Finite Group Databasebeta

← #183017

Subgroups — #183017

Order 512.

959 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60208C211yesmaximal, characteristic
2#20441C211yesmaximal, characteristic, complemented
3#17719C211yesmaximal, characteristic
4#17563C211yesmaximal, characteristic
5#17334C211yesmaximal, characteristic, complemented
6#8258C211yesmaximal, characteristic, complemented
7#8257C211yesmaximal, characteristic, complemented
8#8256C211yesmaximal, characteristic, complemented
9#8255C211yesmaximal, characteristic, complemented
10#8254C211yesmaximal, characteristic, complemented
11#8253C211yesmaximal, characteristic, complemented
12#8252C211yesmaximal, characteristic, complemented
13#8251C211yesmaximal, characteristic, complemented
14#8094C211yesmaximal, characteristic, complemented
15#8082C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2971—21no—
18#2971C412yescomplemented
19#2970—21no—
20#2970C412yescomplemented
21#2969—21no—
22#2969—21no—
23#2969C412yescomplemented
24#2969C412yescomplemented
25#2960C2211yescharacteristic
26#2960C2211yescharacteristic
27#2931C2211yescharacteristic
28#2931C2211yescharacteristic
29#2930—21no—
30#2930—21no—
31#2923—21no—
32#2920C2211yescharacteristic
33#2920C2211yescharacteristic
34#2920C2211yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36#2875—21no—
37C2×#405—21no—
38C2×C4×M16C2211yescharacteristic
39#2285C2211yescharacteristic
40#1771—21no—
41#1771—21no—
42#1770C2211yescharacteristic
43#1770C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46#1729—21no—
47#1552C412yescomplemented
48#1551C412yescomplemented
49#1550C412yescomplemented
50#1547C412yescomplemented
51#1542C2211yescharacteristic
52#1542C2211yescharacteristic
53#1540C2211yescharacteristic
54#1540C2211yescharacteristic
55#1513C414yes—
56#1512C414yes—
57#1503C2211yescharacteristic
58#1503C2211yescharacteristic
59#1502C2211yescharacteristic
60#1502C2211yescharacteristic
61#1455—21no—
62#1455—21no—
63#1453—21no—
64#1453—21no—
65#1452C2211yescharacteristic
66#1452C2211yescharacteristic
67#1450C2211yescharacteristic
68#1450C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1303C2211yescharacteristic
72#1303C2211yescharacteristic
73#1301C2211yescharacteristic
74#1301C2211yescharacteristic
75#1301C2211yescharacteristic
76#1301C2211yescharacteristic
77C2×#131—21no—
78C2×#131—21no—
79C22×M16D411yescharacteristic
80C22×M16D411yescharacteristic
81C23×C8—21no—
82#533C2×C411yescharacteristic
83#533C2×C411yescharacteristic
84#533C2311yescharacteristic
85#532C2×C411yescharacteristic
86#532C2×C411yescharacteristic
87#532C2311yescharacteristic
88#529—21no—
89#529—21no—
90#529—21no—
91#529—21no—
92#529C2×C412yes—
93#529C2×C412yes—
94#525—21no—
95#525—21no—
96#525C2×C412yes—
97C4×#41—21no—
98C4×#41—21no—
99C4×#41C2×C412yes—
100C2×#117—21no—
101C22×C42C2311yescharacteristic
102#446—21no—
103#446—21no—
104#446C2×C412yes—
105#446C2×C412yes—
106C8×Q8—21no—
107C8×Q8—21no—
108C8×Q8C2×C412yes—
109C8×Q8C2×C412yes—
110#436—22no—
111#436—22no—
112#436—22no—
113#436—22no—
114#436—21no—
115#436—21no—
116#436C2×C412yes—
117#436C2×C412yes—
118#435—22no—
119#435—22no—
120#435—22no—
121#435—22no—
122C8×D4—21no—
123C8×D4—21no—
124C8×D4C2×C412yes—
125C8×D4C2×C412yes—
126#433—22no—
127#433—22no—
128#432—21no—
129#432—21no—
130#432—21no—
131#432—22no—
132#432—22no—
133#432C2×C412yes—
134#432C2×C412yes—
135#432C2×C412yes—
136#431—21no—
137#431—21no—
138#431—21no—
139#431C2×C412yes—
140#431C2311yescharacteristic
141#431C2311yescharacteristic
142#431C2311yescharacteristic
143#431C2311yescharacteristic
144#424—21no—
145#424—21no—
146#423—21no—
147#423C2×C412yes—
148#423C2311yescharacteristic
149#423C2311yescharacteristic
150C2×C4⋊C8—21no—
151C2×C4⋊C8—21no—
152C2×C4⋊C8—21no—
153C2×C4⋊C8C2×C412yes—
154C2×C4⋊C8C2×C412yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157#408—21no—
158#408—21no—
159#407—21no—
160#407—21no—
161#407C2×C412yes—
162#405—21no—
163#405—21no—
164#405—22no—
165#405—22no—
166#405—44no—
167C4×M16—21no—
168C4×M16—21no—
169C4×M16—21no—
170C4×M16—22no—
171C4×M16C2×C411yescharacteristic
172C4×M16C2×C411yescharacteristic
173C4×M16C2×C411yescharacteristic
174C4×M16C2×C411yescharacteristic
175C4×M16D411yescharacteristic
176C4×M16D411yescharacteristic
177C4×M16D411yescharacteristic
178C4×M16D411yescharacteristic
179C2×#97—22no—
180C2×#97—21no—
181C2×#97—21no—
182C2×#97D411yescharacteristic
183C2×#97D411yescharacteristic
184C2×C4×C8—21no—
185C2×C4×C8—21no—
186C2×C4×C8C2311yescharacteristic
187C2×C4×C8C2311yescharacteristic
188#384—21no—
189#382C2×C412yes—
190#380—21no—
191#380C2×C412yes—
192C4×C4⋊C4—21no—
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2311yescharacteristic
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#333C2×C412yes—
200#333C2×C412yes—
201#332C2×C412yes—
202#332C2×C412yes—
203#332C2×C412yes—
204#332C2×C412yes—
205#331C2×C412yes—
206#331C2×C412yes—
207#328C2×C414yes—
208#328C2×C414yes—
209#327C2×C414yes—
210#327C2×C414yes—
211#322—21no—
212#322—21no—
213#322—21no—
214#322—21no—
215#322—21no—
216#322—21no—
217C82—21no—
218C82—21no—
219C2×#41—21no—
220C2×#41—21no—
221C2×#41C4212yes—
222C23×C4C22×C411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4#4111yescharacteristic
225#131—42no—
226#131—42no—
227#131—42no—
228#131—42no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—22no—
232C2×M16—22no—
233C2×M16—22no—
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—22no—
237C2×M16—41no—
238C2×M16—41no—
239C2×M16C22⋊C411yescharacteristic
240C2×M16C22⋊C411yescharacteristic
241C2×M16C22⋊C411yescharacteristic
242C2×M16C22⋊C411yescharacteristic
243C2×M16C22⋊C411yescharacteristic
244C2×M16C22⋊C411yescharacteristic
245C2×M16C22⋊C411yescharacteristic
246C2×M16C22⋊C411yescharacteristic
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8C2×D411yescharacteristic
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C4⋊Q8C22×C411yescharacteristic
260C4⋊Q8C22×C411yescharacteristic
261C4⋊Q8C22×C411yescharacteristic
262#127C22×C411yescharacteristic
263#126—22no—
264#126—22no—
265#125—21no—
266#125—21no—
267#125C4212yes—
268#124—21no—
269#124—21no—
270#124C4212yes—
271#123—22no—
272#123—22no—
273#123—22no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8C4212yes—
279C22⋊Q8C22×C412yes—
280#121—21no—
281#121—21no—
282#121—21no—
283#121—21no—
284#121C4212yes—
285#121C22×C412yes—
286#120—22no—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8C4212yes—
292C4×Q8C22×C412yes—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4C4212yes—
304C4×D4C22×C412yes—
305#117—21no—
306#117—21no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—42no—
311#117—42no—
312#117C4212yes—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42#4111yescharacteristic
330C2×C42#4111yescharacteristic
331C2×C42#4111yescharacteristic
332C2×C42C2411yescharacteristic
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339C4⋊C8—22no—
340C4⋊C8—22no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8C22×C412yes—
348C4⋊C8C22×C412yes—
349C4⋊C8C22×C412yes—
350C4⋊C8C22×C412yes—
351C4⋊C8C22×C412yes—
352C4⋊C8C22×C412yes—
353C22⋊C8—22no—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C22⋊C8—22no—
357C22⋊C8—21no—
358C22⋊C8—21no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—22no—
362C22⋊C8—22no—
363C22⋊C8C22×C412yes—
364C22⋊C8C22×C412yes—
365#97—22no—
366#97—22no—
367#97—22no—
368#97—22no—
369#97—22no—
370#97—42no—
371#97—42no—
372#97C22⋊C411yescharacteristic
373#97C22⋊C411yescharacteristic
374#97C22⋊C411yescharacteristic
375#97C22⋊C411yescharacteristic
376#97C22⋊C411yescharacteristic
377#97C22⋊C411yescharacteristic
378#97C22⋊C411yescharacteristic
379#97C22⋊C411yescharacteristic
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—22no—
383C4×C8—21no—
384C4×C8—21no—
385C4×C8—21no—
386C4×C8—21no—
387C4×C8—42no—
388C4×C8—42no—
389C4×C8C22×C411yescharacteristic
390C4×C8C22×C411yescharacteristic
391C4×C8C22×C411yescharacteristic
392C4×C8C22×C411yescharacteristic
393C4×C8C2×D411yescharacteristic
394C4×C8C2×D411yescharacteristic
395C4×C8C2×D411yescharacteristic
396C4×C8C2×D411yescharacteristic
397#95—22no—
398#95C22×C412yes—
399#95C22×C412yes—
400C24C4×D411yescharacteristic
401#41—41no—
402#41—41no—
403#41—41no—
404#41—41no—
405#41—82no—
406#41—82no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8C2×C4212yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4C2×C4212yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4C2×C4211yescharacteristic
428C22×C4C2×C4211yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4#11711yescharacteristic
432C22×C4#11711yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4111yescharacteristic
445M16—22no—
446M16—22no—
447M16—22no—
448M16—22no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—44no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—44no—
459M16—82no—
460M16—82no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—44no—
483C2×C8C2×C22⋊C411yescharacteristic
484C2×C8C2×C22⋊C411yescharacteristic
485C2×C8C2×C22⋊C411yescharacteristic
486C2×C8C2×C22⋊C411yescharacteristic
487C2×C8C2×C22⋊C411yescharacteristic
488C2×C8C2×C22⋊C411yescharacteristic
489C2×C8C2×C22⋊C411yescharacteristic
490C2×C8C2×C22⋊C411yescharacteristic
491C2×C8AGL(1,ℤ/8)12yes—
492C2×C8#13712yes—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—82no—
507C4⋊C4C2×C4212yes—
508C4⋊C4C2×C4212yes—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—44no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42C2×C4211yescharacteristic
527C42C2×C4211yescharacteristic
528C42C2×C22⋊C411yescharacteristic
529C42C2×C22⋊C411yescharacteristic
530C42#11711yescharacteristic
531C42#11711yescharacteristic
532C42C4×D411yescharacteristic
533C42C4×D411yescharacteristic
534C42C4×D411yescharacteristic
535C42C4×D411yescharacteristic
536C42C4×D411yescharacteristic
537C42C4×D411yescharacteristic
538C42C23×C411yescharacteristic
539C42C23×C411yescharacteristic
540C42C22×D411yescharacteristic
541C42C2×#4111yescharacteristic
542C23—21no—
543C23—21no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23C4×C22⋊C411yescharacteristic
548C23C4×C22⋊C411yescharacteristic
549C23C2×C4×D411yescharacteristic
550Q8—42no—
551Q8—41no—
552Q8—41no—
553Q8—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—41no—
558D4—41no—
559D4—42no—
560D4—82no—
561D4—82no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4C4×C22⋊C411yescharacteristic
598C2×C4C4×C22⋊C411yescharacteristic
599C2×C4C4×C22⋊C411yescharacteristic
600C2×C4C4×C22⋊C411yescharacteristic
601C2×C4C4×C22⋊C411yescharacteristic
602C2×C4C4×C22⋊C411yescharacteristic
603C2×C4C22×C4211yescharacteristic
604C2×C4C22×C22⋊C411yescharacteristic
605C2×C4C2×#11711yescharacteristic
606C2×C4C2×C4×D411yescharacteristic
607C2×C4C2×C4×D411yescharacteristic
608C2×C4C2×C4×D411yescharacteristic
609C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
610C2×C4C2×#13711yescharacteristic
611C2×C4#57511yescharacteristic
612C2×C4#57511yescharacteristic
613C8—21no—
614C8—21no—
615C8—21no—
616C8—21no—
617C8—21no—
618C8—21no—
619C8—42no—
620C8—42no—
621C8—42no—
622C8—42no—
623C8—42no—
624C8—42no—
625C8—42no—
626C8—42no—
627C8#41712yes—
628C8#41712yes—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22C2×C4×C22⋊C411yescharacteristic
643C22C4×AGL(1,ℤ/8)11yescharacteristic
644C22C4×#13711yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4C2×#41711yescharacteristic
660C4#289611yescharacteristic
661C4C4×AGL(1,ℤ/8)11yescharacteristic
662C4C4×#13711yescharacteristic
663C4#295712yes—
664C2—21no—
665C2—21no—
666C2—41no—
667C2—41no—
668C2—82no—
669C2#2040111yescharacteristic
670C2C4×#41711yescharacteristic
671C2#2042211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC22C2×C4×C22⋊C4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20441C2semidirect
5. #17334C2semidirect
6. #8258C2semidirect
7. #8257C2semidirect
8. #8256C2semidirect
9. #8255C2semidirect
10. #8254C2semidirect
11. #8253C2semidirect
12. #8252C2semidirect
13. #8251C2semidirect
14. #8094C2semidirect
15. #8082C2semidirect
18. #2971C4semidirect
20. #2970C4semidirect
23. #2969C4semidirect
24. #2969C4semidirect
47. #1552C4semidirect
48. #1551C4semidirect
49. #1550C4semidirect
50. #1547C4semidirect

Maximal subgroups

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

subgroupindex
#602082
#204412
#177192
#175632
#173342
#82582
#82572
#82562
#82552
#82542
#82532
#82522
#82512
#80942
#80822