Finite Group Databasebeta

← #196502

Subgroups — #196502

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1749C211yesmaximal, characteristic, complemented
4C2×#1530C212yesmaximal, complemented
5#9124C214yesmaximal, complemented
6#9119C214yesmaximal, complemented
7#8034C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525C412yes—
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C2×#432—21no—
13C2×#432—28no—
14C2×#432C2212yes—
15C2×#432C2212yes—
16C22×C4⋊C8—21no—
17C22×C22⋊C8—21no—
18C22×C22⋊C8—21no—
19C22×#97C2212yes—
20C22×#97C2212yes—
21#2857C411yescharacteristic
22#2857C411yescharacteristic
23#2856C411yescharacteristic
24#2856C411yescharacteristic
25#2854—21no—
26#2854—21no—
27#2850C2212yes—
28#2850C2212yes—
29#2847—21no—
30#2847—21no—
31C4×#121—21no—
32C4×#121—21no—
33C4×#121—21no—
34C4×#121—21no—
35#2297C412yes—
36#2285C2211yescharacteristic
37#1749—21no—
38#1749—21no—
39#1749C2212yes—
40#1749C2212yes—
41C2×#336—21no—
42#1530—28no—
43C2×#329C2218yescomplemented
44#1295C2218yescomplemented
45C22×#41C2×C412yes—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51#532—24no—
52#532—24no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#124C2312yes—
56#525—48no—
57C2×#123—22no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×#121C2×C412yes—
68C2×#120—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×#117C2×C412yes—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—28no—
89#432—28no—
90#432—24no—
91#432—28no—
92#432D418yescomplemented
93C2×C4⋊C8—28no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C22⋊C8—28no—
97C2×C22⋊C8—28no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×#97—22no—
103C2×#97—28no—
104C2×#97—24no—
105C2×#97—24no—
106C2×#97—22no—
107C2×#97C2314yes—
108C2×#97C2314yes—
109#391C2×C412yes—
110#391C2×C412yes—
111#390—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390C2×C412yes—
116#390C2×C412yes—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#384C2×C411yescharacteristic
132#384C2×C411yescharacteristic
133#382C2312yes—
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144#336—22no—
145#336—22no—
146#329—216no—
147C25C22⋊C411yescharacteristic
148C2×#41—28no—
149C2×#41—44no—
150C2×#41—44no—
151C22×Q8C22×C412yes—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C22×C412yes—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C22×C8—28no—
188C22×C8—28no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—44no—
192C22×C8—44no—
193C22×C8—44no—
194C22×C8—44no—
195C4⋊Q8—24no—
196#127—24no—
197#124—24no—
198#124—24no—
199#123—48no—
200C22⋊Q8—44no—
201C22⋊Q8—44no—
202#121—44no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—44no—
207#121—44no—
208#121—42no—
209#121—44no—
210#121—42no—
211#120—48no—
212C4×D4—48no—
213C4×D4—48no—
214C4×D4—42no—
215C4×D4—44no—
216C4×D4—42no—
217#117—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D414yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4114yes—
273C2×C42C2411yescharacteristic
274C4⋊C8—28no—
275C4⋊C8—28no—
276C22⋊C8—28no—
277C22⋊C8—28no—
278C22⋊C8—28no—
279C22⋊C8—28no—
280#97—28no—
281#97—24no—
282#97—24no—
283#97—24no—
284#97—44no—
285#97C2×D418yes—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—41no—
298C24—41no—
299C24#10014yes—
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C24C2×C22⋊C411yescharacteristic
303#41—816no—
304C2×Q8—24no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×D4—24no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
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—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—416no—
381C22×C4—416no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4#10414yes—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C4×D414yes—
398C22×C4#12012yes—
399C22×C4#12114yes—
400C22×C4#12312yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C2×C8—28no—
405C2×C8—28no—
406C2×C8—416no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—416no—
411C2×C8—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C42—22no—
435C42—22no—
436C42—21no—
437C42—21no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—44no—
449C42C4×D414yes—
450C42#12012yes—
451C42#12114yes—
452C42#12312yes—
453C42C22×D412yes—
454C42C2×#4112yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—28no—
469C23—21no—
470C23—28no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23#38612yes—
484C23C2×#10012yes—
485C23C2×#10012yes—
486C23C2×#10012yes—
487C23#41312yes—
488C23C2×#10412yes—
489C23#42112yes—
490C23C22×C22⋊C411yescharacteristic
491Q8—84no—
492D4—84no—
493D4—84no—
494D4—88no—
495D4—88no—
496D4—88no—
497D4—84no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—88no—
555C2×C4#38612yes—
556C2×C4#38612yes—
557C2×C4#38612yes—
558C2×C4C2×#10412yes—
559C2×C4C2×#10412yes—
560C2×C4#42112yes—
561C2×C4#42112yes—
562C2×C4C2×C4×D412yes—
563C2×C4C2×#12011yescharacteristic
564C2×C4C2×#12112yes—
565C2×C4C2×#12311yescharacteristic
566C8—48no—
567C8—48no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—416no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22#153014yescomplemented
596C22#190014yes—
597C22C2×#38611yescharacteristic
598C22C22×#10011yescharacteristic
599C22C2×#41311yescharacteristic
600C22C22×#10411yescharacteristic
601C22C2×#42111yescharacteristic
602C22C2×#42111yescharacteristic
603C22C2×#42111yescharacteristic
604C4—22no—
605C4—22no—
606C4—21no—
607C4—21no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4#187114yes—
620C2—24no—
621C2—24no—
622C2—42no—
623C2—162no—
624C2C2×#153011yescharacteristic
625C2#1765111yescharacteristic
626C2#1767511yescharacteristic
627C2#1767711yescharacteristic
628C2C2×#187111yescharacteristic
629C2#2097911yescharacteristic
630C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#432C2semidirect
3. C2×#1749C2semidirect
4. C2×#1530C2semidirect
5. #9124C2semidirect
6. #9119C2semidirect
7. #8034C2semidirect
43. C2×#329C22semidirect
44. #1295C22semidirect
92. #432D4semidirect
595. C22#1530semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#366242
C2×#17492
C2×#15302
#91242
#91192
#80342