Finite Group Databasebeta

← #227908

Subgroups — #227908

Order 512.

810 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23569C211yesmaximal, characteristic
2#23084C211yesmaximal, characteristic, complemented
3#22474C211yesmaximal, characteristic, complemented
4C2×#1580C211yesmaximal, characteristic, complemented
5C2×#1494C211yesmaximal, characteristic, complemented
6#9660C211yesmaximal, characteristic, complemented
7#9555C211yesmaximal, characteristic, complemented
8#9547C211yesmaximal, characteristic, complemented
9#9072C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic, complemented
13#8834C211yesmaximal, characteristic, complemented
14#8208C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×#121C2211yescharacteristic
25#2059C411yescharacteristic
26#2042C411yescharacteristic
27#2036C411yescharacteristic
28#2034C411yescharacteristic
29#2025C411yescharacteristic
30#2012C411yescharacteristic
31#2000C411yescharacteristic
32#1999C411yescharacteristic
33#1984C2211yescharacteristic
34#1984C2211yescharacteristic
35#1955—21no—
36#1954C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1870—21no—
40#1869—21no—
41#1869—21no—
42#1818—21no—
43#1797C2211yescharacteristic
44#1770C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1752—21no—
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1729—21no—
51#1580—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1494—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C23×C8—21no—
75#529—22no—
76#529—22no—
77#529—41no—
78#529—41no—
79C2×#126—21no—
80C2×#125C2×C411yescharacteristic
81C2×#124C2×C411yescharacteristic
82C2×#123—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C2×#117—21no—
92C22×C42C2311yescharacteristic
93C4×SD16—41no—
94C4×SD16—41no—
95C4×D8—22no—
96C4×D8—22no—
97#436—22no—
98#436—41no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—41no—
102C8×D4—41no—
103#432—22no—
104#432—41no—
105C2×#107C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8C2×C411yescharacteristic
128C2×C22⋊C8C2×C411yescharacteristic
129C2×C22⋊C8D411yescharacteristic
130C2×C22⋊C8D411yescharacteristic
131C2×#97—21no—
132C2×#97—21no—
133C2×C4×C8—22no—
134C2×C4×C8—21no—
135C2×C4×C8—41no—
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142#390C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#388—21no—
145#386—21no—
146#380C2×C411yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—21no—
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162#336D411yescharacteristic
163#336D411yescharacteristic
164#335—41no—
165#334—22no—
166Q8⋊C8—22no—
167Q8⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—41no—
171D4⋊C8D414yescomplemented
172#322—22no—
173#322—41no—
174C22×Q8C22×C411yescharacteristic
175C22×D4—21no—
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4111yescharacteristic
184C2×SD16—22no—
185C2×SD16—22no—
186C2×SD16—41no—
187C2×D8—22no—
188C2×D8—22no—
189C2×D8—22no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—81no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—81no—
207C22×C8—81no—
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216#126—42no—
217#126—42no—
218#125—22no—
219#125—41no—
220#124—22no—
221#124—41no—
222#123—42no—
223#123—42no—
224C22⋊Q8—22no—
225C22⋊Q8—41no—
226#121—22no—
227#121—22no—
228#121—41no—
229#121—41no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—42no—
241C4×D4—42no—
242C4×D4—81no—
243C4×D4—81no—
244C4×D4C2×D412yes—
245C4×D4#4112yes—
246#117—42no—
247#117—42no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42C2411yescharacteristic
278#107—22no—
279#106—41no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—81no—
288C4⋊C8C2×D412yes—
289#103—22no—
290#103—41no—
291#103—41no—
292#102—22no—
293#102—22no—
294#102—22no—
295#102—41no—
296#102—41no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—81no—
307C22⋊C8—81no—
308#97—42no—
309#97—81no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—42no—
322C4×C8C2×D412yes—
323#95—21no—
324#95—21no—
325#95—21no—
326C24—21no—
327C24—21no—
328C24C4×D411yescharacteristic
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—81no—
350C2×D4—81no—
351C2×D4C4×D412yes—
352C2×D4#13112yes—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393SD16—44no—
394D8—28no—
395M16—84no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—48no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—84no—
438C2×C8—84no—
439C2×C8#13112yes—
440C2×C8#13112yes—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—81no—
456C4⋊C4C4×D412yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42C22×D411yescharacteristic
480C42C2×#4111yescharacteristic
481C42H(5,𝔽2)11yescharacteristic
482C42H(5,𝔽2)11yescharacteristic
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23#44412yes—
504C23C2×C4×D411yescharacteristic
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509D4—22no—
510D4—22no—
511D4—22no—
512D4—22no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—84no—
518D4—84no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4C2×C4×D411yescharacteristic
567C2×C4#51811yescharacteristic
568C2×C4D4211yescharacteristic
569C2×C4#54611yescharacteristic
570C2×C4#54611yescharacteristic
571C2×C4#55111yescharacteristic
572C2×C4C2×#13111yescharacteristic
573C2×C4C2×#13111yescharacteristic
574C2×C4C2×#13111yescharacteristic
575C2×C4#56811yescharacteristic
576C2×C4#57511yescharacteristic
577C2×C4#57511yescharacteristic
578C2×C4#57611yescharacteristic
579C2×C4#57711yescharacteristic
580C8—22no—
581C8—22no—
582C8—42no—
583C8—42no—
584C8—41no—
585C8—41no—
586C8—41no—
587C8—42no—
588C8—41no—
589C8—41no—
590C8—41no—
591C8—81no—
592C8—81no—
593C8—81no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#235811yescharacteristic
616C22#293111yescharacteristic
617C22#293311yescharacteristic
618C22#294311yescharacteristic
619C22#294611yescharacteristic
620C22C2×#44411yescharacteristic
621C22#295611yescharacteristic
622C4—21no—
623C4—21no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4#292911yescharacteristic
634C4#293511yescharacteristic
635C4#297711yescharacteristic
636C4#298111yescharacteristic
637C4#326711yescharacteristic
638C4#326811yescharacteristic
639C4#330711yescharacteristic
640C4#331911yescharacteristic
641C2—42no—
642C2—82no—
643C2—82no—
644C2#1796311yescharacteristic
645C2#1796811yescharacteristic
646C2#1800211yescharacteristic
647C2#1802011yescharacteristic
648C2#2284711yescharacteristic
649C2#2286011yescharacteristic
650C2#2288611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23084C2semidirect
3. #22474C2semidirect
4. C2×#1580C2semidirect
5. C2×#1494C2semidirect
6. #9660C2semidirect
7. #9555C2semidirect
8. #9547C2semidirect
9. #9072C2semidirect
10. #8951C2semidirect
11. #8945C2semidirect
12. #8843C2semidirect
13. #8834C2semidirect
14. #8208C2semidirect
15. #8145C2semidirect
171. D4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#235692
#230842
#224742
C2×#15802
C2×#14942
#96602
#95552
#95472
#90722
#89512
#89452
#88432
#88342
#82082
#81452