Finite Group Databasebeta

← #401241

Subgroups — #401241

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal, complemented
2C2×#2042C212yesmaximal
3C2×#1983C212yesmaximal
4C2×#1981C211yesmaximal, characteristic, complemented
5C2×#1926C211yesmaximal, characteristic
6#15243C211yesmaximal, characteristic
7#13818C211yesmaximal, characteristic, complemented
8#11322C211yesmaximal, characteristic, complemented
9#11076C211yesmaximal, characteristic
10#11010C211yesmaximal, characteristic, complemented
11#10932C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2066—28no—
30#2042—28no—
31#1983—28no—
32#1981—24no—
33#1926—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1439—21no—
39#1439—21no—
40#1439—22no—
41#1436—22no—
42C4⋊#95—21no—
43C4⋊#95C2211yescharacteristic
44C4×#95—21no—
45C4×#95C2211yescharacteristic
46C23×Q8C2312yes—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×#124—24no—
56C2×#124—24no—
57C2×#124—24no—
58C2×#124—24no—
59C2×#124—42no—
60C2×#124—42no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42—22no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#106—24no—
77C2×#106—22no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8—42no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—22no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102D414yes—
103#391—24no—
104#391—24no—
105#390—22no—
106#390—22no—
107#389—24no—
108#389D418yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—42no—
112C4×C4⋊C4D414yes—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122C2×#95D412yes—
123#340—24no—
124#340—22no—
125#336—22no—
126#336—24no—
127C25—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8#4114yes—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C2×D412yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
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—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8C2×D412yes—
179C4⋊Q8—28no—
180C4⋊Q8—48no—
181C4⋊Q8—44no—
182#124—216no—
183#124—48no—
184#124—48no—
185#124—48no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4#4112yes—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42—82no—
234C2×C42—82no—
235C2×C42C2×D414yes—
236C2×C42C2×D412yes—
237C2×C42#4114yes—
238C2×C42#4112yes—
239#106—44no—
240C4⋊C8—48no—
241C4⋊C8—44no—
242#103—48no—
243#103—48no—
244#103—48no—
245#103—48no—
246#102—28no—
247#102—48no—
248#102—44no—
249#95—24no—
250#95—28no—
251#95—22no—
252#95—22no—
253#95—42no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—81no—
259C24—22no—
260C24—41no—
261C24—42no—
262C24—42no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—42no—
267C24—42no—
268C24—41no—
269C24—41no—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—41no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
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—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
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—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4#12114yes—
365C22×C4#12112yes—
366C22×C4#12114yes—
367C22×C4#12112yes—
368C22×C4#13512yes—
369C22×C4#13712yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4112yes—
376C22×C4C2×#4111yescharacteristic
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C42−1+411yescharacteristic
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—88no—
412C4⋊C4#12114yes—
413C4⋊C4#12714yes—
414C22⋊C4—416no—
415C22⋊C4—416no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C42—22no—
419C42—24no—
420C42—22no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—84no—
426C42—88no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—42no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—82no—
436C23—81no—
437C23—82no—
438C23—81no—
439C23—82no—
440C23—82no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—81no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23C2×#12111yescharacteristic
451C23C2×#12112yes—
452C23D4211yescharacteristic
453C23#54612yes—
454C23#54811yescharacteristic
455C23C2×#13511yescharacteristic
456C23C2×#13711yescharacteristic
457C23#57512yes—
458C23#57711yescharacteristic
459C23#57712yes—
460C23#57811yescharacteristic
461Q8—44no—
462Q8—44no—
463Q8—44no—
464Q8—44no—
465Q8—82no—
466Q8—82no—
467Q8—82no—
468Q8—82no—
469D4—42no—
470D4—44no—
471D4—42no—
472D4—81no—
473D4—81no—
474D4—82no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4#46414yes—
528C2×C4C2×#12111yescharacteristic
529C2×C4#52512yes—
530C2×C4#52511yescharacteristic
531C2×C4#52912yes—
532C2×C4C2×#12711yescharacteristic
533C2×C4#53211yescharacteristic
534C2×C4#53212yes—
535C2×C4#53511yescharacteristic
536C2×C4#53711yescharacteristic
537C2×C4#53812yes—
538C2×C4#54212yes—
539C2×C4D4212yes—
540C2×C4#54612yes—
541C2×C4#54711yescharacteristic
542C2×C4#54812yes—
543C2×C4#55012yes—
544C2×C4#55112yes—
545C2×C4#55311yescharacteristic
546C8—42no—
547C8—81no—
548C8—82no—
549C22—84no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—84no—
554C22—161no—
555C22—161no—
556C22—161no—
557C22—161no—
558C22—162no—
559C22—162no—
560C22#241111yescharacteristic
561C22#241611yescharacteristic
562C22#242412yes—
563C22#242612yes—
564C22#285611yescharacteristic
565C22C2×#46411yescharacteristic
566C22#303512yes—
567C22#303712yes—
568C22#304011yescharacteristic
569C22#304211yescharacteristic
570C22#304212yes—
571C22#304612yes—
572C22#304611yescharacteristic
573C22#326712yes—
574C22#327611yescharacteristic
575C22#328411yescharacteristic
576C22#330112yes—
577C22#331612yes—
578C22#332112yes—
579C22#332712yes—
580C22#336411yescharacteristic
581C22#336911yescharacteristic
582C22#337612yes—
583C4—24no—
584C4—44no—
585C4—44no—
586C4—44no—
587C4—42no—
588C4—44no—
589C4—84no—
590C4—84no—
591C4—161no—
592C4#239211yescharacteristic
593C4#246112yes—
594C4#254011yescharacteristic
595C4#254111yescharacteristic
596C4#255412yes—
597C4#266211yescharacteristic
598C2—162no—
599C2#1481511yescharacteristic
600C2#2448311yescharacteristic
601C2#2449911yescharacteristic
602C2#2472911yescharacteristic
603C2#2474511yescharacteristic
604C2#2508512yes—
605C2#2510012yes—
606C2#2528212yes—
607C2#2528712yes—
608C2#3316511yescharacteristic
609C2#3316811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2066C2semidirect
4. C2×#1981C2semidirect
7. #13818C2semidirect
8. #11322C2semidirect
10. #11010C2semidirect
11. #10932C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#20422
C2×#19832
C2×#19812
C2×#19262
#152432
#138182
#113222
#110762
#110102
#109322