Finite Group Databasebeta

← #6477458

Subgroups — #6477458

Order 512.

3284 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#3195C211yesmaximal, characteristic, complemented
3C22×#483C211yesmaximal, characteristic, complemented
4#35084C211yesmaximal, characteristic
5#33008C212yesmaximal, complemented
6#32982C212yesmaximal, complemented
7#29641C212yesmaximal, complemented
8#29599C212yesmaximal, complemented
9#25561C218yesmaximal, complemented
10#24097C214yesmaximal, complemented
11#23970C214yesmaximal, complemented
12#21762C211yesmaximal, characteristic, complemented
13#21740C211yesmaximal, characteristic
14#20700C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—22no—
17C2×#554—22no—
18C2×#554—28no—
19C2×#554C2214yes—
20C2×#554C2214yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125—21no—
23C22×C22⋊Q8—21no—
24C22×C22⋊Q8—21no—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4—21no—
30C22×C4×D4C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3195—28no—
34C2×#483—28no—
35C2×#483C2212yes—
36C2×#483C2212yes—
37C2×#483C2212yes—
38C2×#483C2212yescomplemented
39C2×#483C2214yescomplemented
40C2×#470C2214yescomplemented
41C2×#451C2212yes—
42C2×#451C2212yescomplemented
43C22×#106C2212yescomplemented
44C22×#103C2212yescomplemented
45C2×#407C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2638—22no—
50#2583—21no—
51#2561—21no—
52#2486C2212yes—
53#2483—21no—
54#2415C2214yes—
55#2409C2212yes—
56#2405—21no—
57#2322—22no—
58#2280C2211yescharacteristic
59#2058C2218yescomplemented
60#2035C2218yescomplemented
61#2024C2218yescomplemented
62#2001C2218yescomplemented
63#1892C2214yescomplemented
64#1878C2218yescomplemented
65#1873C2214yescomplemented
66#1872C2218yescomplemented
67#1870C2214yes—
68#1869C2214yes—
69#1865C2218yescomplemented
70#1853C2218yescomplemented
71#1824C2214yescomplemented
72#1823C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1817C2212yes—
76#1787C2212yes—
77#1779C2212yes—
78#1736C2214yescomplemented
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4D411yescharacteristic
83C24×C4—21no—
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C22×Q16C2314yes—
88C22×M16C2312yes—
89C23×C8C2312yes—
90#554—232no—
91#554—232no—
92#554—216no—
93#554—216no—
94#554—216no—
95#554—216no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8C2314yes—
100C2×#125—216no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#123—22no—
104C2×C22⋊Q8—216no—
105C2×C22⋊Q8—216no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8D418yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2314yes—
124C2×C22⋊Q8C2312yes—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2314yes—
127C2×C4×Q8—24no—
128C2×C4×Q8—24no—
129C2×C4×Q8—22no—
130C2×C4×Q8C2314yes—
131C2×C4×D4—216no—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—24no—
136C2×C4×D4D412yes—
137C2×C4×D4D412yes—
138C2×C4×D4C2312yes—
139C2×C4×D4C2312yes—
140C2×#117C2312yes—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—22no—
144C22×C4⋊C4—24no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—24no—
152C22×C4⋊C4—24no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4D411yescharacteristic
155C22×C4⋊C4D418yes—
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2314yes—
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2312yes—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4D411yescharacteristic
167C22×C22⋊C4D411yescharacteristic
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170#483—28no—
171#483—28no—
172#483—28no—
173#483—216no—
174#483—28no—
175#470—216no—
176#451—28no—
177#451—28no—
178C2×#106—28no—
179C2×#106C2314yes—
180C2×#106C2314yes—
181C2×#106C2314yes—
182C2×#103—28no—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2318yes—
188#407—24no—
189C2×C22⋊C8—24no—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2318yes—
195#398—22no—
196#397—22no—
197#397—24no—
198#395—22no—
199#394—24no—
200#393—22no—
201#391C2314yes—
202#389—24no—
203#389C2314yes—
204#388—24no—
205#387—24no—
206#386—22no—
207#385—24no—
208#385—22no—
209#385—22no—
210#385—22no—
211#381—22no—
212#381—21no—
213#381—22no—
214#380C2312yes—
215#380C2312yes—
216#380C2314yes—
217#379—22no—
218#379—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C25C2×D411yescharacteristic
227C22×Q8—28no—
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8—84no—
237C22×Q8C2×D414yes—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×D4—24no—
241C22×D4—28no—
242C22×D4C2×D412yes—
243C22×D4C2×D412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—28no—
249C23×C4—28no—
250C23×C4—24no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—84no—
260C23×C4—84no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D414yes—
269C23×C4C2×D414yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C2×Q16—28no—
275C2×Q16—48no—
276C2×M16—24no—
277C2×M16—44no—
278C22×C8—24no—
279C22×C8—44no—
280C22×C8C2414yes—
281C22×C8C2414yes—
282C4⋊Q8—216no—
283C4⋊Q8—28no—
284C4⋊Q8—28no—
285#125—232no—
286#125—232no—
287#123—48no—
288C22⋊Q8—216no—
289C22⋊Q8—232no—
290C22⋊Q8—232no—
291C22⋊Q8—232no—
292C22⋊Q8—232no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—28no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C22⋊Q8—48no—
306C22⋊Q8C2×D418yes—
307C22⋊Q8C2×D418yes—
308C4×Q8—216no—
309C4×Q8—28no—
310C4×Q8—28no—
311C4×D4—232no—
312C4×D4—232no—
313C4×D4—28no—
314C4×D4—28no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—24no—
319#117—44no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—216no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D418yes—
362C2×C4⋊C4C2×D418yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—44no—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C42—28no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—24no—
391C2×C42—22no—
392C2×C42C2×D412yes—
393C2×C42C2412yes—
394#106—28no—
395#106—28no—
396#106—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—48no—
402C22⋊C8—24no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—48no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—24no—
413C24—22no—
414C24—24no—
415C24—24no—
416C24—24no—
417C24—24no—
418C24—42no—
419C24—42no—
420C24#12014yes—
421C24C22×D411yescharacteristic
422C24C22×D412yes—
423C24C22×D411yescharacteristic
424C24C22×D411yescharacteristic
425C2×Q8—216no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—48no—
433C2×Q8—44no—
434C2×Q8—48no—
435C2×Q8—48no—
436C2×Q8—42no—
437C2×Q8—82no—
438C2×Q8—88no—
439C2×Q8—84no—
440C2×Q8—816no—
441C2×Q8C22×D414yes—
442C2×D4—232no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4#13718yes—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—216no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—28no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—816no—
492C22×C4—84no—
493C22×C4—84no—
494C22×C4—88no—
495C22×C4—82no—
496C22×C4—816no—
497C22×C4#12014yes—
498C22×C4#12014yes—
499C22×C4#12014yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C2511yescharacteristic
513Q16—88no—
514M16—84no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—28no—
518C2×C8—28no—
519C2×C8—48no—
520C4⋊C4—216no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—216no—
524C4⋊C4—24no—
525C4⋊C4—28no—
526C4⋊C4—28no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—28no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—48no—
543C4⋊C4—84no—
544C4⋊C4C22×D414yes—
545C4⋊C4C22×D414yes—
546C4⋊C4C22×D414yes—
547C22⋊C4—216no—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—82no—
560C22⋊C4—84no—
561C42—216no—
562C42—24no—
563C42—22no—
564C42—22no—
565C42—82no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—28no—
571C23—22no—
572C23—22no—
573C23—24no—
574C23—28no—
575C23—28no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—22no—
580C23—44no—
581C23—416no—
582C23—416no—
583C23—44no—
584C23—44no—
585C23C2×#12012yes—
586C23C2×#12012yes—
587C23C2×#12012yes—
588C23C2×#13714yes—
589C23C2×#13714yes—
590C23#57714yes—
591C23C23×D411yescharacteristic
592Q8—28no—
593Q8—48no—
594Q8—44no—
595Q8—44no—
596Q8—44no—
597Q8—84no—
598Q8—816no—
599D4—216no—
600D4—216no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4—816no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4C2×#12012yes—
650C2×C4C2×#12012yes—
651C2×C4C2×#12012yes—
652C2×C4C2×#13714yes—
653C2×C4C23×D411yescharacteristic
654C2×C4C23×D411yescharacteristic
655C2×C4#58514yes—
656C8—44no—
657C8—44no—
658C22—22no—
659C22—22no—
660C22—28no—
661C22—28no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—28no—
666C22—22no—
667C22—24no—
668C22—22no—
669C22—416no—
670C22—416no—
671C22—42no—
672C22—44no—
673C22—44no—
674C22—42no—
675C22#300414yes—
676C22#300914yes—
677C22C22×#12011yescharacteristic
678C22#348512yes—
679C22C22×#13712yes—
680C22C2×#57712yes—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—24no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—82no—
692C4—84no—
693C4—82no—
694C4#344912yes—
695C4#348512yes—
696C2—24no—
697C2—24no—
698C2—42no—
699C2—44no—
700C2#3508811yescharacteristic
701C2C2×#300411yescharacteristic
702C2C2×#300911yescharacteristic
703C2#6173514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3195C2semidirect
3. C22×#483C2semidirect
5. #33008C2semidirect
6. #32982C2semidirect
7. #29641C2semidirect
8. #29599C2semidirect
9. #25561C2semidirect
10. #24097C2semidirect
11. #23970C2semidirect
12. #21762C2semidirect
14. #20700C2semidirect
38. C2×#483C22semidirect
39. C2×#483C22semidirect
40. C2×#470C22semidirect
42. C2×#451C22semidirect
43. C22×#106C22semidirect
44. C22×#103C22semidirect
59. #2058C22semidirect
60. #2035C22semidirect
61. #2024C22semidirect
62. #2001C22semidirect
63. #1892C22semidirect
64. #1878C22semidirect
65. #1873C22semidirect
66. #1872C22semidirect
69. #1865C22semidirect
70. #1853C22semidirect
71. #1824C22semidirect
78. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#31952
C22×#4832
#350842
#330082
#329822
#296412
#295992
#255612
#240972
#239702
#217622
#217402
#207002