Finite Group Databasebeta

← C2×#30068 #6434741

Subgroups — C2×#30068 #6434741

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2640C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic, complemented
3#30068C2116yesmaximal, direct product factor
4C2×#2097C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic
6C2×#2058C211yesmaximal, characteristic, complemented
7C2×#2057C211yesmaximal, characteristic, complemented
8C2×#2042C211yesmaximal, characteristic
9C2×#2042C211yesmaximal, characteristic
10C2×#2024C211yesmaximal, characteristic, complemented
11C2×#2023C211yesmaximal, characteristic
12C2×#1928C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic
14C2×#1927C211yesmaximal, characteristic, complemented
15C2×#1925C211yesmaximal, characteristic
16C2×#1840C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32#2640—24no—
33#2640C2218yes—
34C2×#397—21no—
35C2×#397—21no—
36C2×#393—21no—
37C2×#393—21no—
38#2331—24no—
39#2331C2218yescomplemented
40C2×#391—21no—
41C2×#391—21no—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#387—21no—
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54#2097C2218yescomplemented
55#2092C2218yes—
56#2058C2218yescomplemented
57#2057C2218yescomplemented
58#2042C2218yes—
59#2042C2218yes—
60#2024C2218yescomplemented
61#2023C2218yes—
62#1928C2218yescomplemented
63#1927C2218yescomplemented
64#1926C2218yes—
65#1925C2218yes—
66#1840C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#336C2211yescharacteristic
72C2×#336C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8—21no—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8—42no—
91C2×C4⋊Q8C2314yes—
92C2×C4⋊Q8C2314yes—
93C2×#124—22no—
94C2×#124—42no—
95C2×#124—42no—
96C2×#124C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42—21no—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C2×#107—42no—
119C2×#107—42no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106—42no—
123C2×#106—42no—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#103—24no—
127C2×#103—24no—
128C2×#103—42no—
129C2×#103—42no—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—42no—
137C2×#102—42no—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142#397—28no—
143#397—28no—
144#393—28no—
145#393—28no—
146#391—22no—
147#391—22no—
148#391—28no—
149#391—28no—
150#391C2314yes—
151#391C2314yes—
152#390—22no—
153#390—22no—
154#390C2314yes—
155#390C2314yes—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2314yes—
160#387—28no—
161#387—28no—
162#382—22no—
163#382C2314yes—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168#376D414yes—
169#376C2314yes—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95D411yescharacteristic
181C2×#95D411yescharacteristic
182C2×#95D411yescharacteristic
183C2×#95D411yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C22×Q8—28no—
196C22×Q8—24no—
197C22×Q8—21no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—44no—
206C22×Q8—82no—
207C22×Q8—82no—
208C22×Q8—82no—
209C22×Q8—82no—
210C22×Q8C2412yes—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4C2412yes—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2411yescharacteristic
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C4⋊Q8—216no—
250C4⋊Q8—28no—
251C4⋊Q8—28no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—48no—
255#124—44no—
256#124—44no—
257#124—48no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4#4114yes—
283C2×C4⋊C4#4114yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—82no—
299C2×C22⋊C4—82no—
300C2×C22⋊C4—82no—
301C2×C22⋊C4—82no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—82no—
311C2×C42—82no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2412yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318#107—44no—
319#107—44no—
320#106—44no—
321#106—44no—
322#103—28no—
323#103—28no—
324#103—44no—
325#103—44no—
326#102—28no—
327#102—28no—
328#102—44no—
329#102—44no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95C2×D414yes—
341#95C2×D414yes—
342#95C2×D414yes—
343#95C2×D414yes—
344C24—22no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24C22×D411yescharacteristic
353C2×Q8—216no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—48no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—84no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×Q8—84no—
372C2×Q8—816no—
373C2×Q8—816no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C22×C4C2511yescharacteristic
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—44no—
460C2×C8—44no—
461C2×C8—44no—
462C2×C8—44no—
463C4⋊C4—216no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4C2×#4114yes—
484C4⋊C4C2×#4114yes—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—84no—
501C42—84no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23—84no—
516C23#52512yes—
517C23#53512yes—
518C23#53612yes—
519C23#57612yes—
520C23#57712yes—
521C23#57712yes—
522C23#57812yes—
523C23C23×D411yescharacteristic
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—816no—
529Q8—816no—
530Q8—84no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—84no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4#52512yes—
589C2×C4#53512yes—
590C2×C4#53612yes—
591C2×C4#54214yes—
592C2×C4#54214yes—
593C2×C4#54214yes—
594C2×C4#54214yes—
595C2×C4C22×#4111yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C8—42no—
602C8—42no—
603C8—42no—
604C8—42no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—168no—
613C22#263214yes—
614C22#311814yes—
615C22#312114yes—
616C22#322414yes—
617C22#322514yes—
618C22#324314yes—
619C22#324414yes—
620C22C2×#52511yescharacteristic
621C22C2×#53511yescharacteristic
622C22C2×#53611yescharacteristic
623C22C2×#57611yescharacteristic
624C22C2×#57711yescharacteristic
625C22C2×#57711yescharacteristic
626C22C2×#57811yescharacteristic
627C4—42no—
628C4—42no—
629C4—42no—
630C4—44no—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4#263214yes—
641C4C2×#54211yescharacteristic
642C4C2×#54211yescharacteristic
643C4C2×#54211yescharacteristic
644C4C2×#54211yescharacteristic
645C2—162no—
646C2#3006818yesdirect product factor
647C2C2×#263211yescharacteristic
648C2C2×#311811yescharacteristic
649C2C2×#312111yescharacteristic
650C2C2×#322411yescharacteristic
651C2C2×#322511yescharacteristic
652C2C2×#324311yescharacteristic
653C2C2×#324411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2331C2semidirect
3. #30068C2direct (N × H)
4. C2×#2097C2semidirect
6. C2×#2058C2semidirect
7. C2×#2057C2semidirect
10. C2×#2024C2semidirect
12. C2×#1928C2semidirect
14. C2×#1927C2semidirect
16. C2×#1840C2semidirect
39. #2331C22semidirect
54. #2097C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
60. #2024C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
66. #1840C22semidirect
646. C2#30068direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26402
C2×#23312
#300682
C2×#20972
C2×#20922
C2×#20582
C2×#20572
C2×#20422
C2×#20242
C2×#20232
C2×#19282
C2×#19262
C2×#19272
C2×#19252
C2×#18402