Finite Group Databasebeta

← #227905

Subgroups — #227905

Order 512.

795 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23569C211yesmaximal, characteristic
2#23084C211yesmaximal, characteristic, complemented
3#22531C211yesmaximal, characteristic, complemented
4C2×SD16⋊C8C211yesmaximal, characteristic, complemented
5C2×#1494C211yesmaximal, characteristic, complemented
6#9659C211yesmaximal, characteristic, complemented
7#9556C211yesmaximal, characteristic, complemented
8#9545C211yesmaximal, characteristic, complemented
9#9072C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8846C211yesmaximal, characteristic, complemented
13#8834C211yesmaximal, characteristic, complemented
14#8209C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
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#1955—21no—
34#1955—21no—
35#1954—21no—
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1954C2211yescharacteristic
39#1939C2211yescharacteristic
40#1924—21no—
41#1869—21no—
42#1798—21no—
43#1797C2211yescharacteristic
44#1770C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1752—21no—
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51SD16⋊C8—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
71C23×C8—21no—
72C23×C8—21no—
73#529—22no—
74#529—22no—
75#529—41no—
76#529—41no—
77C2×#126—21no—
78C2×#125C2×C411yescharacteristic
79C2×#124C2×C411yescharacteristic
80C2×#123—21no—
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×#121C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C2×#117—21no—
90C22×C42C2311yescharacteristic
91#446—22no—
92#446—41no—
93C8×Q8—22no—
94C8×Q8—41no—
95C4×SD16—22no—
96C4×SD16—22no—
97C4×D8—41no—
98C4×D8—41no—
99C8×D4—22no—
100C8×D4—41no—
101#432—22no—
102#432—41no—
103C2×#107C2×C411yescharacteristic
104C2×#106C2×C411yescharacteristic
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
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⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8C2×C411yescharacteristic
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8D411yescharacteristic
132C2×C22⋊C8D411yescharacteristic
133C2×#97—21no—
134C2×#97—21no—
135C2×#97—21no—
136C2×C4×C8—22no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139C2×C4×C8—41no—
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146#390C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#388—21no—
149#386—21no—
150#380C2×C411yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152C4×C22⋊C4—21no—
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#336—21no—
158#336—21no—
159#336—21no—
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162#336D411yescharacteristic
163#336D411yescharacteristic
164#335—22no—
165#334—41no—
166Q8⋊C8—22no—
167Q8⋊C8D414yescomplemented
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—41no—
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—22no—
187C2×D8—22no—
188C2×D8—22no—
189C2×D8—41no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—81no—
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C22×C411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210#126—42no—
211#126—42no—
212#125—22no—
213#125—41no—
214#124—22no—
215#124—41no—
216#123—42no—
217#123—42no—
218C22⋊Q8—22no—
219C22⋊Q8—41no—
220#121—22no—
221#121—22no—
222#121—41no—
223#121—41no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8C2×D412yes—
227C4×Q8#4112yes—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—81no—
239C4×D4—81no—
240#117—42no—
241#117—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42C2411yescharacteristic
272#107—41no—
273#106—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C4⋊C8—81no—
284C4⋊C8C2×D412yes—
285#103—22no—
286#103—22no—
287#103—41no—
288#102—22no—
289#102—22no—
290#102—41no—
291#102—41no—
292#102—41no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—81no—
303C22⋊C8—81no—
304C22⋊C8—81no—
305C22⋊C8—81no—
306#97—42no—
307#97—42no—
308#97—81no—
309C4×C8—21no—
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—42no—
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×Q8C4×D412yes—
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—21no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—81no—
351C2×D4—81no—
352C2×D4#13112yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
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—41no—
375C22×C4—41no—
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—28no—
394D8—44no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—22no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—21no—
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—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—48no—
424C2×C8—41no—
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—84no—
437C2×C8#13112yes—
438C2×C8#13112yes—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
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—81no—
454C4⋊C4C4×D412yes—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—21no—
474C42—21no—
475C42—41no—
476C42—41no—
477C42C22×D411yescharacteristic
478C42C2×#4111yescharacteristic
479C42H(5,𝔽2)11yescharacteristic
480C42H(5,𝔽2)11yescharacteristic
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23#44412yes—
502C23C2×C4×D411yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—22no—
508D4—22no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—84no—
516D4—84no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
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—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4C2×C4×D411yescharacteristic
565C2×C4#51811yescharacteristic
566C2×C4D4211yescharacteristic
567C2×C4#54611yescharacteristic
568C2×C4#54611yescharacteristic
569C2×C4#55111yescharacteristic
570C2×C4C2×#13111yescharacteristic
571C2×C4C2×#13111yescharacteristic
572C2×C4C2×#13111yescharacteristic
573C2×C4#56811yescharacteristic
574C2×C4#57511yescharacteristic
575C2×C4#57511yescharacteristic
576C2×C4#57611yescharacteristic
577C2×C4#57711yescharacteristic
578C8—22no—
579C8—22no—
580C8—42no—
581C8—42no—
582C8—41no—
583C8—41no—
584C8—41no—
585C8—42no—
586C8—41no—
587C8—41no—
588C8—41no—
589C8—81no—
590C8—81no—
591C8—81no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#235811yescharacteristic
614C22#293111yescharacteristic
615C22#293311yescharacteristic
616C22#294311yescharacteristic
617C22#294611yescharacteristic
618C22C2×#44411yescharacteristic
619C22#295611yescharacteristic
620C4—21no—
621C4—21no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4#292911yescharacteristic
632C4#293511yescharacteristic
633C4#297711yescharacteristic
634C4#298111yescharacteristic
635C4#326811yescharacteristic
636C4#327111yescharacteristic
637C4#330911yescharacteristic
638C4#332111yescharacteristic
639C2—42no—
640C2—82no—
641C2—161no—
642C2#1796311yescharacteristic
643C2#1796811yescharacteristic
644C2#1800211yescharacteristic
645C2#1802011yescharacteristic
646C2#2284711yescharacteristic
647C2#2285711yescharacteristic
648C2#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. #22531C2semidirect
4. C2×SD16⋊C8C2semidirect
5. C2×#1494C2semidirect
6. #9659C2semidirect
7. #9556C2semidirect
8. #9545C2semidirect
9. #9072C2semidirect
10. #8952C2semidirect
11. #8947C2semidirect
12. #8846C2semidirect
13. #8834C2semidirect
14. #8209C2semidirect
15. #8145C2semidirect
167. Q8⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#235692
#230842
#225312
C2×SD16⋊C82
C2×#14942
#96592
#95562
#95452
#90722
#89522
#89472
#88462
#88342
#82092
#81452