Finite Group Databasebeta

← C2×#29698 #6435430

Subgroups — C2×#29698 #6435430

Order 512.

3472 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#540C211yesmaximal, characteristic
2C22×#476C211yesmaximal, characteristic, complemented
3C22×#474C211yesmaximal, characteristic, complemented
4C2×#2628C211yesmaximal, characteristic, complemented
5#29698C2116yesmaximal, direct product factor
6C2×#2059C212yesmaximal, complemented
7C2×#2041C212yesmaximal, complemented
8C2×#2034C212yesmaximal, complemented
9C2×#2033C212yesmaximal, complemented
10C2×#1804C211yesmaximal, characteristic, complemented
11C2×#1804C211yesmaximal, characteristic, complemented
12C2×#1798C211yesmaximal, characteristic, complemented
13C2×#540—28no—
14C2×#540—28no—
15C2×#540—24no—
16C2×#540C2218yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#127C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#123—21no—
21C22×#123—21no—
22C22×#121—21no—
23C22×#121—21no—
24C22×#121C2211yescharacteristic
25C22×#121C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C2×#476—28no—
30C2×#476C2218yescomplemented
31C2×#474—28no—
32C2×#474C2218yescomplemented
33C22×#107C2212yes—
34C22×#106C2212yes—
35C22×C4⋊C8C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37C22×#102C2212yes—
38C22×#102C2212yes—
39C22×#102C2212yes—
40C22×#102C2212yes—
41C22×C22⋊C8C2212yes—
42#2628—24no—
43#2628C2218yescomplemented
44C2×#400—21no—
45C2×#399—21no—
46C2×#395—21no—
47C2×#393—21no—
48C2×#389C2212yes—
49C2×#382C2211yescharacteristic
50C2×#380C2212yes—
51#2059C22116yescomplemented
52#2041C22116yescomplemented
53#2034C22116yescomplemented
54#2033C22116yescomplemented
55#1804C2218yescomplemented
56#1804C2218yescomplemented
57#1798C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8—21no—
62C23×D4—21no—
63C23×D4—21no—
64C23×D4—21no—
65C23×D4—21no—
66C23×D4C2311yescharacteristic
67C23×D4C2311yescharacteristic
68C24×C4—21no—
69C24×C4—21no—
70C24×C4C2311yescharacteristic
71C23×C8C2312yes—
72C23×C8C2312yes—
73#540—264no—
74#540—232no—
75#540—232no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8C2314yes—
80C2×#127—24no—
81C2×#127—24no—
82C2×#127—22no—
83C2×#127C2314yes—
84C2×#125—24no—
85C2×#125—24no—
86C2×#125—22no—
87C2×#125C2314yes—
88C2×#123—216no—
89C2×#123—216no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#121—216no—
93C2×#121—216no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121—24no—
100C2×#121—28no—
101C2×#121—28no—
102C2×#121C2314yes—
103C2×#121C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4D418yes—
109C2×C4×D4D418yes—
110C2×C4×D4C2314yes—
111C2×C4×D4C2314yes—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117D414yes—
115C2×#117C2314yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#476—232no—
136#474—232no—
137C2×#107—28no—
138C2×#107C2318yes—
139C2×#106—28no—
140C2×#106C2318yes—
141C2×C4⋊C8—24no—
142C2×C4⋊C8—24no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#102—28no—
146C2×#102—28no—
147C2×#102—44no—
148C2×#102—44no—
149C2×#102C2318yes—
150C2×#102C2318yes—
151C2×#102C2318yes—
152C2×#102C2318yes—
153C2×C22⋊C8—44no—
154C2×C22⋊C8C2318yes—
155#400—28no—
156#399—28no—
157#395—28no—
158#393—28no—
159#389—24no—
160#389C2318yes—
161#382—22no—
162#382C2314yes—
163#380—24no—
164#380C2318yes—
165C2×#95—21no—
166C2×#95—22no—
167C2×#95—22no—
168C2×#95—21no—
169#340C2318yes—
170#340C2318yes—
171#336C2318yes—
172C25—21no—
173C25—21no—
174C25C2×D411yescharacteristic
175C22×Q8—24no—
176C22×Q8—44no—
177C22×Q8—84no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—84no—
199C22×D4—84no—
200C22×D4—84no—
201C22×D4—84no—
202C22×D4C2×D414yes—
203C22×D4C2×D414yes—
204C22×D4C2412yes—
205C22×D4C2412yes—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—84no—
221C23×C4—84no—
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C22×C8—24no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8—24no—
234C22×C8—24no—
235C22×C8—24no—
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—28no—
239C4⋊Q8—28no—
240C4⋊Q8—48no—
241#127—216no—
242#127—28no—
243#127—28no—
244#125—216no—
245#125—28no—
246#125—28no—
247#123—264no—
248#123—264no—
249#121—216no—
250#121—216no—
251#121—264no—
252#121—264no—
253#121—28no—
254#121—28no—
255#121—28no—
256#121—28no—
257C4×D4—216no—
258C4×D4—216no—
259C4×D4—28no—
260C4×D4—28no—
261C4×D4C2×D4116yes—
262C4×D4C2×D4116yes—
263#117—216no—
264#117—28no—
265#117—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—84no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—84no—
306C2×C22⋊C4C2×D414yes—
307C2×C22⋊C4C2×D414yes—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2412yes—
315C2×C42C2412yes—
316#107—216no—
317#106—216no—
318C4⋊C8—28no—
319C4⋊C8—28no—
320#102—216no—
321#102—216no—
322#102—48no—
323#102—48no—
324C22⋊C8—48no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—44no—
343C24—44no—
344C24—44no—
345C24—44no—
346C24C22×D412yes—
347C24C22×D411yescharacteristic
348C2×Q8—48no—
349C2×Q8—88no—
350C2×Q8—816no—
351C2×D4—216no—
352C2×D4—216no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—816no—
384C2×D4—816no—
385C2×D4—816no—
386C2×D4—816no—
387C2×D4C22×D414yes—
388C2×D4C22×D414yes—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—816no—
428C22×C4—816no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4#12014yes—
432C22×C4#12014yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C22×C42−1+412yes—
444C22×C42−1+412yes—
445C22×C4C2511yescharacteristic
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—48no—
453C2×C8—48no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—88no—
470C4⋊C4C22×D414yes—
471C4⋊C4C22×D414yes—
472C22⋊C4—216no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—88no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42#12018yes—
483C42C22×D414yes—
484C42C22×D414yes—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—44no—
493C23—48no—
494C23—48no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—44no—
503C23—48no—
504C23—48no—
505C23—44no—
506C23—48no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—88no—
512C23—88no—
513C23—88no—
514C23—88no—
515C23C2×#12012yes—
516C23#53612yes—
517C23#53712yes—
518C23#57614yes—
519C23#57714yes—
520C23C23×D411yescharacteristic
521Q8—816no—
522D4—24no—
523D4—24no—
524D4—24no—
525D4—24no—
526D4—48no—
527D4—48no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—816no—
533D4—816no—
534D4—816no—
535D4—816no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—816no—
568C2×C4—816no—
569C2×C4—88no—
570C2×C4—88no—
571C2×C4C2×#12014yes—
572C2×C4C2×#12014yes—
573C2×C4C2×#12012yes—
574C2×C4#53612yes—
575C2×C4#53614yes—
576C2×C4#53614yes—
577C2×C4#53712yes—
578C2×C4#53714yes—
579C2×C4#53714yes—
580C2×C4C23×D411yescharacteristic
581C2×C4C23×D411yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C2×C4C2×2−1+411yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C8—44no—
587C8—44no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—88no—
600C22—816no—
601C22—816no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22#261614yes—
607C22#300814yes—
608C22#300914yes—
609C22#324118yes—
610C22#325718yes—
611C22C22×#12011yescharacteristic
612C22C2×#53611yescharacteristic
613C22C2×#53711yescharacteristic
614C22C2×#57612yes—
615C22C2×#57712yes—
616C4—24no—
617C4—24no—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—42no—
622C4—44no—
623C4—84no—
624C4—84no—
625C4#261614yes—
626C4C2×#53611yescharacteristic
627C4C2×#53611yescharacteristic
628C4C2×#53711yescharacteristic
629C4C2×#53711yescharacteristic
630C2—82no—
631C2—84no—
632C2—84no—
633C2#2969818yesdirect product factor
634C2C2×#261611yescharacteristic
635C2C2×#300811yescharacteristic
636C2C2×#300911yescharacteristic
637C2C2×#324112yes—
638C2C2×#325712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#476C2semidirect
3. C22×#474C2semidirect
4. C2×#2628C2semidirect
5. #29698C2direct (N × H)
6. C2×#2059C2semidirect
7. C2×#2041C2semidirect
8. C2×#2034C2semidirect
9. C2×#2033C2semidirect
10. C2×#1804C2semidirect
11. C2×#1804C2semidirect
12. C2×#1798C2semidirect
30. C2×#476C22semidirect
32. C2×#474C22semidirect
43. #2628C22semidirect
51. #2059C22semidirect
52. #2041C22semidirect
53. #2034C22semidirect
54. #2033C22semidirect
55. #1804C22semidirect
56. #1804C22semidirect
57. #1798C22semidirect
633. C2#29698direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5402
C22×#4762
C22×#4742
C2×#26282
#296982
C2×#20592
C2×#20412
C2×#20342
C2×#20332
C2×#18042
C2×#17982