Finite Group Databasebeta

← #227649

Subgroups — #227649

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23569C211yesmaximal, characteristic
2#23083C211yesmaximal, characteristic, complemented
3#22531C211yesmaximal, characteristic, complemented
4C2×SD16⋊C8C211yesmaximal, characteristic, complemented
5C2×#1482C211yesmaximal, characteristic, complemented
6#9665C211yesmaximal, characteristic, complemented
7#9556C211yesmaximal, characteristic, complemented
8#9541C211yesmaximal, characteristic, complemented
9#9076C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8846C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14#8213C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C4×#121C2211yescharacteristic
25#2059C411yescharacteristic
26#2042C411yescharacteristic
27#2036C411yescharacteristic
28#2034C411yescharacteristic
29#2025C411yescharacteristic
30#2012C411yescharacteristic
31#2000C411yescharacteristic
32#1999C411yescharacteristic
33#1955—21no—
34#1954—21no—
35#1954—21no—
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1954C2211yescharacteristic
39#1939C2211yescharacteristic
40#1923—21no—
41#1869—21no—
42#1798—21no—
43#1797C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C8×C22⋊C4—21no—
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51SD16⋊C8—24no—
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1482—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—
72#529—22no—
73#529—22no—
74#529—41no—
75#529—41no—
76C2×#126—21no—
77C2×#125C2×C411yescharacteristic
78C2×#124C2×C411yescharacteristic
79C2×#123—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×#121C2×C411yescharacteristic
82C2×#121C2×C411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4C2311yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×#117—21no—
89C22×C42C2311yescharacteristic
90#446—22no—
91#446—41no—
92C8×Q8—22no—
93C8×Q8—41no—
94C4×SD16—22no—
95C4×SD16—22no—
96C4×D8—41no—
97C4×D8—41no—
98C8×D4—22no—
99C8×D4—41no—
100#431—22no—
101#431—22no—
102#431—41no—
103#431—41no—
104C2×#107C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
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⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8C2×C411yescharacteristic
133C2×C22⋊C8C2×C411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135C2×C22⋊C8D411yescharacteristic
136C2×#97—21no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139C2×C4×C8—21no—
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—41no—
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—81no—
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207#126—42no—
208#126—42no—
209#125—22no—
210#125—41no—
211#124—22no—
212#124—41no—
213#123—42no—
214#123—42no—
215C22⋊Q8—22no—
216C22⋊Q8—41no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—41no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8C2×D412yes—
224C4×Q8#4112yes—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—42no—
234C4×D4—42no—
235C4×D4—81no—
236C4×D4—81no—
237#117—42no—
238#117—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
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×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42C2411yescharacteristic
269#107—41no—
270#106—22no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8—81no—
282C4⋊C8C2×D412yes—
283#103—22no—
284#103—22no—
285#103—41no—
286#102—22no—
287#102—22no—
288#102—22no—
289#102—41no—
290#102—41no—
291C22⋊C8—22no—
292C22⋊C8—41no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301C22⋊C8—81no—
302C22⋊C8—81no—
303C22⋊C8—81no—
304C22⋊C8—81no—
305#97—42no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
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—42no—
318C4×C8—42no—
319C4×C8—81no—
320C4×C8C2×D412yes—
321#95—21no—
322#95—21no—
323#95—21no—
324C24—21no—
325C24—21no—
326C24C4×D411yescharacteristic
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8C4×D412yes—
333C2×D4—24no—
334C2×D4—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—81no—
348C2×D4—81no—
349C2×D4—81no—
350C2×D4#13112yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
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—41no—
372C22×C4—41no—
373C22×C4—41no—
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×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4111yescharacteristic
391SD16—28no—
392D8—44no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
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—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
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—84no—
435C2×C8#13112yes—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—81no—
450C4⋊C4C4×D412yes—
451C4⋊C4#13112yes—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—41no—
473C42—41no—
474C42C22×D411yescharacteristic
475C42C2×#4111yescharacteristic
476C42H(5,𝔽2)11yescharacteristic
477C42H(5,𝔽2)11yescharacteristic
478C23—21no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C2×C4×D411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—41no—
503Q8—41no—
504D4—22no—
505D4—22no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—84no—
513D4—84no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
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—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4C2×C4×D411yescharacteristic
562C2×C4#51811yescharacteristic
563C2×C4D4211yescharacteristic
564C2×C4#54611yescharacteristic
565C2×C4#54611yescharacteristic
566C2×C4#55111yescharacteristic
567C2×C4C2×#13111yescharacteristic
568C2×C4C2×#13111yescharacteristic
569C2×C4C2×#13111yescharacteristic
570C2×C4#56811yescharacteristic
571C2×C4#57511yescharacteristic
572C2×C4#57511yescharacteristic
573C2×C4#57611yescharacteristic
574C2×C4#57711yescharacteristic
575C8—22no—
576C8—42no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—41no—
581C8—42no—
582C8—41no—
583C8—41no—
584C8—41no—
585C8—81no—
586C8—81no—
587C8—81no—
588C8—81no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22#235811yescharacteristic
611C22#292911yescharacteristic
612C22#293511yescharacteristic
613C22#294311yescharacteristic
614C22#294611yescharacteristic
615C22#295511yescharacteristic
616C22#295711yescharacteristic
617C4—21no—
618C4—21no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4#292911yescharacteristic
629C4#293511yescharacteristic
630C4#297611yescharacteristic
631C4#297911yescharacteristic
632C4#326811yescharacteristic
633C4#327111yescharacteristic
634C4#330911yescharacteristic
635C4#332111yescharacteristic
636C2—81no—
637C2—82no—
638C2—161no—
639C2#1790911yescharacteristic
640C2#1791811yescharacteristic
641C2#1807411yescharacteristic
642C2#1808011yescharacteristic
643C2#2284311yescharacteristic
644C2#2285711yescharacteristic
645C2#2288711yescharacteristic
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. #23083C2semidirect
3. #22531C2semidirect
4. C2×SD16⋊C8C2semidirect
5. C2×#1482C2semidirect
6. #9665C2semidirect
7. #9556C2semidirect
8. #9541C2semidirect
9. #9076C2semidirect
10. #8952C2semidirect
11. #8947C2semidirect
12. #8846C2semidirect
13. #8831C2semidirect
14. #8213C2semidirect
15. #8149C2semidirect
167. Q8⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#235692
#230832
#225312
C2×SD16⋊C82
C2×#14822
#96652
#95562
#95412
#90762
#89522
#89472
#88462
#88312
#82132
#81492