22#296612yes—
579C22#296712yes—
580C22#296812yes—
581C22#296912yes—
582C22#297912yes—
583C22#298012yes—
584C22#298112yes—
585C22#298112yes—
586C4—24no—
587C4—42no—
588C4—42no—
589C4—44no—
590C4—42no—
591C4—44no—
592C4—44no—
593C4—82no—
594C4#260012yes—
595C4#260012yes—
596C4#260212yes—
597C4#260212yes—
598C2—82no—
599C2—82no—
600C2#1394011yescharacteristic
601C2#2071611yescharacteristic
602C2#2072011yescharacteristic
603C2#2138112yes—
604C2#2174512yes—
605C2#2278912yes—
606C2#2279012yes—
607C2#2283612yes—
608C2#2283812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1818C2semidirect
3. C2×#1750C2semidirect
4. #11147C2semidirect
6. #10922C2semidirect
7. #10921C2semidirect
8. #10875C2semidirect
9. #10874C2semidirect
10. #10795C2semidirect
11. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
C2×#18182
C2×#17502
#111472
#109642
#109222
#109212
#108752
#108742
#107952
#107132