Finite Group Databasebeta

← #6683861

Subgroups — #6683861

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3387C211yesmaximal, characteristic, complemented
2C2×#3375C211yesmaximal, characteristic
3#39746C211yesmaximal, characteristic
4Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
5#33025C211yesmaximal, characteristic
6#30962C214yesmaximal, complemented
7#30824C214yesmaximal, complemented
8#30277C211yesmaximal, characteristic
9#29869C211yesmaximal, characteristic, complemented
10#28684C211yesmaximal, characteristic, complemented
11#28645C211yesmaximal, characteristic
12#26387C214yesmaximal, complemented
13#26185C214yesmaximal, complemented
14#25932C211yesmaximal, characteristic
15#25837C211yesmaximal, characteristic, complemented
16#25836C211yesmaximal, characteristic, complemented
17#25648C211yesmaximal, characteristic
18#22242C211yesmaximal, characteristic, complemented
19#21119C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×Q82C2211yescharacteristic
22C2×Q82C2211yescharacteristic
23C2×#557C2211yescharacteristic
24#3387—28no—
25#3375—28no—
26C2×#501C2212yes—
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#489C2212yes—
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2212yes—
37C2×#474C2212yes—
38C2×#464C2212yes—
39C2×#460C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×#446C2211yescharacteristic
42C2×#441C2212yes—
43C2×#440C2212yes—
44C2×C4×Q16C2211yescharacteristic
45#2868—21no—
46#2868C2211yescharacteristic
47#2858C2211yescharacteristic
48#2694—24no—
49#2658—21no—
50#2567—21no—
51#2567—21no—
52#2567C2211yescharacteristic
53#2567C2211yescharacteristic
54#2551—24no—
55#2551C2214yes—
56#2548C2211yescharacteristic
57#2547—21no—
58#2546C2211yescharacteristic
59#2545—21no—
60#2454C2214yes—
61#2448C2214yes—
62Q8×C4⋊C4C2214yes—
63#2326—21no—
64C4×C4⋊Q8—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2094C2214yes—
69#2094C2214yes—
70#2088C2214yes—
71#2083C2214yes—
72#2072C2214yes—
73#2067C2214yes—
74#2059C2214yes—
75#2058C2214yes—
76#2042C2214yes—
77#2023C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1982C2211yescharacteristic
84#1943C2214yes—
85#1936C2214yes—
86#1928C2212yes—
87#1926C2214yes—
88#1921C2212yes—
89#1917C2214yes—
90#1917C2214yes—
91#1880C2214yes—
92#1880C2214yes—
93#1878C2212yes—
94#1865C2214yes—
95#1865C2214yes—
96#1863C2212yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2311yescharacteristic
104C22×SD16C2312yes—
105#559—24no—
106#559—42no—
107Q82—24no—
108Q82—24no—
109Q82—42no—
110Q82—42no—
111#557—24no—
112#557—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8—24no—
120C2×C4⋊Q8—22no—
121C2×C4⋊Q8D411yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2312yes—
126C2×C4⋊Q8C2312yes—
127C2×#127C2311yescharacteristic
128C2×#125—22no—
129C2×#125C2312yes—
130C2×#124—21no—
131C2×#124—21no—
132C2×#124—21no—
133C2×#124—21no—
134C2×#124C2312yes—
135C2×#123—22no—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×D4—21no—
148C2×C4×D4C2312yes—
149#501—28no—
150#498—24no—
151#498—42no—
152#489—28no—
153#486—24no—
154#486—42no—
155#479—28no—
156#477—24no—
157#477—24no—
158#475—24no—
159#475—28no—
160#474—28no—
161#464—28no—
162#460—24no—
163#446—24no—
164#446—24no—
165#441—28no—
166#440—28no—
167C4×Q16D418yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2311yescharacteristic
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2314yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2311yescharacteristic
186C2×#103C2311yescharacteristic
187C2×#102C2312yes—
188C2×#102C2314yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2314yes—
192C2×#97C2312yes—
193C2×#97C2312yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2311yescharacteristic
196#399—24no—
197#399—24no—
198#398—22no—
199#397—24no—
200#396—24no—
201#395—22no—
202#395—22no—
203#395—24no—
204#395—24no—
205#394—22no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391D411yescharacteristic
210#391D411yescharacteristic
211#391C2312yes—
212#390—21no—
213#390—22no—
214#390C2314yes—
215#390C2314yes—
216#389—24no—
217#389—24no—
218#389—24no—
219#389D414yes—
220#389C2314yes—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#388—24no—
225#388—24no—
226#387—22no—
227#385—24no—
228#384—21no—
229#384—21no—
230#384—24no—
231#384—21no—
232#384—21no—
233#384D411yescharacteristic
234#384C2314yes—
235#380—24no—
236C4×C4⋊C4—24no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2314yes—
244#376—21no—
245C43—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8C2×D411yescharacteristic
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2411yescharacteristic
266C22×Q8C2411yescharacteristic
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C2412yes—
270C23×C4—22no—
271C2×Q16—24no—
272C2×Q16C2×D414yes—
273C2×SD16—24no—
274C2×SD16—28no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—42no—
290C4⋊Q8—42no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—48no—
295C4⋊Q8—48no—
296C4⋊Q8—84no—
297C4⋊Q8—88no—
298C4⋊Q8—82no—
299#127—22no—
300#127—22no—
301#125—24no—
302#125—42no—
303#125—48no—
304#124—24no—
305#124—42no—
306#124—44no—
307#124—44no—
308#124—82no—
309#124—82no—
310#123—48no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8C2×D414yes—
329C4×Q8C2×D414yes—
330C4×D4—24no—
331C4×D4—42no—
332C4×D4—44no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D412yes—
383C2×C42C2×D411yescharacteristic
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388#107—24no—
389#107—24no—
390#107—24no—
391#107C2×D414yes—
392#106—24no—
393#106—24no—
394#106—24no—
395#106—44no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103C2×D414yes—
406#103C2×D414yes—
407#102—24no—
408#102—24no—
409#102—24no—
410#102—28no—
411#102—44no—
412#97—24no—
413#97—24no—
414C4×C8—22no—
415C4×C8C2×D414yes—
416#95—22no—
417#95—22no—
418#95—21no—
419#95—22no—
420#95—22no—
421#95—24no—
422#95—24no—
423#95—21no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—84no—
449C2×Q8—84no—
450C2×Q8C22×D412yes—
451C2×Q8C22×D412yes—
452C2×Q82−1+414yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—82no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—84no—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C2511yescharacteristic
482Q16—28no—
483SD16—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8C22×D412yes—
494C2×C8C22×D412yes—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4C22×D412yes—
532C4⋊C4C22×D412yes—
533C4⋊C42−1+414yes—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—28no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—42no—
552C42—44no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42#13714yes—
559C42C22×D412yes—
560C42H(5,𝔽2)12yes—
561C23—44no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—24no—
569Q8—22no—
570Q8—22no—
571Q8—44no—
572Q8—44no—
573Q8—42no—
574Q8—84no—
575Q8—162no—
576Q8—162no—
577Q8—164no—
578Q8—164no—
579Q8#53614yes—
580D4—42no—
581D4—42no—
582D4—44no—
583D4—84no—
584D4—162no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—164no—
623C2×C4D4212yes—
624C2×C4D4212yes—
625C2×C4C2×#13714yes—
626C2×C4C2×#13712yes—
627C2×C4#57712yes—
628C2×C4C23×D411yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+412yes—
631C2×C4C2×2−1+412yes—
632C8—22no—
633C8—22no—
634C8—42no—
635C8—42no—
636C8—42no—
637C22—82no—
638C22—82no—
639C22—84no—
640C22—82no—
641C22—84no—
642C22C2×D4211yescharacteristic
643C22#348612yes—
644C22C22×#13711yescharacteristic
645C22C2×#57711yescharacteristic
646C22#358512yes—
647C4—24no—
648C4—24no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#327814yes—
664C4C2×#53611yescharacteristic
665C4#349911yescharacteristic
666C4#355212yes—
667C2—162no—
668C2#3767511yescharacteristic
669C2C2×#327811yescharacteristic
670C2#6108211yescharacteristic
671C2#6183012yes—
672C2#6186012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3387C2semidirect
4. Q8×C4⋊Q8C2semidirect
6. #30962C2semidirect
7. #30824C2semidirect
9. #29869C2semidirect
10. #28684C2semidirect
12. #26387C2semidirect
13. #26185C2semidirect
15. #25837C2semidirect
16. #25836C2semidirect
18. #22242C2semidirect
19. #21119C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33872
C2×#33752
#397462
Q8×C4⋊Q82
#330252
#309622
#308242
#302772
#298692
#286842
#286452
#263872
#261852
#259322
#258372
#258362
#256482
#222422
#211192