Finite Group Databasebeta

← #367015

Subgroups — #367015

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#1967C212yesmaximal
3C2×#1966C211yesmaximal, characteristic, complemented
4C2×#1966C211yesmaximal, characteristic, complemented
5#21146C211yesmaximal, characteristic, complemented
6C2×#1800C211yesmaximal, characteristic
7C4⋊C4×M16C211yesmaximal, characteristic, complemented
8C2×C4×#100C211yesmaximal, characteristic
9#11045C212yesmaximal, complemented
10#10982C212yesmaximal
11#10963C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×C4×D4C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2212yes—
17C2×#423C2211yescharacteristic
18C22×#101C2212yes—
19C22×#101C2212yes—
20C22×#101C2212yes—
21C22×#101C2212yes—
22C22×#100C2211yescharacteristic
23C22×#100C2212yes—
24C22×#100C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C2×C4×M16C2212yes—
27C2×C4×M16C2211yescharacteristic
28#2547—22no—
29#2547—22no—
30#2547—21no—
31#2547—21no—
32#2547—22no—
33C2×#397C414yes—
34C2×#391C2212yes—
35C2×#380C414yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1967—28no—
38#1966—24no—
39#1966—24no—
40#1940—22no—
41#1939—22no—
42#1800—24no—
43#1771—22no—
44#1770—21no—
45C8×C4⋊C4—21no—
46C4×#100—24no—
47C2×#343C2211yescharacteristic
48C2×#343C2212yes—
49C2×#343C2211yescharacteristic
50#1728C2212yes—
51#1728C2212yes—
52C23×Q8C2312yes—
53C23×D4C2311yescharacteristic
54C24×C4C2×C412yes—
55C22×M16C2312yes—
56C22×M16C2311yescharacteristic
57C22×M16C2312yes—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×#124—24no—
70C2×#124—24no—
71C2×#124—22no—
72C2×#124—22no—
73C2×#124—24no—
74C2×#124—42no—
75C2×#124—42no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C22×C4⋊C4C2×C412yes—
85C22×C4⋊C4C2×C412yes—
86C22×C4⋊C4C2×C412yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4C2×C412yes—
89C22×C22⋊C4C2×C418yes—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#423—22no—
93#423—24no—
94#423—22no—
95#423—22no—
96#423—42no—
97#423D414yes—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8—21no—
103C2×#101—24no—
104C2×#101—24no—
105C2×#101—24no—
106C2×#101—24no—
107C2×#101—24no—
108C2×#101—24no—
109C2×#101—24no—
110C2×#101—24no—
111C2×#101—42no—
112C2×#101—42no—
113C2×#101—42no—
114C2×#101—42no—
115C2×#100—22no—
116C2×#100—22no—
117C2×#100—22no—
118C2×#100—22no—
119C2×#100—22no—
120C2×#100—24no—
121C2×#100—24no—
122C2×#100—42no—
123C2×#100D414yes—
124C4×M16—22no—
125C4×M16—24no—
126C4×M16—22no—
127C4×M16—22no—
128C4×M16—42no—
129C4×M16D414yes—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—22no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—22no—
136#397—28no—
137#397—24no—
138#397—24no—
139#391—22no—
140#391—22no—
141#391—24no—
142#380—24no—
143#380—24no—
144#380—28no—
145C4×C4⋊C4D412yes—
146C4×C4⋊C4D412yes—
147C4×C4⋊C4D411yescharacteristic
148C4×C4⋊C4D411yescharacteristic
149C4×C4⋊C4D412yes—
150C2×#95C2×C418yes—
151C2×#95C2×C412yes—
152#343—24no—
153#343—22no—
154#343—22no—
155#336—24no—
156#336—22no—
157#336—22no—
158C25C22×C412yes—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4C2×D412yes—
186C22×D4#4112yes—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—48no—
192C23×C4C22×C414yes—
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C412yes—
195C23×C4C22×C414yes—
196C23×C4C22×C411yescharacteristic
197C23×C4C2411yescharacteristic
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—22no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—22no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—22no—
224C2×M16—22no—
225C2×M16—22no—
226C2×M16—22no—
227C2×M16C2×D412yes—
228C2×M16C2×D414yes—
229C2×M16C2×D412yes—
230C2×M16C2×D412yes—
231C2×M16C2×D412yes—
232C2×M16#4112yes—
233C2×M16#4112yes—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—22no—
238C22×C8—21no—
239C22×C8—22no—
240C4⋊Q8—42no—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243C4⋊Q8—42no—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248#124—44no—
249#124—44no—
250#124—48no—
251#124—44no—
252#124—44no—
253#124—48no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C22⋊C412yes—
278C2×C4⋊C4C22⋊C412yes—
279C2×C4⋊C4C22⋊C414yes—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—48no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D414yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4112yes—
302C2×C42#4112yes—
303C4⋊C8—24no—
304C4⋊C8—24no—
305C4⋊C8—48no—
306C4⋊C8—42no—
307C4⋊C8—42no—
308#101—48no—
309#101—48no—
310#101—48no—
311#101—48no—
312#100—28no—
313#100—28no—
314#100—48no—
315#97—24no—
316#97—44no—
317#97—42no—
318C4×C8—24no—
319C4×C8—44no—
320C4×C8—42no—
321#95—28no—
322#95—28no—
323#95C22⋊C414yes—
324#95C22⋊C412yes—
325#95C22⋊C412yes—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—44no—
333C24—44no—
334C24C4×D414yes—
335C24C23×C411yescharacteristic
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4C2×C22⋊C414yes—
416C22×C4C2×C22⋊C414yes—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C4×D412yes—
422C22×C4C4×D412yes—
423C22×C4C4×D412yes—
424C22×C4C4×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4H(5,𝔽2)11yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C42−1+411yescharacteristic
439M16—22no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—22no—
444M16—22no—
445M16—22no—
446M16—24no—
447M16—22no—
448M16—22no—
449M16—24no—
450M16—42no—
451M16—41no—
452M16—41no—
453M16—42no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—42no—
459M16—42no—
460M16—42no—
461M16—42no—
462M16#12114yes—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—42no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—42no—
477C2×C8—41no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42#12112yes—
503C42#12112yes—
504C42#12112yes—
505C42#12112yes—
506C42#12412yes—
507C42#12412yes—
508C42#12712yes—
509C42#12712yes—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—48no—
516C23—48no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—82no—
524C23—82no—
525C23—84no—
526C23—84no—
527C23—82no—
528C23—82no—
529C23C22×C22⋊C411yescharacteristic
530C23C2×C4×D411yescharacteristic
531C23C2×C4×D411yescharacteristic
532C23C2×C4×D411yescharacteristic
533C23C4×#4111yescharacteristic
534C23#51811yescharacteristic
535C23#52011yescharacteristic
536Q8—82no—
537Q8—82no—
538Q8—82no—
539Q8—82no—
540Q8—82no—
541Q8—82no—
542Q8—82no—
543Q8—82no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—88no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—88no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4#39012yes—
581C2×C4#39012yes—
582C2×C4#39012yes—
583C2×C4#39012yes—
584C2×C4#41314yes—
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12111yescharacteristic
587C2×C4C2×#12111yescharacteristic
588C2×C4#52511yescharacteristic
589C2×C4C2×#12411yescharacteristic
590C2×C4#52911yescharacteristic
591C2×C4C2×#12711yescharacteristic
592C2×C4#53211yescharacteristic
593C2×C4#53511yescharacteristic
594C2×C4#53711yescharacteristic
595C2×C4#54011yescharacteristic
596C2×C4#54211yescharacteristic
597C2×C4D4211yescharacteristic
598C2×C4D4211yescharacteristic
599C2×C4D4211yescharacteristic
600C2×C4#54611yescharacteristic
601C2×C4#54611yescharacteristic
602C2×C4#54611yescharacteristic
603C2×C4#54711yescharacteristic
604C2×C4#54811yescharacteristic
605C2×C4#54811yescharacteristic
606C2×C4#54811yescharacteristic
607C2×C4#55011yescharacteristic
608C2×C4#55011yescharacteristic
609C2×C4#55011yescharacteristic
610C2×C4#55311yescharacteristic
611C2×C4#55311yescharacteristic
612C2×C4#55511yescharacteristic
613C8—24no—
614C8—42no—
615C8—44no—
616C8—42no—
617C8—42no—
618C8—82no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—48no—
628C22—44no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—82no—
635C22—82no—
636C22C2×#39011yescharacteristic
637C22C4×#12111yescharacteristic
638C22#232411yescharacteristic
639C22D4×C22⋊C411yescharacteristic
640C22#234111yescharacteristic
641C22#236111yescharacteristic
642C22#236311yescharacteristic
643C22C2×#41312yes—
644C22#288912yes—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—44no—
649C4—44no—
650C4—84no—
651C4—82no—
652C4#196714yescomplemented
653C4#241411yescharacteristic
654C4#241611yescharacteristic
655C4#242611yescharacteristic
656C4#242811yescharacteristic
657C4#249011yescharacteristic
658C4#249111yescharacteristic
659C4#254011yescharacteristic
660C4#254111yescharacteristic
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—84no—
666C2#1399811yescharacteristic
667C2C2×#196711yescharacteristic
668C2#2122011yescharacteristic
669C2#2241112yes—
670C2#2253712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1966C2semidirect
4. C2×#1966C2semidirect
5. #21146C2semidirect
7. C4⋊C4×M16C2semidirect
9. #11045C2semidirect
11. #10963C2semidirect
652. C4#1967semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#19672
C2×#19662
#211462
C2×#18002
C4⋊C4×M162
C2×C4×#1002
#110452
#109822
#109632