Finite Group Databasebeta

← C2×#8330 #185003

Subgroups — C2×#8330 #185003

Order 512.

2926 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic, complemented
3C22×#328C212yesmaximal, complemented
4C22×#327C212yesmaximal
5C2×#1434C211yesmaximal, characteristic, complemented
6#8330C218yesmaximal, direct product factor
7C2×#529—22no—
8C2×#529—22no—
9C2×#529—28no—
10C2×#529—28no—
11C2×#529C414yescomplemented
12C2×#529C2214yes—
13C22×#126—21no—
14C22×#125C411yescharacteristic, complemented
15C22×#124C411yescharacteristic, complemented
16C22×#123—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×Q8C411yescharacteristic, complemented
20C22×C4×D4—21no—
21C22×C4×D4C411yescharacteristic, complemented
22C22×#117—21no—
23C23×C42C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×#95—21no—
26C22×#95—21no—
27C22×#95C2212yes—
28#1817—22no—
29#1817C2214yescomplemented
30C2×#336—21no—
31C2×#336—21no—
32C2×#328—28no—
33C2×#328C2218yescomplemented
34C2×#327—28no—
35C2×#327C2218yes—
36#1434—22no—
37#1434C2214yescomplemented
38C23×Q8C2×C411yescharacteristic
39C23×D4—21no—
40C23×D4C2×C411yescharacteristic
41C24×C4—21no—
42C24×C4—22no—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47#529—232no—
48#529—232no—
49#529—216no—
50#529—216no—
51#529—216no—
52#529—216no—
53C2×#126—216no—
54C2×#126—24no—
55C2×#126—24no—
56C2×#125—24no—
57C2×#125—24no—
58C2×#125—22no—
59C2×#125C2×C414yescomplemented
60C2×#124—24no—
61C2×#124—24no—
62C2×#124—22no—
63C2×#124C2×C414yescomplemented
64C2×#123—216no—
65C2×#123—24no—
66C2×#123—24no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2×C412yes—
71C2×C22⋊Q8D418yes—
72C2×C22⋊Q8C2312yes—
73C2×#121—21no—
74C2×#121—24no—
75C2×#121—21no—
76C2×#121C2×C412yes—
77C2×#121D418yes—
78C2×#121C2312yes—
79C2×C4×Q8—24no—
80C2×C4×Q8—24no—
81C2×C4×Q8—22no—
82C2×C4×Q8C2×C414yescomplemented
83C2×C4×D4—216no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—22no—
89C2×C4×D4C2×C414yescomplemented
90C2×#117—216no—
91C2×#117—24no—
92C2×#117—24no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C42—21no—
105C22×C42—24no—
106C22×C42—24no—
107C22×C42C2×C411yescharacteristic
108C22×C42C2×C411yescharacteristic
109C22×C42C2×C412yes—
110C22×C42D412yes—
111C22×C42D412yes—
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42C2312yes—
115C2×C22⋊C8—24no—
116C2×C22⋊C8—24no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8C2314yes—
119C2×#95—28no—
120C2×#95—24no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95—22no—
124C2×#95—24no—
125C2×#95—44no—
126C2×#95C2314yes—
127#336—24no—
128#336—24no—
129#328—216no—
130#327—216no—
131C25—21no—
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8C22⋊C414yes—
138C22×Q8C22×C412yes—
139C22×D4—22no—
140C22×D4—28no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—44no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4C22⋊C414yes—
150C22×D4C22×C412yes—
151C23×C4—28no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—28no—
163C23×C4—24no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C22⋊C412yes—
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C412yes—
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C22×C8—24no—
192C22×C8—44no—
193C22×C8—44no—
194C22×C8—44no—
195#126—232no—
196#126—232no—
197#125—216no—
198#125—28no—
199#125—28no—
200#124—216no—
201#124—28no—
202#124—28no—
203#123—232no—
204#123—232no—
205C22⋊Q8—24no—
206C22⋊Q8—28no—
207C22⋊Q8—28no—
208C22⋊Q8—24no—
209C22⋊Q8C22⋊C418yes—
210C22⋊Q8C2×D418yes—
211#121—24no—
212#121—28no—
213#121—28no—
214#121—24no—
215#121C22⋊C418yes—
216#121C2×D418yes—
217C4×Q8—216no—
218C4×Q8—28no—
219C4×Q8—28no—
220C4×D4—232no—
221C4×D4—232no—
222C4×D4—216no—
223C4×D4—28no—
224C4×D4—28no—
225#117—232no—
226#117—232no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C22⋊C414yes—
241C2×C4⋊C4C22⋊C414yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—22no—
256C2×C42—28no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42C22⋊C412yes—
271C2×C42C22⋊C412yes—
272C2×C42C22⋊C412yes—
273C2×C42C22⋊C412yes—
274C2×C42C22×C412yes—
275C2×C42C22×C412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C22⋊C8—24no—
279C22⋊C8—24no—
280C22⋊C8—44no—
281#95—216no—
282#95—24no—
283#95—24no—
284#95—48no—
285#95—44no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C2×C22⋊C411yescharacteristic
299C24C2×C22⋊C411yescharacteristic
300C24#12014yes—
301C24#13514yes—
302C24AGL(1,ℤ/8)12yes—
303C24#13712yes—
304C24C22×D411yescharacteristic
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8C2×C22⋊C414yes—
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—48no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—48no—
324C2×D4—44no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—416no—
330C2×D4—416no—
331C2×D4C2×C22⋊C414yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—216no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—416no—
379C22×C4—48no—
380C22×C4—432no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4#10414yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C412yes—
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4#12014yes—
415C22×C4#12014yes—
416C22×C4#12014yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—44no—
424C2×C8—88no—
425C4⋊C4—216no—
426C4⋊C4—216no—
427C4⋊C4—216no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4C2×C22⋊C414yes—
435C4⋊C4C2×C22⋊C414yes—
436C22⋊C4—216no—
437C22⋊C4—216no—
438C22⋊C4—216no—
439C22⋊C4—28no—
440C22⋊C4—28no—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C42—216no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—24no—
449C42—24no—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—24no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—48no—
478C23—48no—
479C23—48no—
480C23—84no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23#37912yes—
485C23#37912yes—
486C23#41012yes—
487C23#41012yes—
488C23#41612yes—
489C23#41612yes—
490C23#41712yes—
491C23#41712yes—
492C23C2×#10412yes—
493C23#42112yes—
494C23#44918yes—
495C23#45218yes—
496C23C22×C22⋊C411yescharacteristic
497C23C2×#12012yes—
498C23C2×#13512yes—
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×#13711yescharacteristic
501Q8—44no—
502Q8—44no—
503Q8—48no—
504D4—44no—
505D4—44no—
506D4—48no—
507D4—48no—
508D4—48no—
509D4—416no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—432no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—84no—
564C2×C4—832no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4#37912yes—
569C2×C4#37912yes—
570C2×C4C2×#10414yes—
571C2×C4C22×C22⋊C411yescharacteristic
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4C2×#12012yes—
574C8—84no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—216no—
585C22—216no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—82no—
595C22—82no—
596C22—816no—
597C22—82no—
598C22—82no—
599C22#150014yes—
600C22#150914yes—
601C22#151214yes—
602C22#152114yes—
603C22#178214yes—
604C22#178814yes—
605C22#178814yes—
606C22#178914yes—
607C22C2×#37911yescharacteristic
608C22C2×#41011yescharacteristic
609C22C2×#41011yescharacteristic
610C22C2×#41611yescharacteristic
611C22C2×#41711yescharacteristic
612C22C22×#10411yescharacteristic
613C22C2×#42111yescharacteristic
614C22C2×#44912yes—
615C22C2×#45212yes—
616C4—22no—
617C4—21no—
618C4—21no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—42no—
627C4—44no—
628C4—88no—
629C2—22no—
630C2—22no—
631C2—24no—
632C2—84no—
633C2#833018yesdirect product factor
634C2C2×#150011yescharacteristic
635C2C2×#150911yescharacteristic
636C2C2×#151211yescharacteristic
637C2C2×#152111yescharacteristic
638C2C2×#178211yescharacteristic
639C2C2×#178811yescharacteristic
640C2C2×#178911yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1817C2semidirect
3. C22×#328C2semidirect
5. C2×#1434C2semidirect
6. #8330C2direct (N × H)
11. C2×#529C4semidirect
14. C22×#125C4semidirect
15. C22×#124C4semidirect
19. C22×C4×Q8C4semidirect
21. C22×C4×D4C4semidirect
29. #1817C22semidirect
33. C2×#328C22semidirect
37. #1434C22semidirect
59. C2×#125C2×C4semidirect
63. C2×#124C2×C4semidirect
82. C2×C4×Q8C2×C4semidirect
89. C2×C4×D4C2×C4semidirect
633. C2#8330direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#18172
C22×#3282
C22×#3272
C2×#14342
#83302