Finite Group Databasebeta

← #106162

Subgroups — #106162

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2361C211yesmaximal, characteristic
2C2×#1519C212yesmaximal
3#10846C211yesmaximal, characteristic, complemented
4#10730C212yesmaximal, complemented
5#7907C211yesmaximal, characteristic, complemented
6C22×#121C2211yescharacteristic
7C22×C4×D4—21no—
8C22×C4×D4C411yescharacteristic, complemented
9C23×C4⋊C4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2361—24no—
12#2361—24no—
13C2×#390C411yescharacteristic, complemented
14C2×#389C411yescharacteristic, complemented
15C2×#388—21no—
16C2×#386—21no—
17C2×#385—21no—
18C2×#380C2212yes—
19C2×C4×C4⋊C4C411yescharacteristic, complemented
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×#342—21no—
22C2×#342—21no—
23C2×#340—24no—
24#1519—24no—
25#1519C418yescomplemented
26C2×#327—24no—
27#1280—21no—
28#1280—21no—
29C23×D4—21no—
30C23×D4C2×C411yescharacteristic
31C24×C4—21no—
32C24×C4—21no—
33C24×C4C2311yescharacteristic
34C23×C8—22no—
35C2×#121—22no—
36C2×#121—22no—
37C2×#121—21no—
38C2×#121—21no—
39C2×#121C2×C414yes—
40C2×#121D414yes—
41C2×#121D412yes—
42C2×#121Q812yes—
43C2×C4×D4—28no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—44no—
47C2×C4×D4—44no—
48C2×C4×D4—42no—
49C2×C4×D4—42no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—44no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4D414yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8C2×C414yescomplemented
75#390—22no—
76#390—22no—
77#389—22no—
78#389—22no—
79#388—28no—
80#386—28no—
81#385—28no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380C2×C414yescomplemented
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95D411yescharacteristic
94C2×#95D411yescharacteristic
95#342—42no—
96#342—42no—
97#340—48no—
98#327—216no—
99C25—21no—
100C25C2×D411yescharacteristic
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—42no—
109C22×D4—42no—
110C22×D4—41no—
111C22×D4—41no—
112C22×D4—42no—
113C22×D4—42no—
114C22×D4—41no—
115C22×D4—41no—
116C22×D4C4212yes—
117C22×D4C22⋊C412yes—
118C22×D4C4⋊C412yes—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4—42no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×Q811yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4#4111yescharacteristic
149C23×C4#4112yes—
150C22×C8—24no—
151C22×C8—44no—
152C22×C8—44no—
153C22×C8—42no—
154C22×C8—42no—
155#121—24no—
156#121—24no—
157#121—24no—
158#121—24no—
159#121—24no—
160#121—24no—
161#121C22⋊C418yes—
162#121C4⋊C418yes—
163C4×D4—416no—
164C4×D4—416no—
165C4×D4—48no—
166C4×D4—44no—
167C4×D4—44no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C4212yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C4⋊C412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4D812yes—
216C2×C22⋊C4SD1612yes—
217C2×C42—24no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C22⋊C8—24no—
227C22⋊C8—24no—
228C22⋊C8—42no—
229C22⋊C8—42no—
230#95—24no—
231#95—24no—
232#95—22no—
233#95—22no—
234#95—24no—
235#95—42no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24C2×C22⋊C411yescharacteristic
252C24C4×D412yes—
253C24#12012yes—
254C24#12111yescharacteristic
255C24C22⋊Q811yescharacteristic
256C24C2×D811yescharacteristic
257C24C2×SD1611yescharacteristic
258C24#13512yes—
259C24AGL(1,ℤ/8)11yescharacteristic
260C24AGL(1,ℤ/8)12yes—
261C24#13711yescharacteristic
262C2×D4—28no—
263C2×D4—28no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4#9514yes—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—44no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—88no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4#10212yes—
359C22×C4#10212yes—
360C22×C4C2×C4211yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C4⋊C411yescharacteristic
365C22×C4C2×C4⋊C412yes—
366C22×C4#11711yescharacteristic
367C22×C4#11711yescharacteristic
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4#12012yes—
372C22×C4#12112yes—
373C22×C4#12111yescharacteristic
374C22×C4C22⋊Q812yes—
375C22×C4C22⋊Q811yescharacteristic
376C22×C4#12312yes—
377C22×C4#12312yes—
378C2×C8—48no—
379C2×C8—44no—
380C2×C8—82no—
381C2×C8—82no—
382C2×C8—84no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—84no—
394C4⋊C4#9514yes—
395C4⋊C4#9914yes—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—84no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C42—42no—
410C42—42no—
411C42—48no—
412C42—42no—
413C42—42no—
414C23—21no—
415C23—22no—
416C23—21no—
417C23—22no—
418C23—21no—
419C23—22no—
420C23—21no—
421C23—22no—
422C23—24no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23C4×C22⋊C411yescharacteristic
464C23#37911yescharacteristic
465C23#38011yescharacteristic
466C23#38512yes—
467C23#38612yes—
468C23C2×#10211yescharacteristic
469C23#41611yescharacteristic
470C23#41711yescharacteristic
471C23#41811yescharacteristic
472C23C4×D812yes—
473C23C4×SD1612yes—
474C23#44012yes—
475C23#44212yes—
476C23#44712yes—
477C23#44912yes—
478C23#45012yes—
479C23#45212yes—
480C23#45911yescharacteristic
481C23#46111yescharacteristic
482C23#46312yes—
483C23#47411yescharacteristic
484C23#47611yescharacteristic
485C23#47812yes—
486D4—44no—
487D4—44no—
488D4—48no—
489D4—48no—
490D4—48no—
491D4—48no—
492D4—44no—
493D4—44no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—816no—
545C2×C4—88no—
546C2×C4—88no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—162no—
557C2×C4—162no—
558C2×C4—162no—
559C2×C4—162no—
560C2×C4C2×#9511yescharacteristic
561C2×C4C4×C22⋊C411yescharacteristic
562C2×C4#38011yescharacteristic
563C2×C4#38111yescharacteristic
564C2×C4#38512yes—
565C2×C4#38612yes—
566C2×C4C2×#9912yes—
567C2×C4#41012yes—
568C8—84no—
569C22—22no—
570C22—21no—
571C22—21no—
572C22—22no—
573C22—22no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—22no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—21no—
586C22—21no—
587C22—22no—
588C22—22no—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22C22⋊#9511yescharacteristic
615C22C4×#10211yescharacteristic
616C22#176211yescharacteristic
617C22#178111yescharacteristic
618C22#178211yescharacteristic
619C22#178611yescharacteristic
620C22#178811yescharacteristic
621C22#179511yescharacteristic
622C22#179611yescharacteristic
623C22#180211yescharacteristic
624C22#180411yescharacteristic
625C22#186212yes—
626C22#186412yes—
627C22#187712yes—
628C22#187912yes—
629C4—22no—
630C4—21no—
631C4—21no—
632C4—22no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4—164no—
641C4C4×#9912yes—
642C4#179912yes—
643C2—22no—
644C2—21no—
645C2—21no—
646C2—21no—
647C2—21no—
648C2—22no—
649C2—22no—
650C2—22no—
651C2—84no—
652C2#837511yescharacteristic
653C2#837611yescharacteristic
654C2#837711yescharacteristic
655C2#837811yescharacteristic
656C2#1076111yescharacteristic
657C2#1077511yescharacteristic
658C2#1078811yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10846C2semidirect
4. #10730C2semidirect
5. #7907C2semidirect
8. C22×C4×D4C4semidirect
13. C2×#390C4semidirect
14. C2×#389C4semidirect
19. C2×C4×C4⋊C4C4semidirect
25. #1519C4semidirect
74. C2×C22⋊C8C2×C4semidirect
85. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23612
C2×#15192
#108462
#107302
#79072