Finite Group Databasebeta

← #6625538

Subgroups — #6625538

Order 512.

3077 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q82C211yesmaximal, characteristic
2C2×#3387C212yesmaximal, complemented
3#36545C211yesmaximal, characteristic
4#32960C212yesmaximal
5#32954C211yesmaximal, characteristic
6#25847C214yesmaximal, complemented
7#25837C212yesmaximal, complemented
8#25816C212yesmaximal
9#25811C211yesmaximal, characteristic
10#24345C214yesmaximal, complemented
11#24333C212yesmaximal, complemented
12#24302C211yesmaximal, characteristic
13#24301C212yesmaximal
14#22525C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic
16#21455C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C2×Q82—216no—
19C2×Q82—21no—
20C2×Q82—21no—
21C2×Q82—22no—
22C2×Q82C2212yes—
23C2×Q82C2214yes—
24C2×Q82C2212yes—
25C2×#557C2211yescharacteristic
26C22×C4⋊Q8—21no—
27C22×C4⋊Q8—22no—
28C22×C4⋊Q8C2211yescharacteristic
29C22×C4⋊Q8C2212yes—
30C22×C4×Q8—21no—
31C22×C4×Q8C2211yescharacteristic
32C22×C4×Q8C2212yes—
33C22×C4×Q8C2211yescharacteristic
34#3387—216no—
35C2×#501C2214yes—
36C2×#498C2212yes—
37C2×#479C2214yes—
38C2×#477C2212yes—
39C2×#475C2214yes—
40C2×#475C2212yes—
41C2×#451C2211yescharacteristic
42C2×#446C2212yes—
43C2×#441C2214yes—
44C2×C4×Q16C2212yes—
45#2857C2211yescharacteristic
46#2851C2212yes—
47#2584C2212yes—
48#2582C2211yescharacteristic
49#2581C2212yes—
50#2546C2211yescharacteristic
51#2473—22no—
52#2462—22no—
53#2446—21no—
54#2434—21no—
55#2365—21no—
56#2360—21no—
57#2355C2211yescharacteristic
58C4×C22⋊Q8C2211yescharacteristic
59C4×C22⋊Q8C2212yes—
60#2058C2214yes—
61#2058C2214yes—
62#2058C2214yes—
63#2057C2214yes—
64#2057C2212yes—
65#2024C2214yes—
66#2024C2212yes—
67#1993C2214yes—
68#1990C2212yes—
69#1988C2214yes—
70#1987C2212yes—
71#1985C2214yes—
72#1984C2212yes—
73#1878C2214yes—
74#1878C2212yes—
75#1872C2211yescharacteristic
76#1872C2212yes—
77#1865C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1863C2214yes—
81#1863C2212yes—
82#1853C2212yes—
83#1853C2212yes—
84#1853C2212yes—
85#1852C2212yes—
86#1852C2211yescharacteristic
87#1805C2214yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90#1797C2212yes—
91#1787C2212yes—
92#1787C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95#1749C2212yes—
96C4×C22⋊C8C2211yescharacteristic
97C23×Q8—22no—
98C23×Q8—21no—
99C23×Q8D411yescharacteristic
100C23×Q8C2311yescharacteristic
101C22×Q16C2312yes—
102Q82—216no—
103Q82—232no—
104Q82—216no—
105Q82—216no—
106Q82—28no—
107Q82—28no—
108Q82—216no—
109Q82—28no—
110Q82D4116yes—
111#557—28no—
112C2×C4⋊Q8—216no—
113C2×C4⋊Q8—232no—
114C2×C4⋊Q8—216no—
115C2×C4⋊Q8—28no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—24no—
119C2×C4⋊Q8—24no—
120C2×C4⋊Q8—22no—
121C2×C4⋊Q8—24no—
122C2×C4⋊Q8—24no—
123C2×C4⋊Q8—22no—
124C2×C4⋊Q8—24no—
125C2×C4⋊Q8—22no—
126C2×C4⋊Q8—21no—
127C2×C4⋊Q8—22no—
128C2×C4⋊Q8—21no—
129C2×C4⋊Q8C2311yescharacteristic
130C2×C4⋊Q8C2314yes—
131C2×C4⋊Q8C2312yes—
132C2×C4⋊Q8C2314yes—
133C2×C4⋊Q8C2312yes—
134C2×#125—22no—
135C2×#125C2312yes—
136C2×C22⋊Q8—21no—
137C2×C22⋊Q8C2311yescharacteristic
138C2×C4×Q8—216no—
139C2×C4×Q8—24no—
140C2×C4×Q8—22no—
141C2×C4×Q8—22no—
142C2×C4×Q8—28no—
143C2×C4×Q8—216no—
144C2×C4×Q8—28no—
145C2×C4×Q8—21no—
146C2×C4×Q8—22no—
147C2×C4×Q8—21no—
148C2×C4×Q8—22no—
149C2×C4×Q8—21no—
150C2×C4×Q8D411yescharacteristic
151C2×C4×Q8D411yescharacteristic
152C2×C4×Q8D412yes—
153C2×C4×Q8C2312yes—
154C2×C4×Q8C2311yescharacteristic
155C2×C4×Q8C2314yes—
156C2×C4×Q8C2312yes—
157C2×C4×Q8C2312yes—
158C2×C4×Q8C2314yes—
159C2×C4×Q8C2312yes—
160C2×C4×Q8C2311yescharacteristic
161C2×C4×Q8C2311yescharacteristic
162C22×C4⋊C4—22no—
163C22×C4⋊C4—21no—
164C22×C4⋊C4—22no—
165C22×C4⋊C4—21no—
166C22×C4⋊C4D411yescharacteristic
167C22×C4⋊C4D412yes—
168C22×C4⋊C4C2312yes—
169C22×C4⋊C4C2311yescharacteristic
170C22×C42—22no—
171C22×C42—21no—
172C22×C42C2312yes—
173C22×C42C2311yescharacteristic
174#501—216no—
175#498—28no—
176#479—216no—
177#477—28no—
178#475—216no—
179#475—28no—
180#451Q818yescomplemented
181#446—28no—
182#441—216no—
183C4×Q16—28no—
184C2×#107C2314yes—
185C2×#107C2312yes—
186C2×#106C2314yes—
187C2×#106C2314yes—
188C2×#106C2314yes—
189C2×C4⋊C8C2312yes—
190C2×C4⋊C8C2314yes—
191C2×#103C2312yes—
192C2×#103C2314yes—
193C2×#103C2314yes—
194C2×#103C2312yes—
195C2×#103C2314yes—
196C2×#103C2312yes—
197C2×#103C2312yes—
198C2×C22⋊C8C2312yes—
199C2×C22⋊C8C2311yescharacteristic
200C2×C22⋊C8C2311yescharacteristic
201C2×#97C2314yes—
202C2×C4×C8C2312yes—
203#400—22no—
204#398—22no—
205#398—22no—
206#398—22no—
207#395—22no—
208#395—22no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391C2311yescharacteristic
214#391C2312yes—
215#389—21no—
216#389—22no—
217#389—22no—
218#389—21no—
219#389C2312yes—
220#389C2311yescharacteristic
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#387—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—21no—
231#385—22no—
232#385—22no—
233#385—22no—
234#385—21no—
235#384C2311yescharacteristic
236#382C2312yes—
237#380—21no—
238#380C2311yescharacteristic
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4—22no—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2312yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#336C2312yes—
255#336C2314yes—
256#336C2314yes—
257#336C2312yes—
258C22×Q8—24no—
259C22×Q8—24no—
260C22×Q8—24no—
261C22×Q8—24no—
262C22×Q8—216no—
263C22×Q8—28no—
264C22×Q8—24no—
265C22×Q8—21no—
266C22×Q8—24no—
267C22×Q8—22no—
268C22×Q8—24no—
269C22×Q8—22no—
270C22×Q8—21no—
271C22×Q8—22no—
272C22×Q8—84no—
273C22×Q8—82no—
274C22×Q8—84no—
275C22×Q8—82no—
276C22×Q8C2×D412yes—
277C22×Q8C2×D411yescharacteristic
278C22×Q8C2×D411yescharacteristic
279C22×Q8C2×Q814yes—
280C22×Q8C2411yescharacteristic
281C22×Q8C2412yes—
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—22no—
285C23×C4C2×D412yes—
286C23×C4C2×D412yes—
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C2×Q16—24no—
291C2×Q16—24no—
292C2×Q16C2×Q818yescomplemented
293C22×C8C2412yes—
294C22×C8C2414yes—
295C22×C8C2412yes—
296C4⋊Q8—232no—
297C4⋊Q8—232no—
298C4⋊Q8—232no—
299C4⋊Q8—232no—
300C4⋊Q8—216no—
301C4⋊Q8—216no—
302C4⋊Q8—28no—
303C4⋊Q8—216no—
304C4⋊Q8—28no—
305C4⋊Q8—216no—
306C4⋊Q8—232no—
307C4⋊Q8—28no—
308C4⋊Q8—28no—
309C4⋊Q8—28no—
310C4⋊Q8—24no—
311C4⋊Q8—28no—
312C4⋊Q8—24no—
313C4⋊Q8—44no—
314C4⋊Q8—44no—
315C4⋊Q8C2×D4116yes—
316C4⋊Q8C2×D418yes—
317#125—28no—
318#125—48no—
319C22⋊Q8—22no—
320C22⋊Q8—44no—
321C22⋊Q8C2×Q814yes—
322C4×Q8—232no—
323C4×Q8—216no—
324C4×Q8—216no—
325C4×Q8—28no—
326C4×Q8—216no—
327C4×Q8—28no—
328C4×Q8—216no—
329C4×Q8—24no—
330C4×Q8—24no—
331C4×Q8—28no—
332C4×Q8—24no—
333C4×Q8—28no—
334C4×Q8—24no—
335C4×Q8—28no—
336C4×Q8—24no—
337C4×Q8—28no—
338C4×Q8—24no—
339C4×Q8—28no—
340C4×Q8—24no—
341C4×Q8—28no—
342C4×Q8—44no—
343C4×Q8C2×D418yes—
344C4×Q8C2×D4116yes—
345C4×Q8C2×D418yes—
346C2×C4⋊C4—216no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—216no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—28no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4C2×D412yes—
378C2×C4⋊C4C2×D412yes—
379C2×C4⋊C4C2×D411yescharacteristic
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2411yescharacteristic
385C2×C4⋊C4C2412yes—
386C2×C4⋊C4C2411yescharacteristic
387C2×C4⋊C4C2412yes—
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2414yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—21no—
394C2×C42—216no—
395C2×C42—28no—
396C2×C42—24no—
397C2×C42—22no—
398C2×C42—24no—
399C2×C42—22no—
400C2×C42—24no—
401C2×C42—28no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42C2×D412yes—
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D412yes—
409C2×C42C2×D411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2412yes—
412C2×C42C2412yes—
413C2×C42C2411yescharacteristic
414#107—24no—
415#107—28no—
416#106—28no—
417#106—28no—
418#106—28no—
419C4⋊C8—24no—
420C4⋊C8—28no—
421#103—24no—
422#103—28no—
423#103—28no—
424#103—24no—
425#103—24no—
426#103—28no—
427#103C2×Q818yescomplemented
428C22⋊C8—24no—
429C22⋊C8—22no—
430C22⋊C8C2×Q814yes—
431#97—28no—
432C4×C8—24no—
433#95—21no—
434#95—22no—
435#95—21no—
436#95—22no—
437#95—21no—
438#95—22no—
439#95—22no—
440#95—21no—
441#95—22no—
442#95—21no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448C24C22×D411yescharacteristic
449C2×Q8—24no—
450C2×Q8—216no—
451C2×Q8—232no—
452C2×Q8—24no—
453C2×Q8—216no—
454C2×Q8—216no—
455C2×Q8—28no—
456C2×Q8—24no—
457C2×Q8—22no—
458C2×Q8—216no—
459C2×Q8—28no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—42no—
475C2×Q8—84no—
476C2×Q8—82no—
477C2×Q8—84no—
478C2×Q8—82no—
479C2×Q8—82no—
480C2×Q8—84no—
481C2×Q8—84no—
482C2×Q8—82no—
483C2×Q8—832no—
484C2×Q8—816no—
485C2×Q8#12014yes—
486C2×Q8#13714yes—
487C2×Q8C22×D414yes—
488C2×Q8C22×Q812yes—
489C2×Q8C22×Q814yes—
490C2×Q82−1+414yes—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—28no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—22no—
510C22×C4—24no—
511C22×C4—22no—
512C22×C4—21no—
513C22×C4—22no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—42no—
517C22×C4—41no—
518C22×C4—42no—
519C22×C4—44no—
520C22×C4—42no—
521C22×C4—44no—
522C22×C4#13714yes—
523C22×C4C22×D412yes—
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D412yes—
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×Q812yes—
533C22×C42−1+412yes—
534C22×C4C2511yescharacteristic
535Q16—216no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—24no—
540C2×C8—22no—
541C2×C8—24no—
542C2×C8—22no—
543C2×C8—24no—
544C2×C8C22×Q814yes—
545C4⋊C4—216no—
546C4⋊C4—28no—
547C4⋊C4—216no—
548C4⋊C4—28no—
549C4⋊C4—28no—
550C4⋊C4—24no—
551C4⋊C4—216no—
552C4⋊C4—28no—
553C4⋊C4—216no—
554C4⋊C4—28no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—24no—
559C4⋊C4—22no—
560C4⋊C4—24no—
561C4⋊C4—22no—
562C4⋊C4—24no—
563C4⋊C4—21no—
564C4⋊C4—24no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—24no—
568C4⋊C4—42no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—44no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—81no—
578C4⋊C4—81no—
579C4⋊C4—82no—
580C4⋊C4#12014yes—
581C4⋊C4#12018yes—
582C4⋊C4C22×D414yes—
583C4⋊C4C22×D418yes—
584C4⋊C4C22×Q812yes—
585C22⋊C4—24no—
586C22⋊C4—44no—
587C22⋊C4—42no—
588C42—216no—
589C42—28no—
590C42—216no—
591C42—28no—
592C42—24no—
593C42—22no—
594C42—24no—
595C42—22no—
596C42—22no—
597C42—24no—
598C42—44no—
599C42—42no—
600C42—42no—
601C42—82no—
602C42C22×D414yes—
603C42C22×D418yes—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23D4×Q812yes—
609C23C2×#13712yes—
610C23#57712yes—
611C23C23×D411yescharacteristic
612Q8—216no—
613Q8—216no—
614Q8—24no—
615Q8—24no—
616Q8—24no—
617Q8—28no—
618Q8—28no—
619Q8—24no—
620Q8—42no—
621Q8—816no—
622Q8—88no—
623Q8—88no—
624Q8—816no—
625Q8—84no—
626Q8D4×Q818yes—
627C2×C4—216no—
628C2×C4—216no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—24no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—21no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—28no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—28no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—21no—
657C2×C4—48no—
658C2×C4—44no—
659C2×C4—416no—
660C2×C4—48no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4C2×#12014yes—
682C2×C4C2×#12014yes—
683C2×C4C2×#12012yes—
684C2×C4C2×#12012yes—
685C2×C4D4×Q812yes—
686C2×C4C2×#13714yes—
687C2×C4C2×#13712yes—
688C2×C4C2×#13712yes—
689C2×C4C2×#13712yes—
690C2×C4C23×D412yes—
691C2×C4C23×Q811yescharacteristic
692C2×C4C2×2−1+412yes—
693C2×C4C2×2−1+411yescharacteristic
694C8—24no—
695C8—42no—
696C8—44no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22#338714yescomplemented
704C22C22×#12011yescharacteristic
705C22C2×D4×Q811yescharacteristic
706C22#348611yescharacteristic
707C22C22×#13711yescharacteristic
708C22C22×#13711yescharacteristic
709C22C2×#57711yescharacteristic
710C22#358511yescharacteristic
711C4—28no—
712C4—22no—
713C4—24no—
714C4—24no—
715C4—22no—
716C4—24no—
717C4—22no—
718C4—42no—
719C4—42no—
720C4—41no—
721C4—42no—
722C4—48no—
723C4—44no—
724C4—82no—
725C4—81no—
726C4—82no—
727C4—81no—
728C4#300414yes—
729C4C2×D4×Q812yes—
730C4#348612yes—
731C2—24no—
732C2Q8×#12011yescharacteristic
733C2C2×#300411yescharacteristic
734C2#6037311yescharacteristic
735C2C2×#338711yescharacteristic
736C2Q8×#13711yescharacteristic
737C2#6183011yescharacteristic
738C2#6183811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3387C2semidirect
6. #25847C2semidirect
7. #25837C2semidirect
10. #24345C2semidirect
11. #24333C2semidirect
14. #22525C2semidirect
180. #451Q8semidirect
292. C2×Q16C2×Q8semidirect
427. #103C2×Q8semidirect
703. C22#3387semidirect

Maximal subgroups

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

subgroupindex
C22×Q822
C2×#33872
#365452
#329602
#329542
#258472
#258372
#258162
#258112
#243452
#243332
#243022
#243012
#225252
#225102
#214552
C4×#4512