Finite Group Databasebeta

← #369223

Subgroups — #369223

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1983C211yesmaximal, characteristic
5#14153C211yesmaximal, characteristic
6#11067C212yesmaximal, complemented
7#11066C212yesmaximal
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10835C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C2×#489—24no—
15C2×#489—24no—
16C2×#489C418yescomplemented
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×#97C2211yescharacteristic
25#2369—24no—
26#2369—24no—
27C2×#391—22no—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#389—21no—
31C2×#389C2212yes—
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1988—24no—
38#1983—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1439—21no—
46#1439—21no—
47#1437—22no—
48C23×Q8—21no—
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8D414yes—
60C2×#124—22no—
61C2×#124—22no—
62C2×#124—22no—
63C2×#124—22no—
64C2×#124—22no—
65C2×#124C2×C414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—21no—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83#489—232no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103C2×C418yes—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102C2×C418yes—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97C2×C414yes—
104#391—216no—
105#391—24no—
106#391—42no—
107#390—24no—
108#390—42no—
109#389—28no—
110#389—24no—
111#389—24no—
112C4×C4⋊C4—28no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D414yes—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95D412yes—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#336—42no—
133#336—42no—
134C25—21no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—24no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—81no—
148C22×Q8—81no—
149C22×Q8—81no—
150C22×Q8—81no—
151C22×Q8C22×C412yes—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4C22×C412yes—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C22×C414yes—
189C4⋊Q8—28no—
190C4⋊Q8—28no—
191C4⋊Q8—28no—
192C4⋊Q8C22⋊C4116yescomplemented
193#124—28no—
194#124—28no—
195#124—44no—
196#124—44no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
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—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C22×C414yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4#4114yes—
226C2×C4⋊C4#4114yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—28no—
247C2×C42—44no—
248C2×C42—42no—
249C2×C42—81no—
250C2×C42—81no—
251C2×C42—82no—
252C2×C42—82no—
253C2×C42C22×C412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C4⋊C8—44no—
260#103—216no—
261#103—44no—
262#103—44no—
263#102—216no—
264#102—44no—
265#102—44no—
266#97—28no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—44no—
272#95—44no—
273#95—42no—
274C24—22no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24C22×D411yescharacteristic
286C2×Q8—216no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4C4×D412yes—
378C22×C4C4×D412yes—
379C22×C4#12012yes—
380C22×C4#12012yes—
381C22×C4#12112yes—
382C22×C4#12112yes—
383C22×C4#13714yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)12yes—
393C22×C42−1+412yes—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—216no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—88no—
427C4⋊C4C2×C22⋊C418yes—
428C4⋊C4#12414yes—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C42—48no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—84no—
439C42—88no—
440C42C2×C22⋊C414yes—
441C42C4×D414yes—
442C42#12114yes—
443C42#12314yes—
444C23—44no—
445C23—44no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—82no—
461C23—81no—
462C23—81no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23C2×C4×D411yescharacteristic
468C23C2×#12011yescharacteristic
469C23C2×#12111yescharacteristic
470C23#54612yes—
471C23#54812yes—
472C23C2×#13712yes—
473C23#57512yes—
474C23#57611yescharacteristic
475C23#57611yescharacteristic
476C23#57711yescharacteristic
477C23#57711yescharacteristic
478Q8—42no—
479Q8—44no—
480Q8—42no—
481Q8—816no—
482Q8—82no—
483Q8—81no—
484Q8—81no—
485D4—42no—
486D4—44no—
487D4—42no—
488D4—82no—
489D4—81no—
490D4—81no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#38614yes—
545C2×C4#38614yes—
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C4×#4112yes—
549C2×C4#52012yes—
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4C2×#12411yescharacteristic
553C2×C4#52912yes—
554C2×C4#53211yescharacteristic
555C2×C4#53511yescharacteristic
556C2×C4#53611yescharacteristic
557C2×C4#54212yes—
558C2×C4#54812yes—
559C2×C4#54812yes—
560C2×C4#55012yes—
561C2×C4#55012yes—
562C2×C4#55612yes—
563C2×C4#55912yes—
564C8—44no—
565C8—81no—
566C8—81no—
567C22—82no—
568C22—84no—
569C22—84no—
570C22—82no—
571C22—82no—
572C22—161no—
573C22—161no—
574C22—162no—
575C22—162no—
576C22—161no—
577C22—161no—
578C22C2×#38611yescharacteristic
579C22#236712yes—
580C22#242612yes—
581C22#243111yescharacteristic
582C22#243611yescharacteristic
583C22C4×#13712yes—
584C22#294611yescharacteristic
585C22#294611yescharacteristic
586C22#300411yescharacteristic
587C22#300511yescharacteristic
588C22#301211yescharacteristic
589C22#301211yescharacteristic
590C22#303712yes—
591C22#304111yescharacteristic
592C22#304211yescharacteristic
593C22#331412yes—
594C22#331612yes—
595C22#331912yes—
596C22#332112yes—
597C22#336912yes—
598C22#337012yes—
599C22#337512yes—
600C22#337612yes—
601C4—24no—
602C4—24no—
603C4—44no—
604C4—48no—
605C4—44no—
606C4—84no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4C4×#12411yescharacteristic
611C4#232611yescharacteristic
612C4#234112yes—
613C4#244812yes—
614C4#254511yescharacteristic
615C4#254811yescharacteristic
616C2—162no—
617C2#1402611yescharacteristic
618C2#2092611yescharacteristic
619C2#2092911yescharacteristic
620C2#2323012yes—
621C2#2323212yes—
622C2#2528512yes—
623C2#2528712yes—
624C2#2557811yescharacteristic
625C2#2557911yescharacteristic
626C2#2594011yescharacteristic
627C2#2594211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
3. C2×#1988C2semidirect
6. #11067C2semidirect
8. #10932C2semidirect
10. #10835C2semidirect
11. #10703C2semidirect
16. C2×#489C4semidirect
192. C4⋊Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C2×#23692
C2×#19882
C2×#19832
#141532
#110672
#110662
#109322
#109312
#108352
#107032