Finite Group Databasebeta

← #6625540

Subgroups — #6625540

Order 512.

2993 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#3387C212yesmaximal, complemented
3Q8×C22⋊Q8C211yesmaximal, characteristic
4#32957C211yesmaximal, characteristic
5#32954C212yesmaximal
6#25848C214yesmaximal, complemented
7#25836C212yesmaximal, complemented
8#25819C212yesmaximal
9#25811C211yesmaximal, characteristic
10#24348C214yesmaximal, complemented
11#24332C212yesmaximal, complemented
12#24307C212yesmaximal
13#24301C211yesmaximal, characteristic
14#22524C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic
16#21455C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C2×#557—216no—
20C2×#557—21no—
21C2×#557—22no—
22C2×#557—21no—
23C2×#557C2212yes—
24C2×#557C2214yes—
25C2×#557C2212yes—
26C22×C4⋊Q8—21no—
27C22×C4⋊Q8C2211yescharacteristic
28C22×#125—22no—
29C22×#125C2212yes—
30C22×C4×Q8—21no—
31C22×C4×Q8C2212yes—
32C22×C4×Q8C2211yescharacteristic
33C22×C4×Q8C2211yescharacteristic
34#3387—216no—
35C2×#501C2214yes—
36C2×#498C2212yes—
37C2×#479C2214yes—
38C2×#477C2212yes—
39C2×#475C2212yes—
40C2×#475C2214yes—
41C2×#451C2211yescharacteristic
42C2×#446C2212yes—
43C2×#441C2214yes—
44C2×C4×Q16C2212yes—
45#2857C2212yes—
46#2857C2211yescharacteristic
47#2582C2211yescharacteristic
48#2582C2212yes—
49#2567C2211yescharacteristic
50#2567C2212yes—
51#2451—22no—
52#2451—21no—
53#2440—21no—
54#2440—22no—
55Q8×C4⋊C4—21no—
56Q8×C4⋊C4C2211yescharacteristic
57Q8×C22⋊C4—21no—
58C4×C22⋊Q8C2211yescharacteristic
59C4×C22⋊Q8C2212yes—
60#2059C2214yes—
61#2059C2214yes—
62#2059C2214yes—
63#2058C2214yes—
64#2058C2212yes—
65#2026C2214yes—
66#2024C2212yes—
67#1992C2214yes—
68#1990C2212yes—
69#1989C2214yes—
70#1988C2212yes—
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#1803C2212yes—
88#1803C2214yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92#1787C2212yes—
93#1763C2214yes—
94C4×#103C2212yes—
95#1749C2212yes—
96C4×C22⋊C8C2211yescharacteristic
97C23×Q8—21no—
98C23×Q8D411yescharacteristic
99C23×Q8C2311yescharacteristic
100C22×Q16C2312yes—
101Q82—28no—
102#557—216no—
103#557—232no—
104#557—216no—
105#557—28no—
106#557—216no—
107#557—28no—
108#557—216no—
109#557—28no—
110#557D4116yes—
111C2×C4⋊Q8—216no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8—28no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2312yes—
124C2×#125—232no—
125C2×#125—216no—
126C2×#125—24no—
127C2×#125—24no—
128C2×#125—24no—
129C2×#125—24no—
130C2×#125—22no—
131C2×#125—22no—
132C2×#125C2314yes—
133C2×#125C2314yes—
134C2×C22⋊Q8—21no—
135C2×C22⋊Q8—22no—
136C2×C22⋊Q8C2311yescharacteristic
137C2×C4×Q8—216no—
138C2×C4×Q8—216no—
139C2×C4×Q8—28no—
140C2×C4×Q8—28no—
141C2×C4×Q8—21no—
142C2×C4×Q8—21no—
143C2×C4×Q8—22no—
144C2×C4×Q8—24no—
145C2×C4×Q8—22no—
146C2×C4×Q8—22no—
147C2×C4×Q8—22no—
148C2×C4×Q8—21no—
149C2×C4×Q8—21no—
150C2×C4×Q8D411yescharacteristic
151C2×C4×Q8D412yes—
152C2×C4×Q8D411yescharacteristic
153C2×C4×Q8C2311yescharacteristic
154C2×C4×Q8C2312yes—
155C2×C4×Q8C2312yes—
156C2×C4×Q8C2314yes—
157C2×C4×Q8C2314yes—
158C2×C4×Q8C2312yes—
159C2×C4×Q8C2311yescharacteristic
160C2×C4×Q8C2312yes—
161C2×C4×Q8C2311yescharacteristic
162C22×C4⋊C4—22no—
163C22×C4⋊C4—21no—
164C22×C4⋊C4—22no—
165C22×C4⋊C4—21no—
166C22×C4⋊C4—22no—
167C22×C4⋊C4D411yescharacteristic
168C22×C4⋊C4D412yes—
169C22×C4⋊C4C2311yescharacteristic
170C22×C4⋊C4C2312yes—
171C22×C42—22no—
172C22×C42—21no—
173C22×C42C2312yes—
174C22×C42C2311yescharacteristic
175#501—216no—
176#498—28no—
177#479—216no—
178#477—28no—
179#475—28no—
180#475—216no—
181#451Q818yescomplemented
182#446—28no—
183#441—216no—
184C4×Q16—28no—
185C2×#107C2314yes—
186C2×#107C2312yes—
187C2×#106C2314yes—
188C2×#106C2314yes—
189C2×#106C2314yes—
190C2×C4⋊C8C2312yes—
191C2×C4⋊C8C2314yes—
192C2×#103C2312yes—
193C2×#103C2314yes—
194C2×#103C2314yes—
195C2×#103C2312yes—
196C2×#103C2314yes—
197C2×#103C2312yes—
198C2×#103C2312yes—
199C2×C22⋊C8C2312yes—
200C2×C22⋊C8C2311yescharacteristic
201C2×C22⋊C8C2311yescharacteristic
202C2×#97C2314yes—
203C2×C4×C8C2312yes—
204#398—22no—
205#398—22no—
206#398—22no—
207#395—22no—
208#395—22no—
209#395—22no—
210#391—21no—
211#391—22no—
212#391—21no—
213#391—22no—
214#391—22no—
215#391—21no—
216#391C2311yescharacteristic
217#391C2312yes—
218#389—21no—
219#389—22no—
220#389—21no—
221#389—22no—
222#389—22no—
223#389—21no—
224#389C2311yescharacteristic
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#387—22no—
230#387—21no—
231#385—21no—
232#385—22no—
233#384—21no—
234#384—22no—
235#384C2311yescharacteristic
236#384C2312yes—
237#380—22no—
238#380—21no—
239#380C2311yescharacteristic
240#380C2312yes—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—21no—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#336C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2314yes—
258C22×Q8—24no—
259C22×Q8—24no—
260C22×Q8—28no—
261C22×Q8—24no—
262C22×Q8—24no—
263C22×Q8—21no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—21no—
268C22×Q8—22no—
269C22×Q8—42no—
270C22×Q8—44no—
271C22×Q8—44no—
272C22×Q8C2×D411yescharacteristic
273C22×Q8C2×D412yes—
274C22×Q8C2×D411yescharacteristic
275C22×Q8C2×Q814yes—
276C22×Q8C2411yescharacteristic
277C22×Q8C2412yes—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—22no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D412yes—
283C23×C4C2×D412yes—
284C23×C4C2×D411yescharacteristic
285C23×C4C2411yescharacteristic
286C2×Q16—24no—
287C2×Q16—24no—
288C2×Q16C2×Q818yescomplemented
289C22×C8C2412yes—
290C22×C8C2414yes—
291C22×C8C2412yes—
292C4⋊Q8—232no—
293C4⋊Q8—216no—
294C4⋊Q8—28no—
295C4⋊Q8—28no—
296C4⋊Q8—216no—
297C4⋊Q8—28no—
298C4⋊Q8—24no—
299C4⋊Q8—28no—
300C4⋊Q8—24no—
301C4⋊Q8—44no—
302C4⋊Q8—48no—
303C4⋊Q8C2×D418yes—
304#125—232no—
305#125—232no—
306#125—232no—
307#125—216no—
308#125—216no—
309#125—232no—
310#125—28no—
311#125—28no—
312#125—28no—
313#125C2×D4116yes—
314C22⋊Q8—22no—
315C22⋊Q8—82no—
316C22⋊Q8—84no—
317C22⋊Q8C2×Q814yes—
318C4×Q8—216no—
319C4×Q8—216no—
320C4×Q8—232no—
321C4×Q8—28no—
322C4×Q8—216no—
323C4×Q8—28no—
324C4×Q8—24no—
325C4×Q8—28no—
326C4×Q8—24no—
327C4×Q8—216no—
328C4×Q8—28no—
329C4×Q8—24no—
330C4×Q8—28no—
331C4×Q8—24no—
332C4×Q8—28no—
333C4×Q8—24no—
334C4×Q8—24no—
335C4×Q8—28no—
336C4×Q8—24no—
337C4×Q8—28no—
338C4×Q8—44no—
339C4×Q8—44no—
340C4×Q8C2×D418yes—
341C4×Q8C2×D4116yes—
342C4×Q8C2×D418yes—
343C2×C4⋊C4—216no—
344C2×C4⋊C4—28no—
345C2×C4⋊C4—216no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—216no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
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—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C2×D411yescharacteristic
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2×D412yes—
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2412yes—
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2414yes—
391C2×C4⋊C4C2411yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C42—216no—
396C2×C42—28no—
397C2×C42—24no—
398C2×C42—24no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42—28no—
402C2×C42—24no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42C2×D412yes—
408C2×C42C2×D412yes—
409C2×C42C2×D411yescharacteristic
410C2×C42C2×D411yescharacteristic
411C2×C42C2412yes—
412C2×C42C2411yescharacteristic
413C2×C42C2412yes—
414C2×C42C2411yescharacteristic
415#107—24no—
416#107—28no—
417#106—28no—
418#106—28no—
419#106—28no—
420C4⋊C8—24no—
421C4⋊C8—28no—
422#103—24no—
423#103—28no—
424#103—28no—
425#103—24no—
426#103—24no—
427#103—28no—
428#103C2×Q818yescomplemented
429C22⋊C8—24no—
430C22⋊C8—22no—
431C22⋊C8C2×Q814yes—
432#97—28no—
433C4×C8—24no—
434#95—22no—
435#95—21no—
436#95—22no—
437#95—21no—
438#95—22no—
439#95—21no—
440#95—22no—
441#95—21no—
442#95—22no—
443#95—21no—
444#95—22no—
445#95—22no—
446#95—22no—
447C24C22×D411yescharacteristic
448C2×Q8—24no—
449C2×Q8—216no—
450C2×Q8—24no—
451C2×Q8—216no—
452C2×Q8—28no—
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—28no—
456C2×Q8—24no—
457C2×Q8—22no—
458C2×Q8—24no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—42no—
469C2×Q8—416no—
470C2×Q8—44no—
471C2×Q8—432no—
472C2×Q8—44no—
473C2×Q8—42no—
474C2×Q8—44no—
475C2×Q8—44no—
476C2×Q8—44no—
477C2×Q8—82no—
478C2×Q8—81no—
479C2×Q8—82no—
480C2×Q8—81no—
481C2×Q8#12014yes—
482C2×Q8C22×D414yes—
483C2×Q8C22×Q812yes—
484C2×Q8C22×Q814yes—
485C2×Q82−1+414yes—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—28no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—21no—
510C22×C4—41no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—42no—
517C22×C4#13714yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D412yes—
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D412yes—
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×Q812yes—
528C22×C42−1+412yes—
529C22×C4C2511yescharacteristic
530Q16—216no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—24no—
535C2×C8—22no—
536C2×C8—24no—
537C2×C8—22no—
538C2×C8—24no—
539C2×C8C22×Q814yes—
540C4⋊C4—28no—
541C4⋊C4—24no—
542C4⋊C4—216no—
543C4⋊C4—216no—
544C4⋊C4—28no—
545C4⋊C4—28no—
546C4⋊C4—216no—
547C4⋊C4—216no—
548C4⋊C4—28no—
549C4⋊C4—216no—
550C4⋊C4—28no—
551C4⋊C4—216no—
552C4⋊C4—24no—
553C4⋊C4—22no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—22no—
559C4⋊C4—24no—
560C4⋊C4—21no—
561C4⋊C4—22no—
562C4⋊C4—24no—
563C4⋊C4—24no—
564C4⋊C4—22no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—42no—
568C4⋊C4—44no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—82no—
572C4⋊C4—81no—
573C4⋊C4—82no—
574C4⋊C4—81no—
575C4⋊C4—82no—
576C4⋊C4—82no—
577C4⋊C4—82no—
578C4⋊C4#12014yes—
579C4⋊C4#12018yes—
580C4⋊C4C22×D418yes—
581C4⋊C4C22×D414yes—
582C4⋊C4C22×Q812yes—
583C22⋊C4—24no—
584C22⋊C4—44no—
585C22⋊C4—42no—
586C42—216no—
587C42—28no—
588C42—216no—
589C42—28no—
590C42—24no—
591C42—24no—
592C42—22no—
593C42—22no—
594C42—24no—
595C42—22no—
596C42—44no—
597C42—44no—
598C42—42no—
599C42—42no—
600C42C22×D414yes—
601C42C22×D418yes—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23D4×Q812yes—
607C23C2×#13712yes—
608C23#57712yes—
609C23C23×D411yescharacteristic
610Q8—216no—
611Q8—24no—
612Q8—24no—
613Q8—24no—
614Q8—28no—
615Q8—24no—
616Q8—48no—
617Q8—416no—
618Q8—42no—
619Q8—44no—
620Q8—416no—
621Q8—164no—
622Q8—162no—
623Q8D4×Q818yes—
624C2×C4—216no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—24no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—21no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—28no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—48no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—48no—
658C2×C4—416no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—41no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4C2×#12014yes—
680C2×C4C2×#12014yes—
681C2×C4C2×#12012yes—
682C2×C4C2×#12012yes—
683C2×C4D4×Q812yes—
684C2×C4C2×#13712yes—
685C2×C4C2×#13712yes—
686C2×C4#57512yes—
687C2×C4C23×D412yes—
688C2×C4C23×Q811yescharacteristic
689C2×C4C2×2−1+412yes—
690C2×C4C2×2−1+411yescharacteristic
691C8—24no—
692C8—42no—
693C8—44no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22#338714yescomplemented
701C22C22×#12011yescharacteristic
702C22C2×D4×Q811yescharacteristic
703C22#348611yescharacteristic
704C22C22×#13711yescharacteristic
705C22C2×#57511yescharacteristic
706C22C2×#57711yescharacteristic
707C22#358611yescharacteristic
708C4—28no—
709C4—22no—
710C4—22no—
711C4—24no—
712C4—24no—
713C4—22no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—48no—
718C4—42no—
719C4—41no—
720C4—44no—
721C4—82no—
722C4—81no—
723C4—82no—
724C4—81no—
725C4#300714yes—
726C4C2×D4×Q812yes—
727C4#348612yes—
728C2—24no—
729C2Q8×#12011yescharacteristic
730C2C2×#300711yescharacteristic
731C2#6037411yescharacteristic
732C2C2×#338711yescharacteristic
733C2#6136911yescharacteristic
734C2#6182911yescharacteristic
735C2#6183611yescharacteristic
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. #25848C2semidirect
7. #25836C2semidirect
10. #24348C2semidirect
11. #24332C2semidirect
14. #22524C2semidirect
181. #451Q8semidirect
288. C2×Q16C2×Q8semidirect
428. #103C2×Q8semidirect
700. C22#3387semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#33872
Q8×C22⋊Q82
#329572
#329542
#258482
#258362
#258192
#258112
#243482
#243322
#243072
#243012
#225242
#225102
#214552
C4×#4512