Finite Group Databasebeta

← C2×#27302 #6440855

Subgroups — C2×#27302 #6440855

Order 512.

2070 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
1C2×#2508C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic
3#27302C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#1942C211yesmaximal, characteristic, complemented
9C2×#1920C211yesmaximal, characteristic, complemented
10C2×#1916C211yesmaximal, characteristic, complemented
11C2×#1864C211yesmaximal, characteristic
12C2×#1864C211yesmaximal, characteristic
13C2×#1862C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic
15C2×#1804C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2508—24no—
31#2508C2218yescomplemented
32C2×#400—21no—
33C2×#398—21no—
34#2349—24no—
35#2349C2218yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#387—21no—
43C2×#385—21no—
44C2×#385—21no—
45C2×#382—21no—
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#380—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C22×#95—21no—
52#2101C2218yescomplemented
53#2088C2218yescomplemented
54#2085C2218yescomplemented
55#2085C2218yescomplemented
56#1942C2218yescomplemented
57#1920C2218yescomplemented
58#1916C2218yescomplemented
59#1864C2218yes—
60#1864C2218yes—
61#1862C2218yes—
62#1862C2218yes—
63#1804C2218yes—
64C4×#102C2218yes—
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C24×C4—21no—
80C24×C4—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4×D4—24no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C2×C4×D4C2314yes—
95C2×C4×D4C2314yes—
96C2×C4×D4C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#107—24no—
120C2×#107—42no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106—24no—
124C2×#106—42no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8—42no—
128C2×C4⋊C8C2314yes—
129C2×#102—24no—
130C2×#102—24no—
131C2×#102—24no—
132C2×#102Q818yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C4×C8—42no—
138C2×C4×C8C2314yes—
139#400—28no—
140#398—28no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389C2314yes—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#387—28no—
154#385—28no—
155#385—28no—
156#382—22no—
157#382—22no—
158#382—28no—
159#382C2314yes—
160#382C2314yes—
161#380—28no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4D414yes—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—28no—
178C2×#95D411yescharacteristic
179C2×#95D411yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25—21no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4C2×Q814yes—
198C22×D4#4114yes—
199C22×D4#4114yes—
200C22×D4C2412yes—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—82no—
217C23×C4—82no—
218C23×C4—82no—
219C23×C4—82no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8C2×Q814yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C4×D4—216no—
244C4×D4—28no—
245C4×D4—28no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—48no—
251C4×D4—48no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×Q814yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—44no—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2412yes—
314C2×C42C2412yes—
315C2×C42C2412yes—
316#107—28no—
317#107—44no—
318#106—28no—
319#106—44no—
320C4⋊C8—44no—
321#102—28no—
322#102—28no—
323#102—28no—
324#102C2×Q8116yes—
325C4×C8—44no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24C22×D411yescharacteristic
348C2×D4—24no—
349C2×D4—28no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4#125116yes—
358C2×D4C22×Q814yes—
359C2×D4C2×#4114yes—
360C2×D4C2×#4114yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—48no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—816no—
416C22×C4—816no—
417C22×C4—816no—
418C22×C4#13514yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q812yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C22×C4C2511yescharacteristic
433C2×C8—28no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C2×C8C22×Q814yes—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4C22×Q814yes—
477C4⋊C4C2×#4114yes—
478C22⋊C4—216no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—48no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—216no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23#52512yes—
506C23#54612yes—
507C23D4×Q812yes—
508C23C2×#13512yes—
509C23#57512yes—
510C23#57612yes—
511C23#57712yes—
512C23C23×D411yescharacteristic
513D4—24no—
514D4—24no—
515D4—24no—
516D4—216no—
517D4C2×#12518yes—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—816no—
565C2×C4—816no—
566C2×C4—816no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4#52512yes—
573C2×C4C2×#12514yes—
574C2×C4#53914yes—
575C2×C4#54612yes—
576C2×C4D4×Q812yes—
577C2×C4#55214yes—
578C2×C4#55414yes—
579C2×C4C2×#13514yes—
580C2×C4C23×Q811yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C8—24no—
587C8—42no—
588C8—42no—
589C8—42no—
590C22—28no—
591C22—44no—
592C22—44no—
593C22—432no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22#247314yes—
599C22#310214yes—
600C22#311814yes—
601C22#329614yes—
602C22#331914yes—
603C22#338014yes—
604C22#338814yes—
605C22C2×#52511yescharacteristic
606C22C2×#54611yescharacteristic
607C22C2×D4×Q811yescharacteristic
608C22C22×#13511yescharacteristic
609C22C2×#57511yescharacteristic
610C22C2×#57611yescharacteristic
611C22C2×#57711yescharacteristic
612C4—24no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—44no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4#247314yes—
626C4C22×#12511yescharacteristic
627C4C2×#53911yescharacteristic
628C4C2×#55211yescharacteristic
629C4C2×#55411yescharacteristic
630C2—48no—
631C2#2730218yesdirect product factor
632C2C2×#247311yescharacteristic
633C2C2×#310211yescharacteristic
634C2C2×#311811yescharacteristic
635C2C2×#329611yescharacteristic
636C2C2×#331911yescharacteristic
637C2C2×#338011yescharacteristic
638C2C2×#338811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2508C2semidirect
3. #27302C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2088C2semidirect
6. C2×#2085C2semidirect
7. C2×#2085C2semidirect
8. C2×#1942C2semidirect
9. C2×#1920C2semidirect
10. C2×#1916C2semidirect
31. #2508C22semidirect
52. #2101C22semidirect
53. #2088C22semidirect
54. #2085C22semidirect
55. #2085C22semidirect
56. #1942C22semidirect
57. #1920C22semidirect
58. #1916C22semidirect
631. C2#27302direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25082
C2×#23492
#273022
C2×#21012
C2×#20882
C2×#20852
C2×#19422
C2×#19202
C2×#19162
C2×#18642
C2×#18622
C2×#18042
C2×C4×#1022